Abstract
We study how the typical gradient and typical height of a random surface are modified by the addition of quenched disorder in the form of a random independent external field. The results provide quantitative estimates, sharp up to multiplicative constants, in the following cases. It is shown that for real-valued random-field random surfaces of the \(\nabla \phi \) type with a uniformly convex interaction potential: (i) The gradient of the surface delocalizes in dimensions \(1\le d\le 2\) and localizes in dimensions \(d\ge 3\). (ii) The surface delocalizes in dimensions \(1\le d\le 4\) and localizes in dimensions \(d\ge 5\). It is further shown that for the integer-valued random-field Gaussian free field: (i) The gradient of the surface delocalizes in dimensions \(d=1,2\) and localizes in dimensions \(d\ge 3\). (ii) The surface delocalizes in dimensions \(d=1,2\). (iii) The surface localizes in dimensions \(d\ge 3\) at low temperature and weak disorder strength. The behavior in dimensions \(d\ge 3\) at high temperature or strong disorder is left open. The proofs rely on several tools: Explicit identities satisfied by the expectation of the random surface, the Efron–Stein concentration inequality, a coupling argument for Langevin dynamics (originally due to Funaki and Spohn (Comm Math Phys 185(1): 1-36, 1997) and the Nash–Aronson estimate.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
1.1 Main results
The seminal work of Imry and Ma [46] predicted that the addition of a quenched random external field eliminates the magnetization phase transition of low-dimensional spin systems. This was argued to be a generic phenomenon in two dimensions and to occur also in three and four dimensions in systems with continuous symmetry. These predictions were confirmed for a broad class of spin systems in the celebrated work of Aizenman and Wehr [4, 5]. While the Imry-Ma phenomenon has mostly been studied in the spin system context, it has been recognized that related effects occur also for random surfaces of the \(\nabla \phi \) type subjected to quenched disorder, in suitable ways [13, 14, 25, 26, 48, 49, 67] (see Sect. 1.2). In this work we study the way in which the fluctuations of random surfaces are enhanced by the addition of quenched randomness in the form of an independent external field, focusing on the localization and delocalization behavior of the gradient and heights of real- and integer-valued surfaces. Quantitative estimates are obtained in all cases studied, complementing other recent quantitative studies of the Imry–Ma phenomenon [2, 3, 23, 28, 32, 33].
Real-valued random-field random surfaces: The first class of random surfaces that we consider are real-valued random surfaces of the \(\nabla \phi \) type with a uniformly convex interaction potential, which are subjected to a quenched independent external field in the sense of (1.1) below.
We start with a few definitions. Let \(\mathbb {Z}^d\) be the standard d-dimensional integer lattice, in which vertices are adjacent if they are equal in all but one coordinate and differ by one in that coordinate. For \(\Lambda \subset \mathbb {Z}^d\), let \(\partial \Lambda \) be the external vertex boundary of \(\Lambda \), \(E(\Lambda )\) be the set of edges of \(\mathbb {Z}^d\) with both endpoints in \(\Lambda \), and \(\Lambda ^+:=\Lambda \cup \partial \Lambda \).
Let \(\Lambda \subset \mathbb {Z}^d\) be finite and let \(\eta :\Lambda \rightarrow \mathbb {R}\). The Hamiltonian \(H_\Lambda ^\eta \) of the random surface on \(\Lambda \) with external field \(\eta \) associates to each \(\phi :\Lambda ^+\rightarrow \mathbb {R}\) the energy
where \(V:\mathbb {R}\rightarrow \mathbb {R}\) is a measurable function satisfying \(V(x)=V(-x)\) for all x, \(\lambda >0\) is the coupling strength of the external field \(\eta \) and \(V(\nabla \phi (e)):=V(\phi (x) - \phi (y))\) for an edge \(e=\{x,y\}\) (noting that the orientation of e is immaterial, as V is an even function). We assume throughout that the potential V is uniformly convex, i.e., that it is twice continuously differentiable and there exist \(c_-,c_+\) satisfying
The probability distribution of the random surface, with zero boundary conditions, is then defined by
where dx indicates Lebesgue measure on \(\mathbb {R}\), \(\delta _0\) is the Dirac delta measure at 0, and
\(Z_{\Lambda }^\eta \) is called the partition function, and normalizes \(\mu _\Lambda ^\eta \) to be a probability measure. We denote the expectation with respect to \(\mu _\Lambda ^\eta \) by \(\left\langle \cdot \right\rangle _{\mu _\Lambda ^\eta }\), and refer to it as the thermal expectation.
A natural question pertaining to random surfaces is whether their fluctuations diverge on sequences of domains \(\left( \mu _{\Lambda _n} \right) _{n\ge 1}\) which increase to \(\mathbb {Z}^d\). In the absence of an external field (i.e., when \(\eta \equiv 0\)) the following facts are known: In dimensions \(d = 1,2\) the variance of the height at a fixed vertex diverges as n tends to infinity [18]; the random surface is delocalized or rough. In dimensions \(d \ge 3\), the Brascamp-Lieb concentration inequality [16, 17] shows that the variance of the height remains bounded uniformly in n; the random surface is localized or smooth. The Brascamp-Lieb inequality further implies that the fluctuations of the (discrete) gradient of the surface remain bounded in n in every dimension \(d \ge 1\).
In this paper, we study the effect that a quenched random independent field has on the localization and delocalization properties of the random surface and its gradient. Explicitly, we assume that the external field \(\eta \) is random, with the random variables \((\eta (x))\) independent and satisfying various additional assumptions, and we study the fluctuations of \(\mu _{\Lambda }^{\eta }\) for a typical realization of \(\eta \). We shall denote the probability measure over the random field, its expectation, and its variance by \(\mathbb {P}\), \(\mathbb {E}\), \({{\,\textrm{Var}\,}}\), respectively.
A specific case of interest is the random-field Gaussian free field, i.e., the model (1.3) with the quadratic potential \(V(x) = \frac{1}{2}x^2\) (see also Sect. 7.2). In this situation, the quenched random surface has a multivariate Gaussian distribution whose covariance structure can be explicitly calculated as a function of the realization of the random field \(\eta \). If \(\eta \) is random, independent and each \(\eta (x)\) has zero mean and unit variance, one can prove that, for almost every realization of the random field, the gradient of the random surface delocalizes if \(d \le 2\) and localizes if \(d \ge 3\), and that the height of the surface delocalizes if \(d \le 4\) and localizes if \(d \ge 5\). The result can be quantified and the typical height of the random surface and its gradient can be estimated in every dimension; we refer the reader to [25, Appendix A.1], where the qualitative delocalization of the random-field Gaussian free field is discussed in dimensions \(d = 3,4\), and to [67, Section 1.2] where the gradient fluctuations are quantified (see also Sect. 4.1 for the calculations in the zero-temperature limit). Our analysis of the real-valued, random-field random surfaces extends these results to the class of potentials satisfying (1.2), for which the law of the random surface is not explicitly known.
Before stating the theorems, we introduce some notation. Write \(\Lambda _L := \left\{ -L, \ldots , L \right\} ^d\) and let \(\left| \Lambda _L \right| =(2L+1)^d\) be its cardinality. In the next two results we consider dimensions \(d\ge 1\), integer \(L\ge 2\), disorder strength \(\lambda >0\), ellipticity parameters \(0<c_-\le c_+<\infty \) and a twice-continuously differentiable \(V:\mathbb {R}\rightarrow \mathbb {R}\) satisfying \(V(x)=V(-x)\) for all x and the uniform convexity assumption (1.2). We suppose \(\eta :\Lambda _L\rightarrow \mathbb {R}\) are independent random variables with moment assumptions as stated below, and that \(\phi \) is sampled from the measure \(\mu _{\Lambda _L}^\eta \) given by (1.3).
Each of our theorems introduces its own positive constants C, c. Intuitively, C stands for a generic large value while c stands for a generic small value.
Our first theorem addresses the fluctuations of the gradient of the random surface.
Theorem 1
(Gradient fluctuations, real-valued). Suppose \(\mathbb {E}\left[ \eta (x) \right] =0\) and \({{\,\textrm{Var}\,}}\left[ \eta (x) \right] =1\) for all \(x \in \Lambda _L\). There exist \(C , c>0\) depending only on the dimension d and the ratios \(c_+/ c_-\) and \(\lambda / c_-\) such that the quantity
satisfies
We remark that in this theorem, as well as in many of our subsequent results, the obtained bounds are with respect to the disorder-averaged (annealed) measure \({{\overline{\mu }}} (d \phi ) := \int {\mathbb {P}}(d \eta ) \mu ^{\eta }_{\Lambda _L} (d \phi )\).
We further remark that our techniques for controlling the gradient fluctuations are applicable for general external fields \(\eta \); see Theorem 6. In addition, our proof of Theorem 1 applies under significant relaxations of the uniform convexity assumption (1.2) (in particular, the proof applies to certain non-convex V); see Remark 4.2.
Our second theorem concerns the fluctuations of individual heights in the random surface.
Theorem 2
(Height fluctuations, real-valued). Suppose \({{\,\textrm{Var}\,}}\left[ \eta (x) \right] =1\) for all \(x \in \Lambda _L\). There exists \(C>0\) depending only on the dimension d and the ratios \(c_+/ c_-\) and \(\lambda / c_-\) such that, for any \(y \in \Lambda _L\),
Additionally, there exists \(c>0\) depending only on the dimension d and the ratios \(c_+/ c_-\) and \(\lambda / c_-\) such that, for any \(y \in \Lambda _{L/2}\),
Let us make a few remarks about these results.
The measure \(\mu _{\Lambda }^\eta \) does not have an explicit dependence on an inverse temperature parameter \(\beta \); a more standard setup would have considered the probability measure \(\mu _{\beta , \Lambda _L}^{\eta }\) defined by
However, the effect of \(\beta \) can be mimicked in the model (1.3) by multiplying V and \(\lambda \) by \(\beta \) and thus the previous results are applicable also to the model (1.15). Moreover, the lower bounds in Theorem 1 and all bounds of Theorem 2 hold uniformly in \(\beta \), since the constants C, c in Theorem 1 and Theorem 2 depend only on the ratio \(\lambda / c_-\) and the ellipticity ratio \(c_+/c_-\). The upper bound of Theorem 1 also depends on \(\frac{1}{c_-}\) and thus improves as \(\beta \) increases. In particular, taking the limit \(\beta \rightarrow \infty \) implies that the finite-volume ground configuration of the random-field \(\nabla \phi \) model (see (1.32) below for its explicit formula when \(\lambda =1\)) satisfies the inequalities stated in Theorem 1 and Theorem 2.
The proof of Theorem 2 does not require \(\eta \) to have mean zero (unlike the proof of Theorem 1). It is worth noting, however, that if \(\eta \) is symmetric (i.e., \(\eta \) has the same distribution as \(-\eta \)), then
Thus, a symmetry assumption can be used to upgrade the conclusion of Theorem 2 from a variance bound to a bound on the \(L^2\)-norm (in the random field) of the thermal expectation \(\left\langle \phi (0) \right\rangle _{\mu _{\Lambda _L}^\eta }\). It is plausible that such an \(L^2\) bound also holds if the symmetry assumption is weakened to requiring that \(\eta \) has mean zero, but this is not proven here; see also Sect. 7.4.
Theorem 2 estimates the extent to which the thermal expectation of the height fluctuates as the random field changes. It is also natural to consider the full fluctuations of the height, as a result of both thermal fluctuations and the randomness of the field. By the law of total variance, this can be decomposed as
with the second term on the right-hand side estimated by Theorem 2. The first term is estimated by the Brascamp-Lieb inequality [16, 17], which, in our setting, reads
We thus see that the first term on the right-hand side of (1.16) is not larger than the second term in every dimension (up to a multiplicative factor). We further remark that the Brascamp-Lieb inequality can also be used to obtain Gaussian concentration estimates for the thermal fluctuations (see [31, Section 2.2.1] and [40, Theorem 4.9 and Remark 4.1]).
While the variance upper bounds of Theorem 2 hold pointwise for every vertex of \(\Lambda _L\), matching lower bounds cannot be expected to hold for vertices arbitrarily close to the boundary and are thus stated for vertices which are at a certain distance of the boundary.
Theorem 1 and Theorem 2 are related to the detailed investigations of Cotar and Külske on the existence and uniqueness of translation-covariant gradient Gibbs measures for disordered random surfaces [25, 26]. This is discussed further in Sect. 1.2 but we already point out here that the upper bounds in Theorem 1 in dimensions \(d\ge 3\) may be deduced from [25, Proposition 2.1 and Lemma 3.5].
Integer-valued random-field Gaussian free field: In this section we study the fluctuations of the integer-valued Gaussian free field when subjected to a quenched independent external field, as we now define. Let \(\Lambda \subset \mathbb {Z}^d\) be finite and let \(\eta :\Lambda \rightarrow \mathbb {R}\). The Hamiltonian \(H^{{{\,\textrm{IV}\,}}, \eta }_\Lambda \) of the integer-valued Gaussian free field on \(\Lambda \) with external field \(\eta \) associates to each \(\phi :\Lambda ^+\rightarrow \mathbb {Z}\) the energy
where \(\lambda >0\) is the coupling strength of the external field. The probability distribution for this surface, at inverse temperature \(\beta > 0\) and with zero boundary conditions, assigns probability
to each \(\phi : \Lambda ^+ \rightarrow \mathbb {Z}\) satisfying \(\phi \equiv 0\) on \(\partial \Lambda \), where
the partition function, normalizes \(\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda }\) to be a probability measure. We denote the (thermal) expectation with respect to the measure \(\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda }\) by \(\left\langle \cdot \right\rangle _{\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda }}\) .
Our results show that the gradient of the integer-valued random-field Gaussian free field shares the delocalization/localization properties of the real-valued surfaces discussed above in all dimensions, and that the two models share similar height fluctuations in dimensions \(d=1,2\). It is further shown that the integer-valued model localizes in all dimensions \(d\ge 3\) at low temperature and weak disorder, thus exhibiting a different behavior from the real-valued surfaces in dimensions \(d=3,4\).
The behavior of the integer-valued model in dimensions \(d\ge 3\) at high temperature or strong disorder is left open; See Sect. 7.
In the next theorems, we consider dimensions \(d\ge 1\), integer \(L\ge 2\), inverse temperature \(\beta >0\) and disorder strength \(\lambda >0\). We suppose \(\eta :\Lambda _L\rightarrow \mathbb {R}\) are independent random variables. \(\phi \) is sampled from the measure \(\mu _{\Lambda _L}^{{{\,\textrm{IV}\,}}, \beta , \eta }\) given by (1.19).
Theorem 3
(Gradient fluctuations, integer-valued). Suppose \(\mathbb {E}\left[ \eta (x) \right] =0\) and \({{\,\textrm{Var}\,}}\left[ \eta (x) \right] =1\) for all \(x \in \Lambda _L\). There exist \(C , c>0\) depending only on the dimension such that the quantities: \(c_\lambda := c \lambda ^2\), \(C_\lambda : = C \lambda ^2\), \(C_\beta := C \left( 1 + \beta ^{-1} \right) \) and
satisfy
Remark 1.1
The lower bound in (1.23) is trivial when \(\lambda \) is small. This is improved in Remark 6.4, where we establish that, for any dimension \(d \ge 1\) and any disorder strength \(\lambda > 0\), there exists a constant \(c_1 >0\) depending on the dimension, the law of the random field and the disorder strength, such that
Theorem 4
(Height fluctuations, integer-valued, \(d=1,2\)). Suppose that the \((\eta (x))\) are identically distributed with \(\mathbb {E}\left[ \eta (x) \right] =0\) and \({{\,\textrm{Var}\,}}\left[ \eta (x) \right] =1\) for all \(x \in \Lambda _L\). There exist a constant \(c>0\) depending on the common distribution of the \((\eta (x))\) and an absolute constant \(C>0\) such that the quantities: \(c_\lambda := c e^{- \frac{1}{c \lambda ^2}}\), \(C_{\lambda , \beta } := C \left( 1 + \lambda ^2 +\beta ^{-1} \right) \) and
satisfy
The theorems determine the order of magnitude of the norms of the gradient and the height of the surface as a function of L, and also estimate the dependence on the disorder strength \(\lambda \). Comparing to Theorem 1, one notices that the dependence of the gradient norm on the disorder strength is the same as in the real-valued case. In contrast, compared with Theorem 2, the lower bound at weak disorder for the height norm is significantly smaller (in its dependence on \(\lambda \)) in the integer-valued case than in the real-valued case. We expect that the two models indeed behave differently. For instance, the proof of Theorem 5 below shows that at zero temperature, in order for \(\phi \) to be non-zero at the origin it is necessary that there exists a connected subset with connected complement of \(\Lambda _L\) containing the origin in which the sum of \(\lambda \) times the disorder \(\eta \) exceeds a constant multiple of its boundary size. A recent result of Ding and Wirth [32, Proposition 2.2] shows that in two dimensions it is unlikely that there exist such subsets if \(L\le \exp (\lambda ^{-4/3+o(1)})\) (with the o(1) term referring to the limit \(\lambda \downarrow 0\)).
We also remark that the assumption that the \((\eta (x))\) are identically distributed in Theorem 4 is only required for the proof of the lower bounds on the height norm, and may be replaced by an assumption that a \((2+\delta )\)-moment of the \((\eta (x))\) admits a uniform upper bound.
Theorem 5
(Height fluctuations, integer-valued, \(d\ge 3\), low temperature and weak disorder) Suppose \(d \ge 3\) and assume that \(\eta (x)\) has the standard Gaussian distribution N(0, 1) for all \(x \in \Lambda _L\). There exist \(\beta _0, \lambda _0, c>0\) such that for all \(\beta \in (\beta _0,\infty )\), \(\lambda \in (0, \lambda _0)\), integer \(L\ge 2\), \(v \in \Lambda _L\), and integer \(t >0\),
In particular, for all \(k >0\),
The exponents 1/2 and 1/6 in (1.26) are not sharp (the proof can be optimized to yield slight improvements). However, the \(\lambda ^2\) term appears to be necessary, at least for the case \(t=1\), as it controls the local fluctuations of the field at v when the disorder strength is weak.
As mentioned above, Theorem 5 proves that the behavior of the integer-valued random-field Gaussian free field differs from its real-valued counterpart in dimensions \(d=3,4\).
We additionally note that it is possible to take the zero-temperature limit \(\beta \rightarrow \infty \) in the results of Theorem 3, Theorem 4 and Theorem 5 in order to obtain that the ground state of the integer-valued random-field Gaussian free field satisfies the estimates stated in theses results.
The discussion up until now has been centered around gradient models where the Hamiltonian depends on the discrete gradient of the surface. One can also consider the effects of the addition of a random field on “higher-order" random surfaces. One such example is the random-field membrane model, whose Hamiltonian is given by the formula
where \(\Delta \) indicates the graph Laplacian on \(\Lambda _L\). Explicit computations, which are available for this model, enable us to prove upper and lower bounds on the fluctuation of \(\phi \); see Sect. 7.7 for a more detailed discussion of the model and the results.
1.2 Background
Brascamp, Lieb and Lebowitz [18] initiated the first detailed investigation of the fluctuations of real-valued random surfaces of the form (1.3) without an external field (sometimes called the \(\nabla \phi \)-model) and compared their behavior to the exactly-solvable Gaussian free field (the case \(V(x) = \frac{1}{2}x^2\)). Among other results, their work proved that such surfaces delocalize in two dimensions (the one-dimensional case is classical) and localize in three and higher dimensions under the assumption (1.2). The integer-valued Gaussian free field (without an external field) exhibits similar behavior in dimensions \(d=1\) and \(d\ge 3\) but undergoes a roughening transition in two dimensions as the temperature increases: Localization at low temperatures follows by a version of the classical Peierls argument. Delocalization at high temperatures was proved in the breakthrough work of Fröhlich and Spencer [38, 39] on the Berezinskii-Kosterlitz-Thouless transition (see also [47, 52]). Further details on random surfaces without external field can be found, e.g., in the works of Funaki [40], Velenik [68] and Sheffield [64]. The theory of disordered random surfaces is less developed; we summarize some of the existing mathematical literature below.
Real-valued disordered random surfaces: Külske and Orlandi [48] studied the model (1.3) in two dimensions under the assumption that V is even, twice continuously differentiable and \(\sup _t V''(t)<\infty \) (and, say, \(V(t)/t^{1+\varepsilon }\rightarrow \infty \) as \(t\rightarrow \infty \) for some \(\varepsilon >0\)). Using a Mermin-Wagner type argument they proved that the height fluctuations in \(\Lambda _L\), for any fixed external field \(\eta \), are at least of order \(\sqrt{\log L}\) (which is sharp when \(V(t) = t^2\) and \(\eta \equiv 0\)) and established Gaussian lower bounds on the tail decay. In this sense, the external field can only enhance the height fluctuations. This result is extended in Külske–Orlandi [49] where it is proved that even an arbitrarily strong \(\delta \)-pinning at height zero cannot localize a two-dimensional random surface with i.i.d. random external field \((\eta (x))\) whose common distribution is symmetric with finite non-zero variance. This is then complemented by an analysis of the effects of pinning in dimensions \(d\ge 3\) proving, in particular, that sufficiently strong pinning can localize the surface (under the assumption that \(\inf _t V''(t)>0\)).
Van Enter and Külske [67] considered the model (1.3) when the potential V is assumed to be even, continuously differentiable and to have super-linear growth (\(V(t)/t^{1+\varepsilon }\rightarrow \infty \) as \(t\rightarrow \infty \) for some \(\varepsilon >0\)), and when the \((\eta (x))\) are i.i.d., having a common symmetric distribution of finite non-zero variance. They proved that in two dimensions there are no translation-covariant gradient Gibbs measures \(\mu ^\eta \) satisfying that \(\mathbb {E}\left[ |\left\langle V'(\nabla \phi (e)) \right\rangle _{\mu ^\eta }|\right] <\infty \) for an edge e. They additionally proved that in three dimensions any gradient Gibbs measure \(\mu ^\eta \) has slow decay of correlations in the following sense: For any \(\varepsilon >0\), the correlations \(C_{e,e'}:=\mathbb {E}\left[ \left\langle V'(\nabla \phi (e)) \right\rangle _{\mu ^\eta }\left\langle V'(\nabla \phi (e')) \right\rangle _{\mu ^\eta }\right] \) satisfy \(\lim _{r\rightarrow \infty } \sup _{e,e'} r^{1+\varepsilon }|C_{e,e'}|=\infty \) where the supremum is over all pairs of edges at distance at least r from each other.
Cotar and Külske [25, 26] considered two models of disordered random surfaces: (A) The model (1.3) with even, twice continuously differentiable V and i.i.d. \((\eta (x))\) whose common distribution has finite non-zero second moment. (B) A model in which a collection of i.i.d. random functions \(\left( V_e \right) _{e \in E(\mathbb {Z}^d)}\) is prescribed, and the formal Hamiltonian is given by
Their results show that model (B), subject to suitable assumptions on the random potentials \((V_e)\), behaves similarly to the non-disordered case. Thus, our discussion here pertains to their results on model (A), which are closer to our work.
The work [25] proves the following results on model (A) when V grows quadratically at infinity and satisfies \(\sup _x V''(x)<\infty \): (1) When \(\mathbb {E}\left[ \eta (x)\right] =0\) and \(d\ge 3\), there exist translation-covariant gradient Gibbs measures \(\mu ^\eta \) with any prescribed disorder-averaged tilt, which also satisfy \(\mathbb {E}\left[ \left\langle \nabla \phi (e)^2\right\rangle _{\mu ^\eta }\right] <\infty \); (2) when \(\mathbb {E}\left[ \eta (x) \right] =0\) and \(d\ge 3\), the infinite-volume surface tension exists and is independent of \(\eta \); (3) when \(\mathbb {E}\left[ \eta (x)\right] \ne 0\), there are no translation-covariant gradient Gibbs measures \(\mu ^\eta \) with \(\mathbb {E}\left[ |\left\langle V'(\nabla \phi (e)) \right\rangle _{\mu ^\eta }|\right] <\infty \) (using the techniques of [67]); (4) the infinite-volume surface tension does not exist in dimensions \(d=1,2\) and, when \(\mathbb {E}\left[ \eta (x)\right] \ne 0\), in dimensions \(d\ge 3\).
The work [26] proves the following results on model (A) under the assumption (1.2) and when \(\mathbb {E}\left[ \eta (x)\right] =0\) and, for some of the results, \(\eta (x)\) has a symmetric distribution which satisfies a Poincaré inequality: (1) In dimensions \(d\ge 3\), there is a unique translation-covariant gradient Gibbs measure \(\mu ^\eta \) with prescribed disorder-averaged tilt, satisfying \(\mathbb {E}\left[ \left\langle \nabla \phi (e)^2\right\rangle _{\mu ^\eta }\right] <\infty \), and whose disorder-average is ergodic; (2) for each translation-covariant gradient Gibbs measure \(\mu ^\eta \), \(\text {Cov}\left( \left\langle \nabla \phi (e)\right\rangle _{\mu ^\eta }, \left\langle \nabla \phi (e')\right\rangle _{\mu ^\eta }\right) \le \frac{C}{\text {dist(e,e')}^{d-2}}\) (and more general bounds).
As mentioned above, our Theorem 1 and Theorem 2 are related to the works of Cotar and Külske [25, 26]. It seems that the upper bounds on the gradient fluctuations in Theorem 1 in dimensions \(d\ge 3\) can be deduced from [25, Proposition 2.1 and Lemma 3.5] and the proofs there may possibly extend to dimensions \(d=1,2\). However, the proofs use different arguments and make different assumptions on the potential (neither set of assumptions implies the other): The results of [25] are established for potentials V having quadratic growth at infinity (\(V(t)\ge At^2 -B\) for \(A>0\) and \(B\in \mathbb {R}\)) and uniformly upper bounded second derivative (\(\sup _t V''(t)<\infty \)), while our results apply in the class described in Remark 4.2. We also note that while Cotar and Külske discuss only gradient Gibbs measures, some of the tools which they use in [26] are related to our approach to the height fluctuations in Theorem 2. Specifically, both proofs use a coupling of Langevin dynamics, as originally proposed by Funaki–Spohn [41] (though [26, Section 4] couples the dynamics with the same disorder while we couple them with different disorders). In addition, our argument relies on the Efron–Stein inequality while [26, Proposition 2.4] uses a related covariance bound.
Integer-valued disordered random surfaces: To our knowledge, the integer-valued random-field Gaussian free field has not been studied before. The following related model was treated mathematically by Bovier–Külske [13, 14] (with earlier treatments on a hierarchical lattice by Bovier, Külske and Picco [11, 12, 15]). The Hamiltonian of the model takes the form
where the surface \(\phi \) is integer-valued, the random variables \((\eta _x(k))_{x\in \mathbb {Z}^d, k\in \mathbb {Z}}\) form a \((d+1)\)-dimensional environment and \(\lambda >0\) is the disorder strength. The model provides an approximation to the domain walls in disordered ferromagnetic Ising models when the random variables \((\eta _x(k))\) are i.i.d. (random-bond case) or when the differences \((\eta _x(k) - \eta _x(k-1))\) are i.i.d. (random-field case) (see also [37, Section 5.1] for a physics discussion of related models in the continuum). The work [13] allows these distributions of \(\eta \) (and more general settings), requiring that the i.i.d. ensembles have zero mean, unit variance and satisfy suitable Gaussian tail bounds, and establishes the existence of infinite-volume Gibbs measures in dimensions \(d \ge 3\) at low temperatures and weak disorder (small \(\lambda \)) (building on the renormalization group approach of Bricmont–Kupiainen [19]). The work [14] shows that, when the \((\eta _x(k))\) are i.i.d. with zero mean and unit variance with common distribution having no isolated atoms or having compact support, the model does not admit translation-covariant Gibbs states in dimensions \(d \le 2\), at all positive temperatures and non-zero disorder strength \(\lambda \) (adapting the arguments of Aizenman–Wehr [5]).
1.3 Convention for constants
Throughout this article, the symbols C and c denote positive constants which may vary from line to line, with C increasing and c decreasing. Except where explicitly stated otherwise, these constants may depend only on the dimension d and the ratios \(c_+/c_-\) and \(\lambda /c_-\).
1.4 Strategy of the arguments
In this section, we present some of the main arguments developed in this article.
1.4.1 Gradient fluctuations
The proof of Theorem 1 relies on a quenched comparison principle. For any fixed realization of the disorder \(\eta \), we are able to relate the \(L^2\) norm (1.5) of the gradient of the random surface with a general uniformly elliptic potential V to the analogous \(L^2\) norm of the ground state of the Gaussian random surface (i.e. the case \(V(x) = \frac{1}{2}x^2\)). The result is precisely stated in (4.15). Theorem 1 follows from this comparison principle and the fact that the \(\eta \)-average of the \(L^2\) norm can easily be estimated via explicit formulas in the Gaussian case, as presented in Proposition 4.1.
The proof of the comparison principle (4.15) relies on computations based on the two identities
where we used the notation \(\sum _{e \ni x}\) to sum over the directed edges containing x as an endpoint (see (2.2) and (2.4)). Both identities follow from a simple integration by parts; see Sect. 3.4. We remark that a version of the identity (1.30) was previously used by van Enter and Külske [67, Proposition 2.2] (the “divergence equation”) in their work on gradient Gibbs states in dimensions \(d=2,3\).
1.4.2 Height fluctuations
To highlight the main ideas of the argument of the proof of Theorem 2, we describe the strategy for the upper bounds in the case of the ground state, instead of the thermal expectation of the field under the Gibbs measure \(\mu _{\Lambda _L}^\eta \). Additionally, for notational simplicity, we set the field strength \(\lambda \) to 1 and consider the variance of the height at the origin. To be more precise, the ground state is defined as the minimizer of the energy
among all the mappings \(v : \Lambda _L^+ \rightarrow \mathbb {R}\) whose values are set to 0 on the boundary \(\partial \Lambda _L\). We denote the minimizer of \(H_{\Lambda _L}^{\eta }\) by \(v_{L,\eta } : \Lambda _L^+ \rightarrow \mathbb {R}\) ; equivalently, it can be characterized as the unique solution of the discrete non-linear elliptic equation
We wish to estimate the variance (over the random field \(\eta \)) of the random variable \(v_{L , \eta }(0)\), and prove that it satisfies the bounds required in Theorem 2
The proof of the upper bounds relies on the Efron–Stein concentration inequality (stated in Proposition 3.2): If we consider two independent copies of the random field, which we denote by \(\eta \) and \(\widetilde{\eta }\), and let \(\eta ^{x}\) be the field satisfying \(\eta ^{x}(y) = \eta (y)\) if \(y \ne x\) and \(\eta ^{x}(x) = {{\widetilde{\eta }}}(x)\), then we have the variance estimate
Consequently, it is sufficient, in order to obtain the desired upper bounds, to prove the inequalities, for any point \(x \in \Lambda _L\),
(note that, although the two-dimensional bound is logarithmic, substituting it into (1.33) will bound the sum by a constant multiple of \(L^2\), as stated in Theorem 2.)
The proof of the upper bounds (1.34) is based on the observation that the difference \(w^x := v_{L , \eta } - v_{L , \eta ^x}\) solves a discrete linear elliptic equation of the form
with \(\delta _y\) the Kronecker delta function and with the elliptic operator \(-\nabla \cdot {\textbf{a}}\nabla \) defined in (2.5). Here, the environment \({\textbf{a}}\) is an explicit function of the ground state \(v_{\eta , L}\) and the potential V (in particular, \({\textbf{a}}\) is random) which satisfies the pointwise uniform ellipticity estimates \(c_- \le {\textbf{a}}\le c_+\) almost surely. Using the linearity of the Eq. (1.35), the mapping \(w^{x}\) can be rewritten as
where \(G_{\textbf{a}}: \Lambda _L^+ \rightarrow (0 , \infty )\) is the Green’s function associated with the environment \({\textbf{a}}\) and satisfying Dirichlet boundary condition on the boundary of the box \(\Lambda _L\). The famed Nash-Aronson bounds (Proposition 3.3) provides upper and lower bounds on the Green’s function \(G_{\textbf{a}}\) whenever \({\textbf{a}}\) is uniformly elliptic. They establish that the map \(G_{\textbf{a}}\) is comparable to the standard random walk Green’s function on \({\mathbb {Z}}^d\), whose square has the same order of magnitude as the right-hand side of (1.34). Using that the random variables \(\eta (x)\) and \({{\widetilde{\eta }}}^x(x)\) have the same expectation and are of unit variance allows to deduce the upper bound (1.34).
The proof of the lower bounds relies on a similar, but more involved, strategy that is outlined below. Building upon the techniques used in the proof of the Efron-Stein inequality (see for instance [10, Theorem 3.1]), we consider an enumeration \(x_1, \ldots , x_{(2L+1)^d}\) of the vertices of the box \(\Lambda _L\), and let \(\mathcal {F}_n\) be the \(\sigma \)-algebra generated by the random variables \(\eta (x_1), \ldots , \eta (x_n)\). We then introduce the martingale \(X_n = \mathbb {E}\left[ v_{L,\eta } \, | \, {\mathcal {F}}_n\right] \) and observe that
We are then able to lower bound each of the terms in the right-hand sides using a strategy similar to the one used for the upper bound, relying on the Nash–Aronson estimates (see (3.6) of Proposition 3.3) to provide lower bounds on the Green’s function \(G_{\textbf{a}}\).
The extension of the result from the ground state to thermal expectation over the Gibbs measure \(\mu ^\eta _{\Lambda _L}\), as stated in Theorem 2, is done by appealing to the Langevin dynamics associated with the models (see Sect. 3.1), and extending the argument presented above from the setting of elliptic equations to the one of parabolic equations. The details are developed in Sect. 5.
1.4.3 Integer-valued random-field Gaussian free field
For the proof of Theorem 3, we begin by observing that the Hamiltonian (1.18) of the integer-valued random-field Gaussian free field may be expressed, by completing a square, as
where \(c(\eta )\) depends only on \(\eta \) and \(u_{\Lambda ,\eta }\) is the ground state of the real-valued random-field Gaussian free field (i.e., \(-\Delta u_{\Lambda ,\eta } = \eta \); see Sect. 4.1). This motivates the following quenched upper bound (Lemma 6.1)
(see Sect. 2.2 for the \(L^2\) notations). The bound implies, using Jensen’s inequality, that
(this bound is especially simple to see at zero temperature, since the left-hand side is essentially the Hamiltonian \(H^{{{\,\textrm{IV}\,}}, \eta }_\Lambda \) and one option for \(\phi \) is the integer part of the function \(\lambda u_{\Lambda ,\eta }\)). The upper bounds on the gradient fluctuations of \(\phi \) (the upper bounds in Theorem 3) are then a direct consequence of (1.39) and the upper bounds on \(\nabla u_{\Lambda _L, \eta }\) stated in Proposition 4.1. which shows that \(\phi \) cannot fluctuate significantly more than \(u_{\Lambda , \eta }\) in dimensions \(d=1,2\).
It is possible to also rely on (1.39) to obtain lower bounds on the gradient and height fluctuations of \(\phi \), but the quality of these lower bounds will deteriorate as \(\beta \) tends to zero. To remove the \(\beta \)-dependence we prove that the thermal average of \(\nabla \phi \) is close to \(\lambda \nabla u_{\Lambda , \eta }\) uniformly in the temperature (Lemma 6.2),
The lower bounds on the gradient fluctuations in Theorem 3 are an immediate consequence by the corresponding estimates on \(u_{\Lambda , \eta }\).
For Theorem 4, we use distinct strategies for the upper and lower bounds. The upper bound on the height fluctuations uses the inequality (1.39) and the Poincaré inequality to deduce that
This inequality is then combined with the estimates of Proposition 4.1 to show that the field \(\phi \) cannot fluctuate significantly more than \(u_{\Lambda , \eta }\) in dimensions \(d=1,2\), since the fluctuations of \(u_{\Lambda , \eta }\) are larger than (the square root of) the right-hand side of (1.41). The lower bounds on the height fluctuations are more involved. We first reduce to a suitable estimate for \(u_{\Lambda _L,\eta }\) by noting that
where we used Jensen’s, the triangle and Poincaré inequalities and applied the estimate (1.40) (this computation can be found in (6.24) and (6.25)). Thus,
As the height fluctuations of \(u_{\Lambda _L,\eta }\) are of order \(L^{2-d/2}\) (see Proposition 4.1), it is natural to expect the right-hand side of the last inequality to be uniformly positive in L in dimensions \(d=1,2\) (with a prefactor depending on \(\lambda \) which tends to zero as \(\lambda \) decreases). To establish this rigorously, we prove a Central Limit Theorem for the projection of \(u_{\Lambda _L,\eta }\) in a suitable direction (see (6.27)).
We outline the proof of Theorem 5 for the ground state of the model and in the case \(t=1\) (to simplify the presentation of the argument). Let \(w_{\Lambda _L , \eta } : \Lambda _L \rightarrow \mathbb {Z}\) be the ground state of the integer-valued random-field Gaussian free field. In this scenario, the conclusion of Theorem 5 can be stated as follows: For any vertex \(v \in \Lambda _L\),
The proof of (1.42) relies on the following observation: If \(w_{\Lambda _L , \eta }(v) \ne 0\), then there exists a finite connected with connected complement \(D \subseteq \mathbb {Z}^d\) containing the vertex v such that
The set D can be constructed as follows. Let us assume, without loss of generality, that \(w_{\Lambda _L , \eta }(v)>0\). Let \(D_0\) be the connected component of the vertex v in the set
and set D to be \(D_0\) union all the connected components of \(D_0^c\) except the one containing \(\partial \Lambda _L\) (this ensures that \(D^c\) is connected). The ground state \(w_{\Lambda _L , \eta }\) necessarily has lower energy than the function \(w_{\Lambda _L , \eta } - \mathbbm {1}_{D}\), which implies (1.43) upon rearranging the terms in the Hamiltonian.
We next appeal to a result of Fisher–Fröhlich–Spencer [36], which states that, in dimensions \(d \ge 3\), the following holds: for all sufficiently small \(\lambda \) and any fixed vertex \(v \in \Lambda _L\), the probability (over the disorder \(\eta \)) that there exists a finite connected set \(D \subseteq \mathbb {Z}^d\), containing v and having connected complement, such that (1.43) holds is at most \(e^{-\frac{c}{\lambda ^2}}\). The result implies the inequality (1.42). The argument can be extended from the ground state to the low-temperature case, as stated in Theorem 5, through a Peierls-type argument.
1.5 Organisation of the article
The rest of the article is organized as follows. Section 2 introduces additional notation. Section 3 collects the tools and preliminary results used in the proofs. Sections 4 and 5 treat the case of real-valued random surfaces and are devoted to the proofs of Theorems 1 and 2, respectively. Section 5 is devoted to the integer-valued random-field Gaussian free field, and contains the proofs of Theorems 3, 4 and 5. Appendix 1 provides a proof of the Nash–Aronson estimate used in Sect. 4, for the heat kernel in a time-dependent uniformly elliptic environment in a box with Dirichlet boundary condition.
2 Notation
2.1 General
Given a (simple) graph \(G = (V(G), E(G))\) we let \(\textbf{E}(G)\) be the set of directed edges of G (each edge in E(G) appears in \(\textbf{E}(G)\) with both orientations). We write \(x\sim y\) to denote that \(\{x,y\}\in E(G)\). We often identify subsets \(\Lambda \subset G\) with the induced subgraph of G on \(\Lambda \). In particular, we write G for V(G) and write \(E(\Lambda )\) and \(\textbf{E}(\Lambda )\) for the edges of E(G) and \(\textbf{E}(G)\) having both endpoints in \(\Lambda \), respectively. We let \(\partial \Lambda = \partial _G \Lambda \) be the external vertex boundary of \(\Lambda \) in G,
\(\Lambda ^+:=\Lambda \cup \partial \Lambda \), and \(\left| \Lambda \right| \) be the cardinality of \(\Lambda \), sometimes referred to as discrete volume.
Let \(\mathbb {Z}^d\) be the standard d-dimensional lattice, and let \(\left| \, \cdot \, \right| \) be the \(\ell ^\infty \)-norm on \(\mathbb {Z}^d\). Two vertices \(v,w\in \mathbb {Z}^d\) are adjacent if they are equal in all but one coordinate and differ by one in that coordinate. Given \(M \in \mathbb {N}\) with \(M \ge 1\), we denote by \(M \mathbb {Z}^d\) the set of points of \(\mathbb {Z}^d\) whose coordinates are divisible by M.
We say that a set \(\Lambda \) is connected, if, for any pair of distinct vertices \(x , y \in \Lambda \), there exists a finite collection of points \(x_1 , \ldots , x_N \in \Lambda \) such that \(x = x_1\), \(y = x_N\) and for any \(i \in \{ 1 , \ldots , N - 1 \}\), \(x_{i}\) and \(x_{i+1}\) are adjacent. We say that a set \(\Lambda \) is \(\star -\)connected, if, for any pair of distinct vertices \(x , y \in \Lambda \), there exists a finite collection of points \(x_1 , \ldots , x_N \in \Lambda \) such that \(x = x_1\), \(y = x_N\) and \(\left| x_i - x_{i+1}\right| =1\) for every \(1\le i \le N - 1\).
Write \(\Lambda _L := \{-L, \ldots , L\}^d\subset \mathbb {Z}^d\) for any integer \(L \ge 0\). This is extended to all \(L \in [0 , \infty )\) by setting \(\Lambda _L:=\Lambda _{\lfloor L\rfloor }\) where \(\lfloor L\rfloor \) denotes the floor of L. For \(x \in \mathbb {Z}^d\), we denote by \(x + \Lambda _L\) the box \(\Lambda _L\) translated by the vector x.
Let \(a \wedge b\) be the minimum and \(a \vee b\) be the maximum of \(a , b \in \mathbb {R}\), and by \(\lceil a \rceil \) the ceiling of \(a \in \mathbb {R}\).
2.2 \(L^2\)-Norms
Let G be a finite graph. For a function \(\phi : G \rightarrow \mathbb {R}\), define the \(L^2\) and normalized \(L^2\)-norms of \(\phi \) by the formulae
Define the discrete gradient
In expressions which do not depend on the orientation of the edge, such as \(|\nabla \phi (e)|^2\) or \(V(\nabla \phi (e))\), we allow the edge e to be undirected. For a function \(v:\textbf{E}(G)\rightarrow \mathbb {R}\) satisfying \(v((x,y)) = -v((y,x))\) (such as the function \(\nabla \phi \)) define the \(L^2\) and normalized \(L^2\)-norms of v by the formulae
2.3 Environments and operators
Let G be a graph and introduce the notation, for each vertex \(x \in G\),
A map \({\textbf{a}}: E(G) \rightarrow \mathbb {R}\) is called an environment. Its definition is extended to directed edges by setting \({\textbf{a}}((x,y)):={\textbf{a}}(\{x,y\})\) for \((x,y)\in \textbf{E}(\Lambda )\). The operator \(-\nabla \cdot {\textbf{a}}\nabla \) is defined by the formula
for a function \(\phi : G \rightarrow \mathbb {R}\) and \(x\in G\). Unravelling the definitions shows that, for any pair of functions functions \(\phi , \psi : G \rightarrow \mathbb {R}\), the following discrete integration by parts identity holds:
in the sense that if one side converges absolutely then the other converges absolutely to the same value. Note that the terms inside the sum on the right-hand side are well defined for undirected edges. We say that the environment \({\textbf{a}}\) is uniformly elliptic if there exist \(c_-,c_+\) such that \(c_- \le {\textbf{a}}\le c_+\), pointwise.
The above definitions naturally extend to time-dependent environments \({\textbf{a}}: I \times E(G) \rightarrow \mathbb {R}\), where \(I\subset \mathbb {R}\) is a (time) interval: The operator \(-\nabla \cdot {\textbf{a}}\nabla \) acts on time-dependent functions \(\phi : I \times G \rightarrow \mathbb {R}\) with the same definition (2.5) applied at each fixed time. The identity (2.6) then holds at each fixed time for time-dependent functions \(\phi ,\psi \).
The discrete Laplacian \(\Delta \) is the operator \(\nabla \cdot {\textbf{a}}\nabla \) with \({\textbf{a}}\equiv 1\).
3 Tools
In this section, we collect tools pertaining to random surfaces, concentration inequalities and estimates on the solution of parabolic equations which are used in the proofs of Theorems 1 and 2.
3.1 Langevin dynamics
The Gibbs measure \(\mu _{\Lambda _L}^{\eta }\) (defined in (1.3)) is naturally associated with the following dynamics.
Definition 3.1
(Langevin dynamics). Given an integer \(L \ge 0\), random field strength \(\lambda \), external field \(\eta :\Lambda _L\rightarrow \mathbb {R}\) and a collection of independent standard Brownian motions \(\left\{ B_t(x) \, : \, x \in \Lambda _L\right\} \), define the Langevin dynamics \(\left\{ \phi _t(x) \, : \, x \in \Lambda _L\right\} \) to be the solution of the system of stochastic differential equations
The Langevin dynamics (3.1) is stationary, reversible and ergodic with respect to the Gibbs measure \(\mu _{\Lambda _L}^\eta \); in particular, one has the convergence
where, in a slight abuse of notation, we use the symbol \(\left\langle \cdot \right\rangle \) in the left-hand side to refer to the expectation with respect to the Brownian motions \(\left\{ B_t(x) \, : \, x \in \Lambda _L \right\} \) .
3.2 The Efron–Stein inequality
We record below the Efron–Stein inequality which will be used in the proof of Theorem 2. A proof can be found in [10, Theorem 3.1].
Proposition 3.2
(Efron–Stein inequality). Let \(\Lambda \) be a finite set, let \(\eta ,{{\widetilde{\eta }}}:\Lambda \rightarrow \mathbb {R}\) be independent and identically distributed random vectors with independent coordinates, and let \(f : \mathbb {R}^{\Lambda } \rightarrow \mathbb {R}\) be a measurable map satisfying \(\mathbb {E}\left[ f(\eta )^2 \right] < \infty \). For each \(x \in \Lambda \), set \(\eta ^{x}\) to be the field defined by the formula \(\eta ^x (y) = \eta (y)\) if \(y \ne x\) and \(\eta ^x (x) = {{\widetilde{\eta }}}(x)\). Then
3.3 Heat kernel bounds
Let \(L\ge 0\) be an integer, let \(0<c_-\le c_+<\infty \), let \(s_0\in \mathbb {R}\) and let \(y\in \Lambda _L\). Let \({\textbf{a}}: [s_0 , \infty ) \times E(\Lambda _L) \rightarrow [c_- , c_+]\) be a continuous time-dependent (uniformly elliptic) environment. For each initial time \(s \ge s_0\), denote by \(P_{\textbf{a}}= P_{\textbf{a}}(\cdot ,\cdot ;s,y):[s,\infty )\times \Lambda _L^+\rightarrow [0,1]\) the heat kernel associated with Dirichlet boundary conditions in the box \(\Lambda _L\), i.e., the solution of the parabolic equation
The maximum principle ensures that \(P_{\textbf{a}}\) is non-negative. Summing \(\nabla \cdot {\textbf{a}}\nabla P_\textbf{a}\) over \(\Lambda _L\) and integrating by parts implies that this sum is non-positive, which, in turn, shows that \(\sum _{x\in \Lambda _L} P_{\textbf{a}}(t,x;s,y)\) is a non-increasing function of t on \([s,\infty )\). In particular, since the value \(\sum _{x\in \Lambda _L} P_{\textbf{a}}(s,x;s,y)\) is equal to 1, one has the estimate, for any \(t \ge s\),
Upper and lower bounds on heat kernels are usually referred to as Nash–Aronson estimates. They were first established by Aronson in [6], in the continuous setting and in infinite volume for parabolic equations with time-dependent and uniformly elliptic environment. In the discrete setting, we refer to the article of Delmotte [29] and the references therein for a collection of heat kernel estimates on general graphs in static environment. The case of discrete parabolic equations with dynamic and uniformly elliptic environment is treated by Giacomin–Olla–Spohn in [43, Appendix B]. The proposition stated below is a finite-volume version of their result.
Proposition 3.3
(Nash–Aronson estimates). In the above setup with \(L\ge 1\) there exist positive constants \(C_0 , c_0\) depending only on the dimension d and the ratio of ellipticity \(c_+ / c_-\) such that the following holds. For all \(t\ge s\) and \(x\in \Lambda _L\),
In addition, there exists a constant \(c_1 > 0\) depending only on d and \(c_+ / c_-\) such that for any \(t\ge s\) and any \((x,y)\in \Lambda _L \times \Lambda _{L/2}\) satisfying \(|x - y| \le \sqrt{c_- (t - s)} \le c_1 L\),
The proof of this result is the subject of Appendix 1.
Remark 3.4
The L-dependent term in the right-hand side of (3.5) is a consequence of the Dirichlet boundary condition in the definition (3.3) of the heat kernel \(P_{\textbf{a}}\). It shows that the map \(P_{\textbf{a}}\) decays exponentially fast after a time of order \(L^2\). It is obtained by analytic arguments in Appendix 1 and has a natural probabilistic interpretation: If we denote by \((X_t)_{t \ge s}\) the continuous time random-walk evolving in the time dependent environment \({\textbf{a}}\) (see [43, Section 3.2] for a formal definition of the process) started at time s from the vertex y, then one has the identity
where \(\tau _{\Lambda _L}\) is the random walk hitting time of the boundary \(\partial \Lambda _L\). Under the assumption that the environment \({\textbf{a}}\) is uniformly elliptic, the random walk \(X_t\) will typically hit the boundary \(\partial \Lambda _L\) in a time of order \(L^2\), and the probability that it remains in the box \(\Lambda _L\) for a time \(t-s\) will exhibit an exponential decay of the form \(\exp \left( - c (t-s)/L^2 \right) \). This observation implies the exponential decay of the heat kernel \(P_{\textbf{a}}\) for large \(t-s \ge L^2\) by the identity (3.7). The inequality (3.5), proved in Appendix 1 using equivalent, analytic techniques, can be deduced from a refinement of the previous argument and gives an accurate description of the heat kernel \(P_{\textbf{a}}\) over the entire time line \(t \in (s , \infty )\).
3.4 Probability density identities
Suppose \(f:\mathbb {R}^n\rightarrow [0,\infty )\) is a continuously differentiable probability density such that |y|f(y) tends to zero at infinity and satisfies that \(y \rightarrow (1 + |y|)\nabla f(y)\) is integrable. Integration by parts implies that, for each index \(1\le j\le n\),
Let \(\Lambda \subset \mathbb {Z}^d\) be finite and \(\eta :\Lambda \rightarrow \mathbb {R}\). Applying the above identities to the probability density of \(\mu _\Lambda ^\eta \) (see (1.3)) under the assumption (1.2) shows that, for each \(x\in \Lambda \),
4 Gradient fluctuations in the real-valued case
The objective of this section is to prove the delocalization of the gradient of the real-valued random-field random surfaces in dimensions \(d \le 2\), and its localization in dimensions \(d \ge 3\), proving Theorem 1.
4.1 The ground state of the random-field Gaussian free field
Given a finite \(\Lambda \subset \mathbb {Z}^d\) and a function \(\eta :\Lambda \rightarrow \mathbb {R}\), let \(u_{\Lambda ,\eta } : \Lambda ^+ \rightarrow \mathbb {R}\) be the solution of the Dirichlet problem
One readily checks that \(u_{\Lambda ,\eta }\) is the ground state of the random-field Gaussian free field in \(\Lambda \) with zero boundary conditions and unit disorder strength. That is, \(u_{\Lambda ,\eta }\) minimizes the Hamiltonian \(H_\Lambda ^\eta \) given by (1.1), with \(V(x) = \frac{1}{2}x^2\) and \(\lambda =1\), among all functions \(\phi \) which equal zero on \(\partial \Lambda \). The function \(u_{\Lambda ,\eta }\) will be instrumental in analyzing the gradient fluctuations of the real-valued random-field random surfaces and will also play a role in our analysis of the integer-valued random-field Gaussian free field in Sect. 6.
The next proposition studies the order of magnitude of \(u_{\Lambda ,\eta }\) and its gradient when \(\Lambda = \Lambda _L\) and \(\eta \) is random with independent and normalized values.
Proposition 4.1
Suppose \((\eta (x))_{x\in \Lambda }\) are independent with zero mean and unit variance. Then
-
(i)
The random variables \((u_{\Lambda ,\eta }(x))_{x\in \Lambda }\) and \((\nabla u_{\Lambda ,\eta } (e))_{e\in \textbf{E}(\Lambda )}\) have zero mean.
Now let \(\Lambda =\Lambda _L\) for an integer \(L\ge 2\). There exist constants \(C, c > 0\) depending only on the dimension d such that
-
(ii)
The following upper bounds hold for all \(x\in \Lambda _L\),
$$\begin{aligned} \mathbb {E}\left[ u_{\Lambda _L,\eta }(x)^2 \right] \le \left\{ \begin{aligned}&C L^{4-d}{} & {} 1\le d\le 3,\\&C \ln L{} & {} d =4, \\&C{} & {} d \ge 5; \end{aligned} \right. \end{aligned}$$(4.2) -
(iii)
The following lower bounds hold for all \(x \in \Lambda _{c L}\),
$$\begin{aligned} \mathbb {E}\left[ u_{\Lambda _L,\eta }(x)^2 \right] \ge \left\{ \begin{aligned}&c L^{4-d}{} & {} 1\le d\le 3,\\&c \ln L{} & {} d =4,\\&c{} & {} d \ge 5; \end{aligned} \right. \end{aligned}$$(4.3) -
(iv)
The averaged \(L^2\)-norm of the gradient of \(u_{\Lambda _L,\eta }\) satisfies
$$\begin{aligned} \mathbb {E}\left[ \left\| \nabla u_{\Lambda _L,\eta } \right\| _{{\underline{L}}^2 \left( \Lambda _L^+ \right) }^2 \right] \approx \left\{ \begin{aligned}&L{} & {} d=1,\\&\ln L{} & {} d =2, \\&1{} & {} d \ge 3, \end{aligned} \right. \end{aligned}$$(4.4)where \(a\approx b\) is used here in the sense \(c\cdot a\le b\le C\cdot a\).
Proof
Let \(G_\Lambda :\Lambda ^+\times \Lambda \rightarrow [0,\infty )\) be the elliptic Green’s function with Dirichlet boundary condition, defined by requiring that for all \(y \in \Lambda \),
with \(\delta _y\) the Kronecker delta function. The maximum principle shows that \(G_\Lambda \) is indeed non-negative. The linearity of (4.1) and (4.5) implies that
for \(x \in \Lambda ^+\) and \(e \in \textbf{E} \left( \Lambda _L^+\right) \). Property (i) is an immediate consequence. In addition, our assumptions on \(\eta \) imply that
Now specialize to the case \(\Lambda = \Lambda _L\) with \(L\ge 2\). Denote by \(P_{\Lambda _L}\) the solution of the parabolic Eq. (3.3) in the specific case of the heat equation (i.e., in the case \({\textbf{a}}\equiv 1\)). The elliptic Green’s function \(G_{\Lambda _L}\) is related to the heat kernel \(P_{\Lambda _L}\) through the identity: For any \(x\in \Lambda _L^+\) and \(y \in \Lambda _L\),
The identity (4.8) can be checked by a direct computation: Proposition 3.3 implies that \(P_{\Lambda _L} \left( t , x ;0 , y \right) \) tends to 0 as t tends to infinity for any \(x , y \in \Lambda _L\) and that the right-hand side of (4.8) is well-defined. Taking the discrete Laplacian of the map \(x \mapsto \int _0^\infty P_{\Lambda _L} \left( t , x ;0 , y \right) \, dt\) and using the definition of the heat kernel \(P_{\Lambda _L}\), we obtain
Consequently the map \(x \mapsto \int _0^\infty P_{\Lambda _L} \left( t , x ;0 , y \right) \, dt\) solves the Eq. (4.5) for each \(y\in \Lambda _L\). Since this equation has a unique solution, it implies the identity (4.8). Proposition 3.3 additionally shows the upper bounds
valid for all \(x, y \in \Lambda _L\), and the lower bounds
valid for \(y \in \Lambda _{L/2}\) and \(x \in \Lambda _L\) such that \(|x - y| \le c L\), with \(C,c>0\) depending only on d. Combining the identity (4.7) with the upper and lower bounds (4.10) and (4.11) shows the upper and lower bounds on the value of \(\mathbb {E}\left[ u_{\Lambda _L,\eta }(x)^2 \right] \) stated in (4.2) and (4.3).
We proceed to obtain the estimates on the averaged \(L^2\)-norm of \(\nabla u_{\Lambda _L,\eta }\). The identity (4.7) implies that
Applying (2.6) (with \(G = \Lambda _L^+\), \(a\equiv 1\) and \(\phi =\psi =G_{\Lambda _L}(\cdot ,y)\)) and using (4.5) we see that
for each \(y \in \Lambda _L\). Substituting in (4.12), we obtain
Applying the upper and lower bounds (4.10) and (4.11) with the choice \(x = y\) and using the non-negativity of the Green’s function \(G_{\Lambda _L}\) we obtain the inequalities in (4.4). \(\square \)
4.2 Gradient fluctuations
In this section, we prove Theorem 1 for the uniformly convex \(\nabla \phi \)-model, as a consequence of the following result which holds for any choice of \(\eta \). Theorem 1 follows from the result upon setting \(\Lambda = \Lambda _L\) and letting \(\eta \) be random as in the theorem and then taking an expectation over \(\eta \) and applying the bounds in item (iv) of Proposition 4.1.
Theorem 6
Let \(\Lambda \subset \mathbb {Z}^d\) be finite, \(\lambda >0\) and \(0<c_-\le c_+<\infty \). Let \(\eta :\Lambda \rightarrow \mathbb {R}\) and let \(u_{\Lambda , \eta }:\Lambda ^+\rightarrow \mathbb {R}\) be the solution of (4.1). Suppose \(\phi \) is sampled from the measure \(\mu ^\eta _\Lambda \) of (1.3), where the potential V satisfies (1.2). Then the quantity
satisfies
Proof
We start with the proof of the lower bound. Multiply the identity (3.10) by \(u_{\Lambda ,\eta }(x)\) and sum over all \(x\in \Lambda \) to obtain
The left-hand side is developed by observing that each edge appears in the sum with both orientations and that \(V'\) is an odd function,
(where the boundary terms may be added by noting that \(u_{\Lambda ,\eta }\equiv 0\) on \(\partial \Lambda \)). The right-hand side of (4.16) is developed as
by using the Eq. (4.1) and the equality (2.6). Substituting the last two equalities in (4.16) shows that
An application of the Cauchy-Schwarz inequality to the right-hand side then gives
Finally, as V is an even function satisfying (1.2), it follows that
for all \(x\in \mathbb {R}\). In particular, \(|V'(x)|\le c_+ |x|\). Substituting in (4.18) shows that
The Cauchy-Schwarz inequality shows that \(\langle |\nabla \phi (e)|\rangle _{\mu _\Lambda ^\eta }^2\le \langle (\nabla \phi (e))^2\rangle _{\mu _\Lambda ^\eta }\) and the lower bound of the theorem follows.
We proceed to prove the upper bound. Sum the identity (3.11) over all \(x\in \Lambda \) to obtain
(where the terms with \(x\in \partial \Lambda \) are added by noting that \(\phi \equiv 0\) on \(\partial \Lambda \)). As in the analysis that lead to the lower bound, the left-hand side is developed by summing over the two orientations of each edge, and thus
The right-hand side of (4.20) is developed as
by using the Eq. (4.1) and the equality (2.6). Substituting in (4.20) shows that
The left-hand side of the equality is developed using the lower bound in (4.19) while the right-hand side is developed using the Cauchy-Schwarz inequality, yielding
Recalling the definition of \(\left\| \nabla \phi \right\| _{L^2 \left( \Lambda ^+, \mu _\Lambda ^\eta \right) }\) from (4.14) and using that \(a+b\le 2\max (a,b)\) for \(a,b>0\) and \(\langle \nabla \phi (e)\rangle _{\mu _\Lambda ^\eta }^2\le \langle \nabla \phi (e)^2\rangle _{\mu _\Lambda ^\eta }\) (which follows from Jensen’s inequality), we conclude that
Rearranging we get
which implies the upper bound of the theorem. \(\square \)
Remark 4.2
An inspection of the proof of Theorem 6 shows that the assumption that V satisfies (1.2) may be relaxed. Specifically, the proof of the lower bound in Theorem 6 requires only that V is a sufficiently smooth even function, having sufficient growth at infinity for the identity (3.10) to hold, and that \(|V'(x)|\le c_+ |x|\) for all \(x\in \mathbb {R}\). Similarly, the proof of the upper bound in Theorem 6 requires only that V is a sufficiently smooth even function, having sufficient growth at infinity for the identity (3.11) to hold, and that \(V'(x)\ge c_- x\) for all \(x \ge 0\). Note that V need not be convex for these relaxed assumptions to hold.
Consequently, the lower and upper bounds in Theorem 1 hold under the same relaxed assumptions.
Section 7.5 contains additional discussion on general potentials.
5 Height fluctuations in the real-valued case
In this section, we study the fluctuations of the height of real-valued random-field random surfaces. We prove that the surface delocalizes in dimensions \(1\le d\le 4\) and that it localizes in dimensions \(d\ge 5\), proving Theorem 2. Quantitative upper and lower bounds for the variance of the thermal expectation of the height are obtained. The section is organized as follows. In Sect. 5.1, we establish a quantitative theorem which estimates the difference of the thermal expectations of the height of the random surface at the center of a box with two different external fields. Subsection 5.2 is devoted to the proof of the upper bounds (1.9), (1.10) and (1.11) by combining the results of Subsection 5.1 with the Efron–Stein inequality following the outline presented in Sect. 1.4. In Sect. 5.3, we prove the lower bounds (1.12), (1.13) and (1.14), and thus complete the proof of Theorem 2.
5.1 A quantitative estimate for the thermal expectation of the height of the random surface
This section is devoted to the proof of an upper bound and a lower bound on the difference of the thermal expectations of the height at the center of a box with two different (and arbitrary) external fields. The argument relies on a coupling argument for the Langevin dynamics associated with the random-field \(\nabla \phi \)-model and on the Nash–Aronson estimate for parabolic equation with uniformly convex and time-dependent environment (Proposition 3.3).
Theorem 7
Let \(L\ge 2\), \(\lambda > 0\) and \(\eta , {{{\overline{\eta }}}} : \Lambda _L \rightarrow \mathbb {R}\) be two external fields. Then there exists a constant \(C \in (0 , \infty )\) depending only on d and the ratios \(c_+/c_-\) and \(\lambda /c_-\) such that, for any \(y \in \Lambda _L\),
Moreover, there exist two constants \(c , {{\widetilde{c}}} \in (0 , \infty )\) depending on d, \(c_+/c_-\) and \(\lambda /c_-\) such that, for any \(y \in \Lambda _{L / 2}\), if the fields \(\eta \) and \({{\overline{\eta }}}\) are such that there exists a vertex \(x \in y + \Lambda _{{{\widetilde{c}}} L}\) such that \(\eta = {{{\overline{\eta }}}}\) on \(\Lambda _L \setminus \{ x \}\) and \(\eta (x) \ne {{\overline{\eta }}}(x)\), then
Remark 5.1
We highlight that the left-hand side of (5.2) does not involve an absolute value. This way of stating the result is strictly stronger than with absolute values as it implies that the sign of the difference \(\left\langle \phi (y) \right\rangle _{\mu _{\Lambda _L}^\eta } - \left\langle \phi (y) \right\rangle _{\mu _{\Lambda _L}^{ {{{\overline{\eta }}}}}}\) is the same as the one of \(\eta (x) - {{\overline{\eta }}}(x)\), i.e., that the value of the thermal expectation \(\left\langle \phi (y) \right\rangle _{\mu _{\Lambda _L}^\eta }\) increases with the value of \(\eta (x)\).
It will be clear from the proof that this monotonicity is a direct consequence of the non-negativity of the heat kernel \(P_{\textbf{a}}\) (defined under the environment (5.7) below) and is an important ingredient in the proof of the lower bounds of Theorem 2 in Sect. 5.3.
The proof of this theorem relies on coupling two different Langevin dynamics using the same Brownian motion. This argument was originally used by Funaki and Spohn in [41].
Proof
Let us fix an integer \(L\ge 2\), two external fields \(\eta , {\overline{\eta }} : \Lambda _L \rightarrow \mathbb {R}\), consider a collection of independent Brownian motions \(\left\{ B_t(x) \, : \, x \in \Lambda _L \right\} \), and run two Langevin dynamics (with the same Brownian motions). Explicitly, we set
and
By ergodicity of the Langevin dynamics stated in Sect. 3.1, the two dynamics convergence to the appropriate thermal averages
Taking the difference between the two stochastic differential Eqs. (5.3) and (5.4), and using that the two driving Brownian motions are the same, we obtain
The strategy is then to rewrite the Eq. (5.6) as a discrete linear parabolic equation. To this end, we note that, since the potential V is assumed to be twice continuously differentiable, the following identity is valid, for any edge \(e \in E \left( \Lambda _L\right) \) and any time \(t \ge 0\),
Thus, if we let \({\textbf{a}}: [0 , \infty ) \times E\left( \Lambda _L^+ \right) \rightarrow [0, \infty )\) be the time-dependent environment defined by the formula
then we have the identity, for any point \(y \in \Lambda _L\),
where we used the notation (2.5) for discrete elliptic operators. The inequalities \(c_- \le V'' \le c_+\) imply that the environment \({\textbf{a}}\) is uniformly elliptic and satisfies \(c_- \le {\textbf{a}}(t,y)\le c_+\) for any pair \((t,e) \in [0, \infty ) \times E \left( \Lambda _L \right) \). Denoting by \(w_t (z) := \phi _t(z) - {\overline{\phi }_t}(z)\), we obtain that the map w solves the discrete parabolic equation
Using the notations introduced in Sect. 3.3, we denote by \(P_{\textbf{a}}\) the heat kernel associated with the environment \({\textbf{a}}\). We next claim that one has the identity
The identity (5.9) is a consequence of Duhamel’s formula (see [35, Theorem 2 page 50] where the formula is proved for the continuous heat equation in \({\mathbb {R}^d}\)) applied to the Eq. (5.8). In the present setting, it can be verified directly by computing the time derivative of the function \((t , y) \mapsto \lambda \sum _{x \in \Lambda _L} \left( \eta (x) - {{\overline{\eta }}}(x) \right) \int _{0}^t P_{\textbf{a}}\left( t , y; s , x \right) \, ds\) and observing that this map is a solution of the Eq. (5.8).
By the Nash–Aronson estimate on the heat-kernel \(P_{\textbf{a}}\) stated in Proposition 3.3, we obtain the inequality, for any time \(t \ge 0\),
where the constant C depends on d and the ratio of ellipticity \(c_+ / c_-\). Taking the expectation with respect to the collection of Brownian motions \(\left\{ B_t(y) \, : \, y \in \Lambda _L \right\} \) and using the identities (5.5), we obtain
where the constant C depends on d and the ratios \(c_+ / c_-\) and \(\lambda / c_-\). This is the inequality (5.1). It thus suffices to prove (5.2). Since the two external fields \(\eta \) and \({{{\overline{\eta }}}}\) are equal everywhere except at the vertex x, the identity (5.9) becomes
Let us denote by \(c_1\) the constant which appears in the statement of Proposition 3.3 and set \({{\widetilde{c}}} := c_1/2\). Assuming that the vertex x belongs to the box \(y+ \Lambda _{\widetilde{c} L}\), we may apply the lower bound for the map \(P_{\textbf{a}}\) stated in (3.6) and use that the heat kernel is non-negative to obtain, for any time \(t \ge c_1^2 L^2/c_-\),
The term in the right-hand side can be explicitly computed, and we obtain the lower bound, for any time \(t \ge c_1^2 L^2/c_-\),
A similar computation as in (5.10) completes the proof of the lower bound (5.2). \(\square \)
5.2 Real-valued random-field random surface models: upper bounds
In this section, we establish the upper bounds of the inequalities (1.9), (1.10) and (1.11) by combining the result of Theorem 7 with the Efron–Stein concentration inequality stated in Proposition 3.2.
Proof of Theorem 2:
Upper bounds (1.9), (1.10) and (1.11). Let us fix an integer \(L \ge 2\) and a vertex \(y \in \Lambda _L\). The strategy is to apply the Efron–Stein inequality with the map \(f : \eta \mapsto \left\langle \phi (y)\right\rangle _{\mu _{\Lambda _L}^{\eta }}\). Let us note that, by the inequality (5.1) applied with \({{\overline{\eta }}} = 0\) and the observation \(\left\langle \phi (y) \right\rangle _{\mu _{\Lambda _L}^{0}} = 0\), the \(L^2\)-norm of the mapping \(\eta \mapsto \left\langle \phi (y)\right\rangle _{\mu _{\Lambda _L}^{\eta }}\) is finite. The assumption of Proposition 3.2 is thus satisfied.
We let \(\eta \) and \({{\widetilde{\eta }}}\) be two independent copies of the random field. For each point \(x \in \Lambda _L\), we denote by \(\eta ^x\) the resampled random field defined by the formula \(\eta ^x(z) = \eta (z)\) if \(z \ne x\) and \(\eta ^x(x) = {{\widetilde{\eta }}} (x)\). By Proposition 3.2, one has the estimate, for any \(y \in \Lambda _L\),
Applying the quenched upper bound (5.1) obtained in Theorem 7 with the external fields \(\eta := \eta \) and \({{{\overline{\eta }}}} := \eta ^x\), we obtain the upper bound
Taking the expectation in the previous display (with respect to the random field \(\eta \)) and using that the random variables \(\eta (x)\) and \({{\widetilde{\eta }}}(x)\) have expectation 0 and variance 1 shows the inequality, for any \(x \in \Lambda _L\),
Summing the inequality (5.13) over all the points \(x \in \Lambda _L\) and using the estimate (5.12) completes the proof of the upper bounds of (1.9), (1.10) and (1.11).
5.3 Real-valued random-field random surface models: lower bounds
The objective of this section is to prove the lower bounds for the variance (in the random field) of the thermal average of the heights.
Proof of Theorem 2:
Lower bounds (1.12), (1.13) and (1.14). We fix an integer \(L \ge 2\), a vertex \(y \in \Lambda _{L/2}\) and let \({{\widetilde{c}}}\) be the constant which appears in the statement of Theorem 7. We may assume without loss of generality that \({{\widetilde{c}}} \le 1/2\). We let \(x_1 , \ldots x_{(2L+1)^d}\) be an enumeration of the points of the box \(\Lambda _{L}\) such that the collection \(x_0 , \ldots , x_{\left( 2 \lfloor {{\widetilde{c}}} L \rfloor +1\right) ^d}\) is an enumeration of the vertices of the box \(y + \Lambda _{ {{\widetilde{c}}} L}\). We let \(\eta \) and \({{\widetilde{\eta }}}\) be two independent copies of the random field. For each integer \(k \in \{ 1 , \ldots , (2L+1)^d \}\), we introduce the notations \(\eta _k := \eta (x_k)\), \({{\widetilde{\eta }}}_k := {{\widetilde{\eta }}}(x_k)\), denote by \(\nu _k\) the law of \(\eta _k\), and let \({\mathcal {F}}_k\) be the sigma-algebra generated by the random variables \(\eta _1 , \ldots , \eta _k\). We additionally denote by \(\eta ^k\) the random field satisfying \(\eta ^k(x) = \eta (x)\) if \(x \ne x_k\), and \(\eta ^k(x_k) = \widetilde{\eta }(x_k)\).
For each integer \(k \in \{1 , \ldots , (2L+1)^d \}\), we introduce two random variables. We let \(X_k\) be the conditional expectation of \(\left\langle \phi (y)\right\rangle _{\mu _{\Lambda _L}^{\eta }}\) given the \(\eta _1,\dots \eta _k\) — that is,
The standard Pythagorean identity for martingales tells us that
Next, we define \({\widetilde{X}}_k\) as the conditional expectation of \(\left\langle \phi (y)\right\rangle _{\mu _{\Lambda _L}^{\eta }}\) given \(\eta _1,\dots \eta _{k-1}\) and where the external field at \(x_k\) is set to \({\widetilde{\eta }}_k\). Since \({\widetilde{\eta }}_k\) is independent of \(\eta _k\), we find that
and therefore
Combining this with (5.14) allows us to conclude that
We next prove the lower bound, for any integer \(k \in \left\{ 1 , \ldots , \left( 2 \lfloor {{\widetilde{c}}} L \rfloor + 1 \right) ^d \right\} \),
Using that all the terms in the right side of (5.16) are non-negative, that the collection of points \(x_1 , \ldots x_{2^d \lfloor {{\widetilde{c}}} L \rfloor ^d \vee 1}\) is an enumeration of the points of the box \(y +\Lambda _{{{\widetilde{c}}} L}\) and the estimate (5.17) completes the proof of the lower bounds (1.9), (1.10) and (1.11).
There remains to prove (5.17). Let us note that the two fields \(\eta \) and \(\eta ^k\) are equal on the set \(\Lambda _L \setminus \{ x_k \}\). We can thus apply Theorem 7 to obtain, on the event \(\left\{ \eta _k \ne {{\widetilde{\eta }}}_k \right\} \),
Integrating the inequality (5.18) over the field variables \(\eta _{ k+1}, \ldots , \eta _{L^d}\), we obtain, on the event \(\{ \eta _k \ne {{\widetilde{\eta }}}_k \}\),
The lower bound (5.19) shows, for any realization of the random fields \(\eta \) and \({{\widetilde{\eta }}}\),
Taking the expectation with respect to the external field, using that the two random variables \(\eta _k\) and \({{\widetilde{\eta }}}_k\) are independent and that their variance is equal to 1 completes the proof of (5.17).\(\square \)
Remark 5.2
Theorem 2 is proved for independent \((\eta (x))\) with the sole assumption that \({{\,\textrm{Var}\,}}(\eta (x))=1\) for all \(x\in \Lambda _L\). We mention that, when further assumptions are imposed on \(\eta \), a different approach to the variance lower bounds of the heights of the random surface is available. We present below a simple statement of this kind in the case that each \(\eta (x)\) has a standard Gaussian distribution which yields, in addition to the variance lower bound, the following estimate: There exists \(\varepsilon > 0\) (depending on d, \(c_+/c_-\) and \(\lambda /c_-\)) such that for each \(y\in \Lambda _{L/2}\) and each interval \(I \subseteq \mathbb {R}\) whose side length is equal to \(L^{2-d/2}\) in dimensions \(d = 1 , 2 , 3\), \(\sqrt{\ln L}\) in dimension \(d = 4\) and 1 in dimensions \(d \ge 5\), one has the estimate
Estimates of this form are often referred to as anti-concentration bounds, and we refer to [56, Section 1.1] for related arguments which can yield more detailed anti-concentration bounds.
Let X be a standard Gaussian vector in \(\mathbb {R}^n\). We first recall a useful bound on the total variation distance,
for each deterministic vector \(t\in \mathbb {R}^n\), where we write \(d_{\text {TV}}(X,Y)\) for the total variation distance between the distributions of the random vectors X and Y.
Now suppose \(f:\mathbb {R}^n\rightarrow \mathbb {R}\) satisfies that for each \(1\le x\le n\) there exists \(\ell (x)>0\) so that
where \(\delta _x\in \mathbb {R}^n\) is the vector satisfying \(\delta _x(y) = 0\) for \(y\ne x\) and \(\delta _x(x)=1\). Such a function necessarily satisfies the following: There exists an \(\varepsilon >0\) such that for each interval \(I\subset \mathbb {R}\) satisfying \(|I|\le \varepsilon \Vert \ell \Vert _2\) it holds that
This bound clearly implies that \({{\,\textrm{Var}\,}}(f(\eta ))\ge c\Vert \ell \Vert _2^2\). To see (5.23) set \(t:=\varepsilon \frac{\ell }{\Vert \ell \Vert _2}\) and observe that, by (5.22), it is impossible that for some \(\alpha \), \(f(\alpha )\in I\) and \(f(\alpha +t)\in I\) simultaneously. Consequently, by (5.21),
which implies (5.23) by choosing \(\varepsilon \) sufficiently small.
Since for each \(y\in \Lambda _{L/2}\) the function \(f(\eta ) = \left\langle \phi (y) \right\rangle _{\mu _{\Lambda _L}^\eta }\) satisfies bounds of the form (5.22) by (5.2), the variance lower bounds of Theorem 2 follow from the above argument.
6 Integer-valued random-field Gaussian free field
In this section, we study the integer-valued random-field Gaussian free field. We emphasize that, unlike our discussion of real-valued random surfaces with random fields, we assume here that the potential is \(V(x) = \tfrac{1}{2} x^2\). We further assume that \(\eta \) is independent and satisfies \(\mathbb {E}[\eta (x)]=0\) and \({{\,\textrm{Var}\,}}[\eta (x)] =1\).
6.1 Upper bounds on gradient and height fluctuations
The upper bounds of Theorem 3 and Theorem 4 will follow from the following lemma which compares the gradient of the integer-valued random-field Gaussian free field with the gradient of \(u_{\Lambda ,\eta }\), the ground state of the real-valued random-field Gaussian free field, at any fixed external field \(\eta \).
Lemma 6.1
Let \(d\ge 1\), \(\Lambda \subset \mathbb {Z}^d\) be a finite subset of \(\mathbb {Z}^d\), \(\lambda >0\) be the strength of the random field, \(\beta > 0\) be an inverse temperature, \(\eta :\Lambda \rightarrow \mathbb {R}\) be an external field and \(u_{\Lambda , \eta }:\Lambda ^+\rightarrow \mathbb {R}\) be the solution of (4.1). Suppose \(\phi \) is sampled from the probability distribution \(\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda }\). Then there exists a constant C depending only on the dimension d such that
Proof of Lemma 6.1
We first treat the specific case \(\eta = 0\) and prove the following slightly stronger version of the inequality (6.1): There exists a constant C depending on d such that
To prove the estimate (6.2), we first note that, for each \(\phi : \Lambda ^+ \rightarrow \mathbb {Z}\) normalized to be 0 on the boundary \(\partial \Lambda \), the cardinality of the set \(\left\{ \psi : \Lambda ^+ \rightarrow \mathbb {Z}\, : \, \psi \equiv 0 ~ \text{ on } ~ \partial \Lambda ~ \text{ and } \lfloor \psi /2 \rfloor = \phi \right\} \) is smaller than \(2^{\left| \Lambda \right| }\). From this observation, one deduces the inequality
Using that the difference between a real number and its floor is smaller than 1, and the inequality \((a + b)^2 \le 2a^2 + 2b^2\), we obtain
Bringing a factor of 1/2 ‘into’ the \(L^2\) norm and then applying inequalities (6.3) and (6.4), we can derive the bound
We then expand the thermal expectation as a normalized sum, writing
Using the explicit formula for the partition function \(Z^{{{\,\textrm{IV}\,}}, \beta , 0}_{\Lambda } := \sum _{\phi : \Lambda ^+ \rightarrow \mathbb {Z}} \exp \left( - \frac{\beta }{2} \left\| \nabla \phi \right\| _{L^2 \left( \Lambda ^+ \right) }^2 \right) \) and combining the inequality (6.5) with the identity (6.6) and the inequality \(\left| \Lambda \right| \le \left| \Lambda ^+ \right| \) completes the proof of (6.2).
We now prove the inequality (6.1) for a general external field \(\eta : \Lambda \rightarrow \mathbb {R}\). Applying the estimate (6.2) at inverse temperature \(\frac{5}{4} \beta \), we see that it is sufficient to prove that there exists a constant \(C > 0\) depending on d such that
The rest of the argument is thus devoted to the proof of the inequality (6.7). Using the identity \(-\Delta u_{\Lambda ,\eta } = \eta \) and performing a discrete integration by parts, one obtains, for any integer-valued surface \(\phi : \Lambda ^+ \rightarrow \mathbb {Z}\) normalized to be 0 on \(\partial \Lambda \),
Using the previous computation, we see that
Additionally, from the definition of the partition function \(Z^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda _L}\) and the computation (6.8), we obtain the identity
Let us then introduce the mapping \(w_{\Lambda , \eta } := \lambda u_{\Lambda , \eta } - \lfloor \lambda u_{\Lambda , \eta } \rfloor \). Since the function \(\lfloor \lambda u_{\Lambda , \eta } \rfloor \) is integer-valued, we may perform the discrete change of variable \(\phi \mapsto \phi - \lfloor \lambda u_{\Lambda , \eta } \rfloor \), and write
By definition, the mapping \(w_{ \Lambda , \eta }\) is bounded by 1. Applying the inequality \((a + b)^2 \ge \frac{5}{6} a^2 - 5 b^2\), we obtain
A combination of (6.11) and (6.12) implies
A similar computation, but using this time the inequality \(\left\| \nabla \phi -\nabla w_{ \Lambda , \eta } \right\| _{L^2 \left( \Lambda ^+ \right) }^2 \le \frac{5}{4}\left\| \nabla \phi \right\| _{L^2 \left( \Lambda ^+ \right) }^2 + C \left| \Lambda ^+ \right| \), yields the lower bound
A combination of the identities (6.9), (6.10) with the inequalities (6.13) and (6.14) implies the upper bound
The proof of the inequality (6.7) is complete.
Proof of the upper bounds of Theorem 3 and Theorem 4
Lemma 6.1 and Jensen’s inequality imply that
The upper bound of Theorem 3 now follows from Proposition 4.1 and the triangle inequality. We also obtain in this way a version of the lower bound of Theorem 3 which is, however, suboptimal for small \(\beta \). To obtain a lower bound which is temperature independent (as stated in Theorem 3), we provide a different argument in the following section.
For the upper bound of Theorem 4, one can apply the Poincaré inequality and obtain
Again, applying the triangle inequality and appealing to Proposition 4.1 for an upper bound on \(\Vert u_{\Lambda _L,\eta } \Vert _{{\underline{L}}^2 ( \Lambda _L^+)}\) for \(d=1,2\) completes the proof.
6.2 Lower bounds on gradient and height fluctuations
The lower bounds of Theorem 3 and Theorem 4 are also deduced from a comparison of the gradient \(\nabla \phi \) of the integer-valued random-field Gaussian free field to the gradient \(\nabla u_{\Lambda ,\eta }\) of the ground state of the real-valued random-field Gaussian free field. While it is possible to use Lemma 6.1 to this end, the resulting bounds would deteriorate at high temperatures. Instead, we rely on the following lemma which shows that the thermal expectation \( \left\langle \nabla \phi \right\rangle _{\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda }}\) is close to \(\lambda \nabla u_{\Lambda ,\eta }\) uniformly in the temperature parameter (closeness of the thermal expectation suffices for the lower bounds, due to Jensen’s inequality, but is insufficient for the upper bounds).
Lemma 6.2
Let \(d\ge 1\), \(\Lambda \subset \mathbb {Z}^d\) be a finite subset of \(\mathbb {Z}^d\), \(\lambda >0\) be the strength of the random field, \(\beta > 0\) an inverse temperature, \(\eta :\Lambda \rightarrow \mathbb {R}\) be an external field. Let \(\phi \) be distributed according to the integer-valued random-field Gaussian free field \(\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda }\). Then for any map \(w : \Lambda ^+ \rightarrow \mathbb {Z}\) normalized to be 0 on the boundary \(\partial \Lambda \),
Consequently, there exists a constant C depending only on the dimension d such that
Proof
We only prove the inequality (6.17) in the case \(\eta = 0\); the general case can be obtained by a notational modification of the argument. By performing the discrete change of variable \(\phi \rightarrow \phi + w\), we have the identity
Subtracting the right and left hand sides of the identity (6.19), and expanding the square, we obtain the identity
Using the identity \(e^y - 1 \ge y\), we obtain the estimate
Dividing both sides of the inequality (6.20) by the partition function \(Z^{{{\,\textrm{IV}\,}}, \beta , 0}_{\Lambda }\) yields the inequality: For any \(w : \Lambda \rightarrow \mathbb {Z}\),
Using the definition of the mapping \(u_{\Lambda , \eta }\), the inequality (6.21) can be equivalently rewritten as follows: For any \(w : \Lambda \rightarrow \mathbb {Z}\),
We then deduce the inequality (6.17) by applying inequality (6.22) with either the function w or the function \(-w\).
We next prove (6.18). To this end, let us consider the integer-valued function \(w = \lfloor \lambda u_{\Lambda , \eta } - \left\langle \phi \right\rangle _{\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda }} \rfloor \). Using that the difference between the functions w and \(\lambda u_{\Lambda , \eta } - \left\langle \phi \right\rangle _{\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda }}\) is smaller than 1 in absolute value and the inequality \((a+b)^2 \le \frac{5}{4} a^2 + 5 b^2\), we obtain the inequality
A similar argument yields
where we used the inequality \(ab \le \frac{1}{4C} a^2 + 16C b^2\) in the second line. A combination of the three previous displays implies the inequality (6.18). \(\square \)
Remark 6.3
We point out that the inequality (6.17) also follows from the non-negativity of the Kullback-Leibler divergence between the distribution of \(\phi \) and \(\phi +w\), when \(\phi \) is sampled from \(\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda }\).
Proof of the lower bounds of Theorem 3 and Theorem 4
We first prove the lower bounds of Theorem 3. To this end, we use the triangle and Jensen’s inequalities to write
The lower bounds of Theorem 3 are then obtained by appealing to the inequality (6.18) and to Proposition 4.1 for a lower bound on \(\Vert \nabla u_{\Lambda _L,\eta } \Vert _{{\underline{L}}^2 ( \Lambda _L^+)}\).
Let us now prove the lower bounds of Theorem 4. Fix an integer \(L \ge 1\). Let \(\{X_n\}\) be the simple random walk on \(\mathbb {Z}^d\) and define \(\tau _L:= \min \{n \ge 0\, :\, X_n \in \mathbb {Z}^d\setminus \Lambda _L\}\). For \(x \in \mathbb {Z}^d\), denote by \(\mathbb {P}_x\) and \(\mathbb {E}_x\) the law and expectation with respect to \(\{X_n\}\) started from \(X_0 = x\). Define a map \(P_L : \mathbb {Z}^d\rightarrow \mathbb {R}\) by
The Markov property for the random walk implies that, for any \(x \in \Lambda _{L}\),
rearranging this equation shows that \(-\Delta P_L = 1\) in \(\Lambda _L\). Furthermore, standard estimates on the random walk imply that
Therefore, we find that
where we use discrete integration by parts in the bound on the norm of \(\nabla P_L\).
By Jensen’s and the triangle inequalities,
We next estimate the second term in the right-hand side of the previous display. To this end, we apply Poincaré inequality and the estimate (6.18). We obtain
Thus,
To complete the proof, we wish to show that \(\Vert u_{\Lambda _L,\eta } \Vert _{{\underline{L}}^2(\Lambda _L)}\) has large fluctuations. By the Cauchy-Schwarz inequality, the construction of \(P_L\) and discrete integration by parts, we may write
We will use the Central Limit Theorem to prove that, as L grows,
where N(0, 1) is the standard normal distribution. Indeed, the summands in (6.27) are independent with mean zero, and \(\lambda \eta (x) P_L(x)\) has variance \(\lambda ^2 P_L(x)^2\). Since \(|P_L(x)| \le C L^2\), \(\sum _{x \in \Lambda _L} P_L(x)^2 > cL^{4 +d} \) and the \((\eta (x))\) are identically distributed, the sum (6.27) satisfies the Lindeberg-Feller Central Limit Theorem (see [34, Theorem 3.4.10]).
To connect the two previous displays, we point out that
Thus, combining the previous four displays, we find that
in dimensions \(d=1,2\). Since this bound is uniform in L, we deduce the desired lower bound on the expectation of \(\left\| \phi \right\| _{{\underline{L}}^2 \left( \Lambda _L^+ , \mu ^{{{\,\textrm{IV}\,}}, \beta ,\eta }_{\Lambda _L}\right) }\).
Remark 6.4
We complete this section by establishing the lower bound stated in Remark 1.1. Using that the random field \((\eta (x))_{x \in \mathbb {Z}^d}\) is assumed to be i.i.d. with expectation zero and variance one, we see that there exists a constant \(\kappa > 0\) depending on the law of the random field such that, for any \(x \in \mathbb {Z}^d\),
We may then assume, without loss of generality, that \(\mathbb {P}\left[ \eta (x) \ge 1 \right] \ge \kappa \). We next let M be the smallest integer larger than 1 such that
We assume that \(L \ge 2M\) and consider the random set
We next set
Since the random field is i.i.d., there exists a constant \(c > 0\) depending only on the dimension, the law of the random field, and the disorder strength, such that
We next introduce the function \(w := \mathbbm {1}_{E_{L , \lambda }}\) and observe that, by the definition of the set \(E_{L , \lambda }\),
Combining the bounds of (6.29) with (6.17), the definition of the integer M and the triangle inequality, we deduce that
Using the Cauchy-Schwarz inequality and noting that \(\left| \nabla w(e) \right| \le 1\) for any edge \(e \in E \left( E_{\Lambda _L^+} \right) \) since w is an indicator function, we deduce that
where the constant \(c'\) depends only on dimension (due to the normalization counting vertices instead of edges). Taking the expectation on both sides of the previous inequality and using the estimate (6.28) completes the proof.
6.3 Height fluctuations in dimensions \(d \ge 3\)
We conclude the paper by proving Theorem 5. The argument relies on a Peierls-type argument, and we will make use of the two following propositions. The first is a result of Fisher–Fröhlich–Spencer [36] (see also Chalker [22]).
Proposition 6.5
(Fisher–Fröhlich–Spencer [36]). Let \(d \ge 3\) and \(\left( \eta (x) \right) _{x \in \mathbb {Z}^d}\) be a collection of independent standard Gaussian random variables. There exist two constants \(c, \lambda ' > 0\) such that for any \(\lambda \in (0, \lambda ']\) and any \(v \in \mathbb {Z}^d\), the event
satisfies
We mention that the original proof of Fisher–Fröhlich–Spencer [36] is written for the case \(d = 3\) only, and states that
However, the proof proves the stronger statement that we have claimed above, and the argument applies to any dimension \(d \ge 3\) (as remarked at the end of the proof in [36]). In fact, the exponent 1/3 can be improved to a constant approaching 1 as d grows.
The second result is a classical upper bound on the number of bounded connected subsets of \(\mathbb {Z}^d\) containing a fixed vertex v and with a fixed boundary size.
Proposition 6.6
(Lemma 5.3.5 of [7, 54, 61]) There exists a constant \(\alpha _d > 0\) depending only on the dimension such that for each integer \(N \in \mathbb {N}\) and each \(v \in \mathbb {Z}^d\), the set
satisfies
We are now ready to give the proof of Theorem 5.
Proof of Theorem 5
We need only prove the estimate (1.26) as the bound (1.27) on the k-th moment can be deduced from (1.26) by (suitably) integrating over the t variable. Fix \(d \ge 3\), a side length \(L \ge 2\), a vertex \(v \in \Lambda _L\), and an integer \(t \ge 1\). Let \(\lambda '\) and \(\alpha _d\) be the constants appearing in the statements of Proposition 6.5 and Proposition 6.6. We also define \(s_0 = t^{\frac{d-2}{2(d-1)}}\). Let \(\beta _0 \in (1 , \infty )\), and \(c \in (0 , 1)\) be constants depending only on the dimension whose value will be selected later in the argument. Set \( \lambda _0 := (1/4 \wedge c) \lambda '\) and fix \(\lambda \in (0,\lambda _0)\) and \(\beta \in (\beta _0,\infty )\).
By an application of Proposition 6.5, we see that it is sufficient for the proof of Theorem 5 to prove the inclusion
Let us fix a realization of the random field \(\eta \in {\mathcal {E}}_{4\lambda ,v, t^{1/2}} \cap {\mathcal {E}}_{\lambda /(cs_0),v,1}\), and let \(\phi : \Lambda _L \rightarrow \mathbb {Z}\) be a random surface distributed according to the random-field integer-valued Gaussian free field \(\mu ^{{{\,\textrm{IV}\,}}, \beta , \eta }_{\Lambda _L}\). We denote by \(D_+^0 \left( \phi \right) \) the connected component of v in the set \(\left\{ x \in \Lambda _L \, : \, \phi (x) \ge 1 \right\} \), and set \(D_+^0 \left( \phi \right) = \emptyset \) if \(\phi (v) \le 0\). We then define the set \(D_+(\phi )\) to be the union of the set \(D_+^0(\phi )\) and of all the finite connected components of \(\mathbb {Z}^d\setminus D_+^0(\phi )\). This procedure ensures that the sets \(D_+(\phi )\) and \(\mathbb {Z}^d\setminus D_+(\phi )\) are connected and thus that the boundary \(\partial D_+(\phi )\) is \(\star \)-connected (see [65, Lemma 2]), and will allow us to apply Proposition 6.5. Symmetrically, we let \(D_-^0 \left( \phi \right) \) be the connected component of v in the set \(\left\{ x \in \Lambda _L \, : \, \phi (x) \le -1 \right\} \), set \(D_-^0 \left( \phi \right) = \emptyset \) if \(\phi (v) \ge 0\), and let \(D_-(\phi )\) to be the union of the set \(D_-^0(\phi )\) and of all the finite connected components of \(\mathbb {Z}^d\setminus D_-^0(\phi )\). We note that either \(D_-(\phi )\) or \(D_+(\phi )\) is empty, and both are empty if and only if \(\phi (v) = 0\).
For convenience, we will assume that \(D_+(\phi )\) is nonempty, as the other case follows from symmetry. We say that \(\phi \) is s-steep if there exists a cutset of edges \({\mathcal {S}}\) that separates v from \(\partial D_+(\phi )\) such that \(\nabla \phi (e) \ge s\) for each \(e \in {\mathcal {S}}\). We first claim that, for some constant \(c_d \) depending on dimension only,
Indeed, assume that \(|\partial D_+(\phi )| \le t^{1/2}\) and \(\phi \) is not \(c_d s_0\)-steep and let us show that \(\phi (v) \le t\). Let \(c_d>0\) be sufficiently small for the following arguments. By standard isoperimetric estimates on \({\mathbb {Z}}^d\) (see [55, Lemma 3.1], following [8]), there are at most \(t^{\frac{d}{2(d-1)}}/c_{d}\) edges with at least one endpoint in \(D_{+}(\phi )\). Furthermore, standard duality between paths and cutsets tells us that, whenever \(\phi \) is not \(c_d s_0\)-steep, there exists a path from \(\partial D_+(\phi )\) to v all of whose edges e satisfy \(|\nabla \phi (e)| < c_d s_0\) and have at least one endpoint in \(\partial D_+(\phi )\). Let \(\gamma _{wv}\) be such a path, whose starting point is some \(w\in \partial D_+(\phi )\). Then
where we used \(\phi (w) \le 0\) since \(w\in \partial D_+(\phi )\) and the fact that all edges of \(\gamma _{wv}\) have at least one endpoint in \(D_+(\phi )\). This establishes (6.32).
Thus, the inclusion (6.31) follows from showing that for any \(\eta \in {\mathcal {E}}_{4\lambda ,v, t^{1/2}} \cap {\mathcal {E}}_{\lambda /(cs_0),v,1}\),
We start the proof of (6.33) by estimating the first term on its left-hand side. To this end we decompose the event \(\{ |\partial D_+ \left( \phi \right) | > t^{1/2}\}\) according to the identity
where the sum in the right-hand side is computed over all the bounded connected subsets D containing the vertex v such that \(\partial D\) is \(\star \)-connected and has more than \(t^{1/2}\) vertices. Let us fix such a set \(D \subseteq \Lambda _L\). We have the identity
For each integer-valued \(\phi : \Lambda _L \rightarrow \mathbb {Z}\) (with zero boundary values) such that \(D_+ \left( \phi \right) = D\), we define \({{\widetilde{\phi }}}\) according to the formula \({{\widetilde{\phi }}} := \phi - \mathbbm {1}_D\). Using that, by the identity \(D_+ \left( \phi \right) = D\), the map \(\phi \) satisfies \(\phi \ge 1\) on D and \(\phi \le 0\) on \(\partial D\), we have the inequality, for any edge \(e \in E \left( \Lambda _L^+\right) \),
Summing the inequality (6.36) over the edges \(e \in E \left( \Lambda _L^+ \right) \) and using that the cardinality of the support of the mapping \(\nabla \mathbbm {1}_D\) is at least \(\left| \partial D \right| \) (since an edge belongs to the support of this map if and only if it has exactly one endpoint in D and one endpoint in \(\partial D\)), we obtain
Combining the inequality (6.37) with the assumption \(\eta \in {\mathcal {E}}_{4\lambda ,v, t^{1/2}}\) (and noting that all sets we are interested in here have boundary of at least \(t^{1/2}\) vertices), we obtain
Using the previous computation, we may write
A combination of the identity (6.35) and the inequality (6.38) implies
Putting the estimate (6.39) back into (6.34), using Proposition 6.6, the assumption \(\beta \ge \beta _0\), selecting \(\beta _0 \) large enough and c small enough, and summing over sets with boundary of size at least \(t^{1/2}\), we deduce that
Next, we wish to analyze the second term on the left-hand side of (6.33), i.e., the probability that \(\phi \) is \(c_d s_0\)-steep. The logic is very similar to the proof above, except that the set \(D_+(\phi )\) will be replaced by a set where the gradient of \(\phi \) is large. More specifically, for every \(c_d s_0\)-steep \(\phi \), let \(\overline{{\mathcal {S}}}\) be the outermost cutset of edges separating v from \(\partial D_+(\phi )\) on which \(\nabla \phi (e) \ge c_d s_0\) for every edge e; this is well defined, e.g., using an exploration procedure from the boundary of \(D_+(\phi )\). We then define \({\overline{I}}\) to be the set of vertices in the interior of \(\overline{{\mathcal {S}}}\) – i.e., the vertices u such that every path from u to \(\partial D_+(\phi )\) must include at least one edge in \(\overline{{\mathcal {S}}}\). If \(\phi \) is not \(c_d s_0\)-steep, set \({\overline{I}}\) to be empty. Thus,
where the sum is, again, over the connected subsets of \(\Lambda _L\) containing v whose complement is also connected. Repeating the procedure used above, we define \({\hat{\phi }} = \phi - \lceil c_d s_0 \rceil \cdot \mathbbm {1}_{I}\), and observe that
where we observe that every edge in the support of \(|\nabla \mathbbm {1}_{I}|\) must be an edge of the cutset \(\overline{{\mathcal {S}}}\), and hence \(\nabla \phi (e)\) will be at least \(\lceil c_d s_0\rceil \), granting the final inequality. Summing the inequality (6.42) over all edges, using the assumption \(\eta \in {\mathcal {E}}_{\lambda /(cs_0), v,1}\) and choosing the constant c small enough (e.g., smaller than \(c_d/4\)), we have that
Repeating the earlier logic, we deduce that
Applying Proposition 6.6 and selecting \(\beta _0\) large enough and c small enough yields
where we used the definition of \(s_0\) and the fact that \(d\ge 3\). Plugging (6.40) and (6.43) into (6.33) completes the proof.
7 Discussion and open questions
In this section we provide further discussion and highlight several research directions.
7.1 The integer-valued random-field Gaussian free field and the random-phase sine-Gordon model
Theorem 3 determines that the gradient of the integer-valued random-field Gaussian free field delocalizes in dimensions \(d=1,2\) and localizes in dimensions \(d\ge 3\), at all temperatures (including zero temperature) and all positive disorder strengths \(\lambda \). Our results for the height fluctuations are less complete: Theorem 4 proves delocalization in dimensions \(d=1,2\), again at all temperatures and positive disorder strengths, while Theorem 5 establishes localization in dimensions \(d\ge 3\) at low temperature and weak disorder (unlike the real-valued case, which delocalizes in dimensions \(d=3,4\); see Theorem 2). The height fluctuations in dimensions \(d\ge 3\) in the high temperature or strong disorder regimes remain unclear. We emphasize that delocalization for some \(d\ge 3\), temperature and disorder strength would constitute a roughening transition from the localized behavior proved in Theorem 5, and would thus be of significant interest. Unlike the non-disordered case, where roughening transitions are familiar only in two dimensions [38, 39, 52, 68], we conjecture that such a transition indeed takes place for the three-dimensional integer-valued random-field Gaussian free field, with delocalization occurring at least in the low-temperature and strong disorder regime. We further conjecture that no such transition occurs in dimensions \(d\ge 5\), where we expect the surface to remain localized at all temperatures and disorder strengths (as in the real-valued case; see Theorem 2). These conjectures are supported by a connection of the model with the random-phase sine-Gordon model, on which we elaborate next. We tend to think that a roughening transition also occurs in the intermediate four-dimensional case.
The sine-Gordon model is a model of real-valued surfaces \(\psi \) whose Hamiltonian on a domain \(\Lambda \) with given boundary conditions takes the form
where the \((r(v))_{v \in \Lambda }\) are given elements of the torus \(\mathbb {R}/\mathbb {Z}\) and \(z\ge 0\) is a given activity parameter. In the limit \(z\rightarrow \infty \), configurations \(\psi \) are restricted to satisfying \(\psi _v\in \mathbb {Z}+ r(v)\) at every vertex, and the effective Hamiltonian on this restricted configuration space consists only of the first sum in (7.1). The case where the (r(v)) are (quenched) random is known as the random-phase sine-Gordon model.
The random-phase sine-Gordon model has received much attention in the physics literature (see, e.g., [21, 45, 53, 60, 66]) and the following behavior was predicted for the heights \(\psi \) in the \(z\rightarrow \infty \) limit when the (r(v)) are uniform and independent: In two dimensions, on a box \(\Lambda _L\) with zero boundary conditions, the heights are predicted to delocalize with \(\log L\) variance at high temperature (rough phase), but with \(\log ^2 L\) variance at low temperature (super-rough phase); thus the fluctuations of the model are expected to decrease as the temperature rises! It is further predicted that the heights delocalize with logarithmic variance in three dimensions and are localized when \(d\ge 5\) (see e.g. [44, 58, 59, 69]). These predictions appear to be open in the mathematical literature, apart from the following recent result of Garban and Sepúlveda [42] on the sine-Gordon model: At sufficiently low temperature, for any deterministic choice of (r(v)) and any activity \(z\in [0,\infty ]\), the heights \(\psi \) on a two-dimensional box \(\Lambda _L\) with zero boundary conditions delocalize with variance at least \(\log L\).
Let us now describe the connection between the random-field integer-valued Gaussian free field and the random-phase sine-Gordon model. To this end, observe that the Hamiltonian of the random-field integer-valued Gaussian free field can be written, up to the addition of a constant factor depending only on \(d, \lambda \) and the quenched disorder \(\eta \), as
where we recall that \(u_{\Lambda , \eta }\), the ground state of the real-valued random-field Gaussian free field, is defined in (4.1). Let \(\phi \) be sampled from the integer-valued random-field Gaussian free field and define \(\psi := \phi - \lambda u_{\Lambda , \eta }\). Observe that \(\psi (v)\in \mathbb {Z}+ r(v)\) at every vertex where \(r(v) := -\lambda {{\,\textrm{proj}\,}}u_{\Lambda ,\eta }(v)\) and where \({{\,\textrm{proj}\,}}\) denotes the canonical projection from \(\mathbb {R}\) to \(\mathbb {R}/\mathbb {Z}\). It follows that \(\psi \) is distributed as the random-phase sine-Gordon model in the \(z\rightarrow \infty \) limit with this choice of r(v) (noting that r is a function of the quenched disorder \(\eta \)).
The distribution of the vector r in the above connection is not uniform and independent, though for large \(\lambda \) we expect it to be ‘somewhat close’ to such a distribution, at least when the \((\eta (x))\) are independent standard Gaussian random variables. It is thus suggested that, for large \(\lambda \), the distribution of \(\psi \) is close to that of the random-phase sine-Gordon model with uniform and independent r. This supports the above conjectures regarding the height fluctuations of the integer-valued random-field Gaussian free field in the strong disorder regime.
Further support to the possibility of a roughening transition in dimensions \(d=3,4\) is lent by the following observation, which does not rely on the predictions for the behavior of the random-phase sine-Gordon model. Suppose the Hamiltonian (7.2) is modified by replacing the function \(u_{\Lambda ,\eta }\) with the function \(v_{\Lambda , \eta }:=u_{\Lambda , \eta } + \frac{1}{\lambda } p\) where \(p:\Lambda ^+\rightarrow \mathbb {R}\) is another quenched disorder, consisting of independent random variables which are uniform on \([-1/2, 1/2]\), independently of \(\eta \). We claim that the function \(\varphi :\Lambda ^+\rightarrow \mathbb {Z}\) sampled from this modified Hamiltonian is delocalized in dimensions \(d=3,4\) at all temperatures and disorder strengths. However, we would like to think of \(\frac{1}{\lambda } \nabla p\) as a ‘small perturbation’ of \(\nabla u_{\Lambda , \eta }\) (noting that (7.2) depends on \(u_{\Lambda , \eta }\) only through its gradient) and this is more reasonable at strong disorder, when \(\lambda \) is large, since \(\nabla u_{\Lambda ,\eta }\) is localized in dimensions \(d=3,4\) (Proposition 4.1); thus, consideration of the modified Hamiltonian lends support to delocalization of the random-field integer-valued Gaussian free field in dimensions \(d=3,4\) in the strong disorder regime. The main observation leading to the delocalization claim is that the random phase \(r(v) := -\lambda {{\,\textrm{proj}\,}}v_{\Lambda ,\eta }(v) = -p(v)-\lambda {{\,\textrm{proj}\,}}u_{\Lambda ,\eta }(v)\) is distributed uniformly on the torus \(\mathbb {R}/\mathbb {Z}\), independently between vertices, and that this remains its distribution even after conditioning on \(\eta \) (i.e., for every fixed realization of \(u_{\Lambda , \eta }\), the addition of p to the projection uniformizes the random phase r). It follows, as in the above discussion, that the integer-valued surface \(\varphi \) which follows the modified Hamiltonian is a sum of two independent contributions: The integer part of \(\lambda u_{\Lambda ,\eta }\) and a sample of the random-phase sine-Gordon model with uniform phases r. Thus, regardless of the fluctuations of the random-phase sine-Gordon model, the fluctuations of \(\varphi \) are at least as large as the fluctuations of the integer part of \(\lambda u_{\Lambda ,\eta }\) and thus \(\varphi \) is delocalized in dimensions \(d=3,4\).
7.2 The real-valued random-field Gaussian free field and its relation with the membrane model
The simplest among the random-field \(\nabla \phi \)-models is the random-field Gaussian free field (when the interaction potential is \(V = \frac{1}{2}x^2\)) with the random field \(\eta \) taken to be independent standard Gaussians. The Gaussian nature of this case makes it amenable to exact calculations and, as we will now discuss, relates it to the more familiar membrane model (see [62], and [63] for background and recent results). In the physics literature [37, Section 5.1] such a model (in the continuum) is termed the random-rod model. Precisely, for a finite \(\Lambda \subset \mathbb {Z}^d\) and independent standard Gaussians \(\eta :\Lambda \rightarrow \mathbb {R}\), the Hamiltonian \(H_\Lambda ^{\eta ,\textrm{GFF}}\) of the random-field Gaussian free field associates to each \(\phi :\Lambda ^+\rightarrow \mathbb {R}\) the energy
For simplicity, we study the model with zero boundary conditions, i.e., \(\phi \equiv 0\) on \(\partial \Lambda \) and \(\lambda = 1\). Recall the function \(u_{\Lambda , \eta }\) defined by (4.1) and note that it is precisely the ground state of random-field Gaussian free field (i.e., the minimizer of the Hamiltonian (7.3)). Moreover, the quadratic nature of the Hamiltonian implies that when \(\varphi \) is sampled with the Hamiltonian (7.3) at any positive temperature then the difference \(\varphi - u_{\Lambda , \eta }\) is distributed as a Gaussian free field (with zero external field) at that temperature, independently of \(\eta \). Thus the study of the random-field Gaussian free field decouples to the separate studies of the ground state \(u_{\Lambda , \eta }\) and of the thermal fluctuations given by the Gaussian free field. It turns out that the fluctuations of \(u_{\Lambda , \eta }\) dominate the behavior, so we focus solely on it in the following discussion.
The ground state \(u_{\Lambda , \eta }\) has a joint Gaussian distribution, with covariance matrix given by
and it is simple to check that the right-hand side solves the discrete biharmonic equation
This covariance function coincides with that of the membrane model, and we conclude that \(u_{\Lambda , \eta }\) is equal to it in distribution (to be more precise, Eq. (7.5) uses one of the two standard normalizations for the membrane model. In the other normalization, the requirement \(\Delta v(\cdot , y) = 0\) on \(\partial \Lambda \) is replaced by the requirement that \(v(\cdot ,y)=0\) on a second boundary layer. See [51, Section 2] for a discussion of these two possibilities). This observation allows to directly apply the established results for the membrane model to the study of the random-field Gaussian free field, including precise fluctuation results, behavior of the maximum and entropic repulsion [1, 9, 20, 50, 51]. Of special interest is the scaling limit of the membrane model: The limit, a continuum membrane model, was identified in [24] for the other standard boundary condition. For the boundary condition (7.5), by taking suitable limits of the lattice Green’s function, the following scaling limit result is expected in dimensions \(d\ge 1\): For smooth functions f compactly supported in the continuum box \(\left( -1 , 1\right) ^d\),
where the mapping \(w_f\) is the solution of the continuous biharmonic equation
7.3 Thermodynamic and scaling limits of the random-field \(\nabla \phi \)-model
Theorem 2 establishes quantitative estimates on the height of the random-field \(\nabla \phi \)-model. A natural direction for further research is the study of the infinite-volume limits of the random surface and its gradient. As mentioned, Cotar and Külske [25] proved the existence of translation-covariant gradient Gibbs measures, and their uniqueness for a given tilt [26], in dimensions \(d\ge 3\). The first author [27] proved the convergence along the thermodynamic limit for the random surface in dimensions \(d\ge 5\) and for its gradient in dimensions \(d\ge 4\); convergence of the gradient is also expected in dimension \(d=3\), but remains unproven (the existence and uniqueness of infinite-volume translation-covariant gradient Gibbs measure with a specified tilt was established in [25, 26] in any dimension \(d \ge 3\)).
Beyond infinite-volume Gibbs measures, it is also natural to study the scaling limit of the random-field \(\nabla \phi \)-model. We expect universality of the scaling limit, so that the convergence to the continuum membrane model discussed in (7.6) for the random-field Gaussian free field should continue to hold for the uniformly convex random-field random surface model (1.3).
7.4 Relation between between the ground state of the \(\nabla \phi \) model and its expectation
If we let \(\phi \) be a random-field Gaussian free field with quenched disorder \(\eta \), then we have the identity, for any side length \(L \ge 2\) and any vertex \(x \in \Lambda _L\),
that is, the thermal expectation of the surface is equal to the ground-state of the Hamiltonian associated with the random-field Gaussian free field. We mention the following question: What is the relation, if any, between the ground state of the \(\nabla \phi \)-model and its thermal expectation?
7.5 The random-field \(\nabla \phi \)-model with general potentials V
To what extent are the fluctuations of the random-field \(\nabla \phi \)-model affected by the specific choice of the potential V? Theorem 1 and Theorem 2 show that the order of magnitude in L of the gradient and height fluctuations does not depend on V, as long as the second derivative of V is uniformly bounded from zero and infinity (assumption (1.2)). How generic is this phenomenon? In the non-disordered case, universality of the order of magnitude has been shown for a wide class of potentials, including the family \(V(x) = |x|^p\) with \(p>1\) (see [55, 56] and references within). We observe here that, in the disordered case, such wide universality fails in one dimension. The situation in higher dimensions is not clear.
Let \(d=1\) and consider the model (1.15) with potential \(V(x) = |x|^p\) when the disorder \(\eta \) consists of, say, independent standard Gaussian random variables. We take \(p>1\), as the model is not well defined when \(p\le 1\) (its partition function on \(\Lambda _L\) is infinite with positive probability). When \(p=2\), the ground state \(u_{\Lambda _L,\eta }\) satisfies \(\mathbb {E}\left[ u_{\Lambda _L,\eta }(x)^2 \right] \le C L^3\) for all \(x\in \Lambda _L\) by Proposition 4.1. Now, for general \(p>1\), observe that, on the one hand, any random function \(u_\eta :\Lambda _L^+\rightarrow \mathbb {R}\) with \(u_\eta (x)=0\) on \(\partial \Lambda _L\) which satisfies
for all \(x\in \Lambda _L\) also satisfies the energy estimate
On the other hand, for \(a>0\), the function \(v_\eta :\Lambda _L^+\rightarrow \mathbb {R}\) defined by \(v_\eta (x):=0\) on \(\partial \Lambda _L\) and \(v_\eta (x):={{\,\textrm{sign}\,}}({{\hat{\eta }}})L^a\) for \(x\in \Lambda _L\), where \({{\hat{\eta }}}:=\frac{1}{|\Lambda _L|}\sum _{x\in \Lambda _L} \eta (x)\), satisfies
Since \(|{{\hat{\eta }}}|\) is of order \(L^{-1/2}\) with high probability, we conclude that \(H_{\Lambda _L}^\eta (v_\eta ) \le -c \lambda L^{a+1/2}\) with high probability when \(p<1 + \frac{1}{2a}\) and L is sufficiently large. Thus, taking any \(a>2\), we see that, at least when \(1<p<1+\frac{1}{2a}\), the ground state of the disordered model cannot satisfy the bound (7.7) for large L; it thus exhibits different height fluctuations than the case \(p=2\).
We also point out that the fluctuations of the gradient of the ground state in dimensions \(d\ge 3\) are of order one for the class of potentials \(V(x)=|x|^p\), \(p>1\). Indeed, the ground state in the domain \(\Lambda \) with zero boundary conditions minimizes (using (4.1), discrete integration by parts (2.6) and Hölder’s inequality with \(\frac{1}{p}+\frac{1}{q}=1\))
where we extended the norm notation (2.3) to \(L^p\) in the standard way. Since the energy of the zero function is zero, we conclude that the ground state \(\phi _{\Lambda ,\eta }\) satisfies
Using this inequality with the proof of Proposition 4.1, we see that \(\sup _L \mathbb {E}\left[ \Vert \nabla \phi _{\Lambda _L,\eta }\Vert _{{\underline{L}}^p(\Lambda _L^+)}^p\right] <\infty \) in dimensions \(d\ge 3\) (extending again the norm notation (2.3)), for a class of disorder distributions which includes the case that \(\eta \) consists of independent standard Gaussian random variables.
7.6 Dynamical random-field random surfaces
One may naturally form a dynamics on random-field random surfaces by allowing the random field to evolve in time. When the random field consists of independent standard Gaussians, it is natural to let it evolve via independent Ornstein-Uhlenbeck processes. Note that this gives, in particular, a natural dynamics for the membrane model using its representation as the ground state of the random-field Gaussian free field (see Sect. 7.2).
7.7 The random-field membrane model
In this paper, we considered the effect that a random field can have on the fluctuations of random surfaces of the \(\nabla \phi \) model. The same type of disorder may also be applied to other random surface models. Here, we briefly discuss its effect on the membrane model. The membrane model (in the absence of disorder) is known to fluctuate more strongly than the \(\nabla \phi \) model, delocalizing in dimensions \(1\le d\le 4\) and localizing in dimensions \(d\ge 5\). We will now see that the added disorder causes the surface to delocalize in all dimensions \(d\le 8\).
In this setting, given \(L \ge 0\) and an external field \(\eta : \Lambda _L \rightarrow \mathbb {R}\), the disordered Hamiltonian of a finite-volume \(\phi : \Lambda _L^+ \rightarrow \mathbb {R}\) normalized to 0 on \(\partial \Lambda _L\) is given by the formula
The random-field membrane model is the probability distribution
where the partition function \(Z^{\eta , \Delta }_{\Lambda _L}\) is the constant which makes the measure (7.12) a probability distribution. Let us first consider the ground state of the model, that is, the interface \(v_{\Lambda , \eta } : \Lambda _L \rightarrow \mathbb {R}\) which minimizes the variational problem
Note that it can can be equivalently defined as the solution of the biharmonic equation
In that case, the interface \(v_{\Lambda _L, \eta }\) is a linear functional of \(\eta \), and is given by the explicit formula
(analogously to the discussion in Sect. 7.2, the ground state \(v_{\Lambda _L, \eta }\) has the distribution of a non-disordered random surface model with Hamiltonian proportional to \(\left\| \Delta ^2 \phi \right\| _{L^2 \left( \Lambda _L \right) }^2\). Such a construction may also be applied to higher powers of the Laplacian, and their combination with the gradient operator). Using the upper and lower bounds on the Green’s function stated in (4.10) and (4.11), we obtain that the ground state satisfies the following estimates
It is thus delocalized in dimensions \(d \le 8\) and localized in dimensions \(d \ge 9.\)
To study the probability distribution (7.12), let us observe that if \(\phi : \Lambda _L^+ \rightarrow \mathbb {R}\) is a random surface distributed according to (7.12), then \(\psi := \phi - v_{\Lambda _L, \eta }\) is a membrane model with external field set to 0. The field \(\psi \) has thus a multivariate Gaussian distribution, its mean vector is equal to 0 and its covariance matrix is given by, for any \(x , y \in \Lambda _L\),
Two consequences can be deduced from this observation: First, the expectation of the random variable \(\psi \) is equal to 0, and second the fluctuations of \(\psi \) at the center of the box can be explicitly quantified using the upper and lower bounds on the Green’s function stated in (4.10) and (4.11). We have, for any realization of the random field \(\eta \),
A combination of the two previous sets of estimates shows that the random-field membrane model satisfies the inequalities
The model is delocalized in dimensions \(d \le 8\) and localized in dimensions \(d \ge 9\).
7.8 Random-field Gaussian free field with 1-dependent external field
Theorem 2 states that, if the components of the random field \(\eta \) are independent, then the random-field \(\nabla \phi \)-model is delocalized in dimensions \(d \le 4\) and localized in dimensions \(d \ge 5\). One can thus raise the question whether the independence assumption on the random field can be relaxed without changing the critical dimension for localization. In this section, we show that the independence assumption cannot generally be relaxed to 1-dependence (i.e., independence at distance 2). Specifically, we present a model of a random-field random surface in which the external random field is 1-dependent, which localizes in dimensions \(d \ge 3\) and thus exhibits a different qualitative behaviour.
Let us fix an integer \(L \ge 1\) and let \(\zeta : \Lambda _L \rightarrow \mathbb {R}\) a discrete Gaussian free field in the box \(\Lambda _L\) with Dirichlet boundary condition. We then define the external field \(\eta \) by the formula, for any vertex \(x \in \mathbb {Z}^d\),
The law of the external field \(\eta \) is Gaussian, let us verify that it is 1-dependent. An explicit computation shows, for any points \(x , z \in \Lambda _L\),
Using the identity (7.14) and an explicit computation, we further deduce, for any \(x , y \in \Lambda _L\)
If the vertices x and y are distinct and non-adjacent, then all the terms in the right-hand side are equal to 0 and thus \(\mathbb {E}\left[ \eta (x) \eta (y) \right] =0\). Since the field \(\eta \) is Gaussian, we deduce that the random variables \( \eta (x)\) and \(\eta (y)\) are independent.
Let us now consider the random-field Gaussian free field \(\phi \) with the external field \(\eta \) given by (7.13). Since the random surface \(\phi - \zeta \) is a Gaussian free field with external field set to 0, independently of \(\zeta \), we have,
where \(a\approx b\) is used here in the sense \(c\cdot a\le b\le C\cdot a\). From the inequality (7.15) and the fact that \(\zeta \) is a Gaussian free field with Dirichlet boundary condition in the box \(\Lambda _L\), we obtain
The random surface is thus localized in dimensions 3 and higher.
References
Adams, S., Kister, A., Weber, H.: Sample path large deviations for laplacian models in \((1+ 1) \)-dimensions. Electron. J. Probab. 21, 1–36 (2016)
Aizenman, M., Harel, M., Peled, R.: Exponential decay of correlations in the \(2D\) random field Ising model. J. Stat. Phys. 180(1–6), 304–331 (2020)
Aizenman, M., Peled, R.: A power-law upper bound on the correlations in the \(2D\) random field Ising model. Comm. Math. Phys. 372(3), 865–892 (2019)
Aizenman, M., Wehr, J.: Rounding of first-order phase transitions in systems with quenched disorder. Phys. Rev. Lett. 62(21), 2503 (1989)
Aizenman, M., Wehr, J.: Rounding effects of quenched randomness on first-order phase transitions. Comm. Math. Phys. 130(3), 489–528 (1990)
Aronson, D.G.: Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73, 890–896 (1967)
Balister, P.N., Bollobás, B.: Counting regions with bounded surface area. Comm. Math. Phys. 273(2), 305–315 (2007)
Bollobás, B., Leader, I.: Edge-isoperimetric inequalities in the grid. Combinatorica 11(4), 299–314 (1991)
Bolthausen, E., Cipriani, A., Kurt, N.: Exponential decay of covariances for the supercritical membrane model. Comm. Math. Phys. 353(3), 1217–1240 (2017)
Boucheron, S., Lugosi, G., Massart, P.: Concentration inequalities A nonasymptotic theory of independence. Oxford University Press (2013)
Bovier, A., Külske, C.: Stability of hierarchical interfaces in a random field model. J. Stat. Phys. 69(1), 79–110 (1992)
Bovier, A., Külske, C.: Hierarchical interfaces in random media II: the Gibbs measures. J. Stat. Phys. 73(1), 253–266 (1993)
Bovier, A., Külske, C.: A rigorous renormalization group method for interfaces in random media. Rev. Math. Phys. 6(3), 413–496 (1994)
Bovier, A., Külske, C.: There are no nice interfaces in (2+1)-dimensional SOS models in random media. J. Stat. Phys. 83(3–4), 751–759 (1996)
Bovier, A., Picco, P.: Stability of interfaces in a random environment. A rigorous renormalization group analysis of a hierarchical model. J. Stat. Phys. 62(1), 177–199 (1991)
Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)
Brascamp, H.J., Lieb, E.H.: Some inequalities for gaussian measures and the long-range order of the one-dimensional plasma. In Inequalities, pages 403–416. Springer, (2002)
Brascamp, H.J., Lieb, E.H., Lebowitz, J.L.: The statistical mechanics of anharmonic lattices. Inst. Internat. Statist., 46(1):393–404 (1976), (1975)
Bricmont, J., Kupiainen, A.: Phase transition in the \(3d\) random field Ising model. Comm. Math. Phys. 116(4), 539–572 (1988)
Caravenna, F., Deuschel, J.-D.: Pinning and wetting transition for (1+ 1)-dimensional fields with laplacian interaction. Ann. Prob. 36(6), 2388–2433 (2008)
Cardy, J.L., Ostlund, S.: Random symmetry-breaking fields and the XY model. Phys. Rev. B 25(11), 6899 (1982)
Chalker, J.: On the lower critical dimensionality of the Ising model in a random field. J. Phys. C: Solid State Phys. 16(34), 6615 (1983)
Chatterjee, S.: On the decay of correlations in the random field Ising model. Comm. Math. Phys. 362(1), 253–267 (2018)
Cipriani, A., Dan, B., Hazra, R.S.: The scaling limit of the membrane model. Ann. Probab. 47(6), 3963–4001 (2019)
Cotar, C., Külske, C.: Existence of random gradient states. Ann. Appl. Probab. 22(4), 1650–1692 (2012)
Cotar, C., Külske, C.: Uniqueness of gradient Gibbs measures with disorder. Probab. Theory Relat. Fields 162(3–4), 587–635 (2015)
Dario, P.: Convergence of the thermodynamic limit for random-field random surfaces. arXiv preprint arXiv:2105.03940, (2021)
Dario, P., Harel, M., Peled, R.: Quantitative disorder effects in low dimensional spin systems. arXiv preprint arXiv:2101.01711, (2021)
Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15(1), 181–232 (1999)
Delmotte, T.: Parabolic harnack inequality and estimates of markov chains on graphs. Revista matemática iberoamericana 15(1), 181–232 (1999)
Deuschel, J.-D., Giacomin, G., Ioffe, D.: Large deviations and concentration properties for \(\nabla \phi \) interface models. Probab. Theory Relat. Fields 117(1), 49–111 (2000)
Ding, J., Wirth, M.: Correlation length of two-dimensional random field Ising model via greedy lattice animal. arXiv preprint arXiv:2011.08768, (2020)
Ding, J., Xia, J.: Exponential decay of correlations in the two-dimensional random field Ising model. Invent. Math., pp. 1–47, (2021)
Durrett, R.: Probability—theory and examples, volume 49 of Cambridge Series in Statistical and Probabilistic Mathematics. 5th edition, Cambridge University Press, Cambridge, (2019)
Evans, L.C.: Partial differential equations, vol 19. American Mathematical Soc., (2010)
Fisher, D.S., Fröhlich, J., Spencer, T.: The Ising model in a random magnetic field. J. Statist. Phys. 34(5–6), 863–870 (1984)
Forgacs, G., Lipowsky, R., Nieuwenhuizen, T.M.: The behavior of interfaces in ordered and disordered systems. Phase Trans. Crit. Phenom. 14, 135–363 (1991)
Fröhlich, J., Spencer, T.: Kosterlitz-Thouless transition in the two-dimensional plane rotator and Coulomb gas. Phys. Rev. Lett. 46(15), 1006–1009 (1981)
Fröhlich, J., Spencer, T.: The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas. Comm. Math. Phys. 81(4), 527–602 (1981)
Funaki, T.: Stochastic interface models. In Lectures on Probability Theory and Statistics, volume 1869 of Lecture Notes in Math., pages 103–274. Springer, Berlin, (2005)
Funaki, T., Spohn, H.: Motion by mean curvature from the Ginzburg-Landau \(\nabla \phi \) interface model. Comm. Math. Phys. 185(1), 1–36 (1997)
Garban, C., Sepúlveda, A.: Statistical reconstruction of the Gaussian free field and KT transition. arXiv preprint arXiv:2002.12284, (2020)
Giacomin, G., Olla, S., Spohn, H.: Equilibrium fluctuations for \(\nabla \phi \) interface model. Ann. Probab. 29(3), 1138–1172 (2001)
Giamarchi, T., Le Doussal, P.: Elastic theory of flux lattices in the presence of weak disorder. Phys. Rev. B 52(2), 1242 (1995)
Hwa, T., Fisher, D.S.: Vortex glass phase and universal susceptibility variations in planar arrays of flux lines. Phys. Rev. Lett. 72(15), 2466 (1994)
Imry, Y., Ma, S.-K.: Random-field instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35(21), 1399 (1975)
Kharash, V., Peled, R.: The Fröhlich-Spencer proof of the Berezinskii-Kosterlitz-Thouless transition. arXiv preprint arXiv:1711.04720, (2017)
Külske, C., Orlandi, E.: A simple fluctuation lower bound for a disordered massless random continuous spin model in \(d=2\). Electron. Comm. Probab. 11, 200–205 (2006)
Külske, C., Orlandi, E.: Continuous interfaces with disorder: even strong pinning is too weak in two dimensions. Stoch. Process. Appl. 118(11), 1973–1981 (2008)
Kurt, N.: Entropic repulsion for a class of Gaussian interface models in high dimensions. Stoch. Process. Appl., 117(1), (2007)
Kurt, N.: Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension. Ann. Prob. 37(2), 687–725 (2009)
Lammers, P.: Height function delocalisation on cubic planar graphs. arXiv preprint arXiv:2012.09687, (2020)
Le Doussal, P., Schehr, G.: Disordered free fermions and the Cardy-Ostlund fixed line at low temperature. Phys. Rev. B 75(18), 184401 (2007)
Lebowitz, J.L., Mazel, A.E.: Improved Peierls argument for high-dimensional Ising models. J. Statist. Phys. 90(3–4), 1051–1059 (1998)
Magazinov, A., Peled, R.: Concentration inequalities for log-concave distributions with applications to random surface fluctuations. arXiv preprint arXiv:2006.05393, (2020)
Miłoś, P., Peled, R.: Delocalization of two-dimensional random surfaces with hard-core constraints. Comm. Math. Phys. 340(1), 1–46 (2015)
Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80(4), 931–954 (1958)
Nattermann, T.: Scaling approach to pinning: charge density waves and giant flux creep in superconductors. Phys. Rev. Lett. 64(20), 2454 (1990)
Orland, H., Shapir, Y.: A disorder-dependent variational method without replicas: application to the random phase sine-Gordon model. EPL (Europhys. Lett.) 30(4), 203 (1995)
Ristivojevic, Z., Le Doussal, P., Wiese, K.J.: Super-rough phase of the random-phase sine-gordon model: two-loop results. Phys. Rev. B 86(5), 054201 (2012)
Ruelle, D.: Statistical mechanics: Rigorous results. World Scientific, (1999)
Sakagawa, H.: Entropic repulsion for a Gaussian lattice field with certain finite range interaction. J. Math. Phys. 44(7), 2939–2951 (2003)
Schweiger, F.: On the membrane model and the discrete Bilaplacian. PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2021)
Sheffield, S.: Random surfaces. Astérisque, (304):vi+175, (2005)
Timár, Á.: Boundary-connectivity via graph theory. Proc. Am. Math. Soc. 141(2), 475–480 (2013)
Toner, J., DiVincenzo, D.: Super-roughening: A new phase transition on the surfaces of crystals with quenched bulk disorder. Phys. Rev. B 41(1), 632 (1990)
van Enter, A.C., Külske, C.: Nonexistence of random gradient Gibbs measures in continuous interface models in \(d=2\). Ann. Appl. Probab. 18(1), 109–119 (2008)
Velenik, Y.: Localization and delocalization of random interfaces. Probab. Surv. 3, 112–169 (2006)
Villain, J., Fernandez, J.F.: Harmonic system in a random field. Zeitschrift für Physik B Condens. Matter 54(2), 139–150 (1984)
Acknowledgements
We are indebted to David Huse for a discussion on the possible behavior of the integer-valued random-field Gaussian free field in dimensions \(d\ge 3\), and to Michael Aizenman, Charles M. Newman, Thomas Spencer and Daniel L. Stein for encouragement and helpful conversations on the topics of this work. We are grateful to Florian Schweiger for clarifying to us the different choices of boundary conditions in the membrane model and for pointing out a mistake in a previous draft of Sect. 7. We thank two anonymous referees for useful and constructive comments which helped us improve the manuscript. We are also grateful to Antonio Auffinger, Wei-Kuo Chen, Izabella Stuhl and Yuri Suhov for the opportunity to present these results in online talks and for useful discussions. The research of the authors was supported in part by Israel Science Foundation grants 861/15 and 1971/19 and by the European Research Council starting grant 678520 (LocalOrder).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A. Nash–Aronson estimates for the Dirichlet problem
Appendix A. Nash–Aronson estimates for the Dirichlet problem
In this section, we prove the Nash–Aronson estimate in finite volume stated in Proposition 3.3. The proof builds upon the infinite-volume result of [43, Appendix B] stated below (which itself builds upon the infinite-volume and continuous estimate of Aronson [6]). We first introduce the infinite-volume heat kernel and state the Nash–Aronson estimate for discrete, time-dependent and uniformly elliptic environment of Giacomin–Olla–Spohn [43].
Definition A.1
Let \(s_0 \in \mathbb {R}\). For each continuous, time-dependent, uniformly elliptic environment \({\textbf{a}}: [s_0 , \infty ) \times E(\mathbb {Z}^d) \rightarrow [c_- , c_+]\), each initial time \(s \in [s_0 , \infty )\), and each point \(y \in \Lambda _L\), we introduce the infinite-volume heat kernel \(P_{{\textbf{a}}, \infty }\) to be the solution of the parabolic equation
The next proposition establishes lower and upper bounds on the map \(P_{{\textbf{a}}, \infty }.\)
Proposition A.2
(Nash–Aronson estimates, Propositions B.3 and B.4 of [43]). There exist constants C, c depending on the dimension d and the ellipticity parameters \(c_-, c_+\) such that, for any pair of times \(s , t \in (s_0 , \infty )\) with \(t \ge s\) and any pair of points \(x , y \in \mathbb {Z}^d\),
Under the additional assumption \(|x - y| \le \sqrt{t - s} \), one has the lower bound
Remark A.3
The article of Giacomin–Olla–Spohn [43] only establishes the lower bound of the Nash–Aronson estimate in the on-diagonal case (i.e., under the assumption \(|x - y| \le \sqrt{t - s}\)). While it would be possible to obtain off-diagonal lower bounds (in the case \(|x - y| \ge \sqrt{t - s}\)) by adapting the techniques of [6, 30] (written in either the continuous setting or the discrete setting with a static environment), they are not necessary in the article [43] or in the proof of Theorem 2.
Remark A.4
Under the assumption \(|x - y| \le \sqrt{t - s}\), the ratio \(|x - y| / (1 \vee \sqrt{t- s})\) is smaller than 1 and the right-hand sides of (A.1) and (A.2) are of comparable sizes. The estimates are thus sharp up to multiplicative constants.
The proof of Proposition A.2 given below relies on analytic arguments. We mention that a more probabilistic approach, relying on the introduction of the random-walk whose generator is the operator \(- \nabla \cdot {\textbf{a}}\nabla \) (following [43, Section 3.2]) and on stopping time arguments, would yield the same result.
Proof of Proposition 3.3
First let us note that by the change of variable \((t - s) \rightarrow (t-s)/c_-\), it is sufficient to prove the result when \(c_- = 1\). Let us fix an integer \(L\ge 0\), let \( c_+ \in [1 , \infty )\) be an ellipticity constant, and let \(s_0\in \mathbb {R}\). Let \({\textbf{a}}: [s_0 , \infty ) \times E(\Lambda _L^+) \rightarrow [1 , c_+]\) be a continuous time-dependent (uniformly elliptic) environment. For any \(s \in [s_0 , \infty )\) and any vertex \(y \in \Lambda _L\), we denote by \(P_{\textbf{a}}(\cdot , \cdot ; s, x)\) the solution of the parabolic Eq. (3.3). We extend the environment \({\textbf{a}}\) to the space \([s_0 , \infty ) \times E(\mathbb {Z}^d)\) by setting \({\textbf{a}}(t , e) = c_+\) for any pair \((t , e) \in [s_0 , \infty ) \times \left( E(\mathbb {Z}^d) \setminus E\left( \Lambda _L^+ \right) \right) \), and let \(P_{{\textbf{a}}, \infty }\) be the infinite volume heat kernel associated with the extended environment \({\textbf{a}}\) as defined in Definition A.1. We prove the upper and lower bounds of Proposition 3.3 separately. We will make use of the notation
as well as, for any directed edge \(e = (x , z) \in \textbf{E} \left( \Lambda _L^+ \right) \),
and, following the conventions of Section 2.2,
Proof of the upper bound. We first note that the estimate (3.5) is equivalent to the two following inequalities: There exist constants c, C depending on \(d , c_+ \) such that
We first treat the case \((t - s) \le L^2\). By the maximum principle for the parabolic operator \(\partial _t - \nabla \cdot {\textbf{a}}\nabla \), one has the estimate, for any \(t, s \in [ s_0 , \infty )\) with \(t \ge s\) and any \(x , y \in \Lambda _L\),
Combining the inequality (A.4) with Proposition A.2 yields the upper bound, for any \(s , t \in (s_0 , \infty )\) with \(t \ge s\) and any \(x , y \in \mathbb {Z}^d\),
This is (A.3) in the case \(t \le L^2\). We now focus on the case \(t \ge L^2\). To this end, we denote by \(C_{\textrm{Poinc}}\) the constant which appears in the Poincaré inequality, that is, the smallest constant which satisfies \(\left\| u \right\| _{L^2 \left( \Lambda _\ell \right) }^2 \le C_\textrm{Poinc} \ell ^2 \left\| \nabla u \right\| _{L^2 \left( \Lambda _\ell ^+\right) }^2\) for any side length \(\ell \in \mathbb {N}\) and any function \(u : \Lambda _\ell ^+ \rightarrow \mathbb {R}\) normalized to be 0 on the boundary \(\partial \Lambda _\ell \). We then let \(c_1 := 1/C_\textrm{Poinc}\), and note that this constant depends only on the parameter d.
Using that the heat kernel \(P_{\textbf{a}}\) solves the parabolic Eq. (3.3) and the discrete integration by parts (2.6), we have
Using the lower bound \({\textbf{a}}\ge 1\) on the environment, and the Poincaré inequality in the box \(\Lambda _L\) (which can be applied since the mapping \(x \mapsto P_{\textbf{a}}(t , x ; s , y )\) is equal to 0 on \(\partial \Lambda _L\)), we obtain
A combination of (A.6) and (A.7) with the definition of the constant \(c_1\) above yields
By the Nash inequality (see [57]) and the non-negativity of the map \(P_{\textbf{a}}\), there exists a constant \(C_{\textrm{Nash}}\) depending only on the dimension d such that
Applying the inequality (3.4) obtained in Section 3.3, we may simplify the inequality (A.8) and write
Combining (A.7) and (A.9) yields
Integrating the differential inequality (A.10) and using the identity \(\left\| P_{\textbf{a}}\left( s , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L \right) } = 1\), we obtain that there exists a constant C depending on d and \(c_-\) such that, for any \(t \ge s\),
We now show that the estimate (A.11) implies the inequality (A.3) in the case \((t -s) \ge L^2\). Using the convolution property for the heat kernel \(P_{\textbf{a}}\), we have the identity, for any \(t, s\in [s + L^2 , \infty )\) and any \(x, y \in \Lambda _L\),
The Cauchy-Schwarz inequality then yields
The first term of the right-hand side of (A.12) can be estimated by the upper bound (A.11) applied with the initial time \(\frac{s + t}{2}\) instead of s. We obtain
To estimate the second term in the right-hand side, we use the estimate (A.5), the assumption \(t - s \ge L^2\), the observations that, for any pair of points \(x , z \in \Lambda _L\), \(|x - z| \le C L\), and that the cardinality of the box \(\Lambda _L\) is equal to \((2L+1)^d\). We obtain
A combination of (A.12), (A.13) and (A.14) completes the proof of (A.3) in the case \(t \ge L^2\).
Proof of the lower bound. We first claim that there exists a constant \(c_0 \in (0 , 1)\) depending on the parameters \(d , c_+\) such that, for any \(t , s \in [ s_0 , \infty )\) satisfying \(\sqrt{t - s} \le c_0 L\) and any \(y \in \Lambda _{L/2}\),
The proof of this inequality relies on Proposition A.2. First by the lower bound (A.2), we have the estimate, for any \(t, s \in [s_0 , \infty )\) such that \(t - s \le L^2\),
Let us fix a constant \(c_1 \in (0 , 1)\). Using that for any point \(x \in \partial \Lambda _L\) and any point \(y \in \Lambda _{L/2}\), we have \(|x - y| \ge L/2\) together with the upper bound (A.1), we obtain the estimate, for any \(L \ge c_1^{-1}\) and any \(t, s \in [s_0 , \infty )\) satisfying \(\sqrt{t - s} \le c_1 L\),
Using that the mapping \(t' \mapsto t'^{-d/2} \exp \left( - c/ (2 \sqrt{t'}) \right) \) tends to 0 as \(t'\) tends to 0, we may select a constant \(c_0 \in (0 , 1]\) such that
where the constants c, C are the ones which appear in the right-hand sides of (A.16) and (A.17). Let us note that, since the constants c, C depend only on the dimension d and the ellipticity constant \(c_+\), the constant \(c_0\) may be chosen so that it depends only on \(d , c_+\). Multiplying both sides of the inequality (A.18) by \(L^{-d}\) yields, for any \(t, s \in [s_0 , \infty )\) such that \(\sqrt{t - s} \le c_0 L\),
The proof of (A.15) is complete.
We now deduce the lower bound (3.6) from the inequality (A.15). To this end, let us fix a time \(t \in (s , s + c_0 L^2)\), set \(\varepsilon := \frac{1}{2} \inf _{x \in y + \Lambda _{\sqrt{t-s}}} P_{{\textbf{a}}, \infty } \left( t , x ; s , y \right) \), and define the map \(P_{{\textbf{a}}, \infty }^\varepsilon := P_{{\textbf{a}}, \infty } - \varepsilon \). Let us note that the mapping \(P_{{\textbf{a}}, \infty }^\varepsilon \) solves the parabolic equation
By the definition of the parameter \(\varepsilon \) and the inequality (A.15), the map \(P_{{\textbf{a}}, \infty }^\varepsilon \) satisfies the boundary estimates
Applying the maximum principle for the parabolic operator \(\partial _t - \nabla \cdot {\textbf{a}}\nabla \), we obtain the inequality, for any \((t', x) \in [s , t] \times \Lambda _L\),
Applying the estimate (A.19) at time \(t' = t\) and using the definition of the parameter \(\varepsilon \) yields, for any vertex x satisfying \(|x - y| \le \sqrt{t - s}\),
Combining the estimate (A.20) with the lower bound of Proposition A.2 implies
The proof of the lower bound (3.6) is complete. \(\square \)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Dario, P., Harel, M. & Peled, R. Random-field random surfaces. Probab. Theory Relat. Fields 186, 91–158 (2023). https://doi.org/10.1007/s00440-022-01179-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-022-01179-0