Tightness of the Ising–Kac Model on the Two-Dimensional Torus

Abstract

We consider the sequence of Gibbs measures of Ising models with Kac interaction defined on a periodic two-dimensional discrete torus near criticality. Using the convergence of the Glauber dynamic proven by Mourrat and Weber (Commun Pure Appl Math 70:717–812, 2017) and a method by Tsatsoulis and Weber employed in (arXiv:1609.08447 2016), we show tightness for the sequence of Gibbs measures of the Ising–Kac model near criticality and characterise the law of the limit as the $\Phi ^4_2$ measure on the torus. Our result is very similar to the one obtained by Cassandro et al. (J Stat Phys 78(3):1131–1138, 1995) on $\mathbb {Z}^2$, but our strategy takes advantage of the dynamic, instead of correlation inequalities. In particular, our result covers the whole critical regime and does not require the large temperature/large mass/small coupling assumption present in earlier results.

1 Introduction

Let $N>0$ be a positive integer and consider the (periodic) lattice $\Lambda _N= \{1-N,\dots ,N\}^2$. For $\gamma > 0$, let $\mathfrak {K}: \mathbb {R}^2 \rightarrow [0,1]$ be a twice differentiable, non negative, isotropic function supported on a ball of radius 3 and define $\kappa _\gamma$ for $x \in \Lambda _N$

$$\begin{aligned} \kappa _\gamma (x) \simeq \gamma ^2 \mathfrak {K}(\gamma |x|)\ ,\qquad \sum _{z \in \Lambda _N\setminus \{0\}} \kappa _\gamma (z) = 1. \end{aligned}$$

(1.1)

Consider a spin system formed by a set of spins parametrized by the lattice $\Lambda _N$. Each spin can assume the value $+\,1$ or $-\,1$ representing two possible states of the magnetization and we will denote with $\Sigma _N=\{-\,1,1\}^{\Lambda _N}$ the set of all possible configurations. The Ising–Kac model on the two-dimensional lattice with periodic boundary condition and external magnetization $b \in \mathbb {R}$ is given by the following Hamiltonian

$$\begin{aligned} \mathcal {H}^{\gamma }_{\beta ,b} (\sigma ) = \frac{\beta }{2} \sum _{x, y \in \Lambda _N} \kappa _\gamma (x-y) \sigma _x \sigma _y + b \sum _{x \in \Lambda _N} \sigma _x\, \qquad \sigma \in \Sigma _N. \end{aligned}$$

(1.2)

The Gibbs measure over $\Sigma _N$ associated to the potential (1.1), with inverse temperature $\beta$ and external magnetic field b, is given by

$$\begin{aligned} \mathbf {P}^{\gamma }_{\beta ,b} (\sigma ){\mathop {=}\limits ^{\tiny \text{ def }}}\left( Z_{\gamma ,\beta ,b}^N\right) ^{-1} \exp \left( \mathcal {H}^{\gamma }_{\beta ,b} (\sigma )\right) , \end{aligned}$$

(1.3)

where $Z_{\gamma ,\beta ,b}^N$ is the partition function that makes (1.3) a probability measure. We will also denote with $\mathbf {E}^{\gamma }_{\beta ,b}$ the expectation under $\mathbf {P}^{\gamma }_{\beta ,b}$. For technical reason, we set $\kappa _\gamma (0) = 0$ and we remark that its precise value doesn’t affect (1.3).

As in [12], we will let the inverse temperature $\beta$ converge in a precise way as $\gamma \rightarrow 0$ to $\beta _c=1$ the critical value of the mean-field system. The purpose of this paper is to prove the tightness of the magnetisation fluctuation field

$$\begin{aligned} \gamma ^{-1} \left( \sigma _{\lfloor N \cdot \rfloor } - \mathbf {E}^{\gamma }_{\beta ,b} \left[ \sigma _{\lfloor N \cdot \rfloor }\right] \right) \end{aligned}$$

(1.4)

in the strong topology of $\mathcal {S}'(\mathbb {T}^2)$ as both $N \rightarrow \infty$ and $\gamma \rightarrow 0$ in a precise way that we shall describe later.

Moreover, in case $b=0$, we are also able to characterise the limit as the $\Phi ^4(\mathbb {T}^2)$ measure, formally described by

$$\begin{aligned} \mathcal {Z}^{-1}\exp \left( -\int _{\mathbb {T}^2} \frac{1}{2} |\nabla \Phi (x)|^2 + \frac{1}{12}\Phi ^{:4:}(x) - \frac{A}{2}\Phi ^{:2:}(x)\ dx\right) d \Phi , \end{aligned}$$

(1.5)

where $\Phi ^{:4:}$ denotes the Wick renormalisation of the fourth power of the field. For a more detailed and formal definition, see for example [6, 14].

The Ising–Kac model is a mean-field model with ferromagnetic long range potential that has been introduced in statistical mechanics for its simplicity and because it provides a framework to recover rigorously the van der Waals theory [9] of phase transition. It has been then developed by Lebowitz and Penrose in [11], see also [15] for more details. The model has already been useful to study the $\Phi ^4_d$ theory, see [7], where a renormalisation group approach has been used to approximate $\Phi ^4_d$ with generalised Ising models, and [16] with classical Ising spins.

The present work is mainly built upon the article of Weber and Mourrat [12], where the Glauber dynamic on a periodic two-dimensional lattice is shown to converge to the solution of the two-dimensional stochastic Allen–Cahn equation on the torus. Their approach however doesn’t imply the tightness for the invariant measure of the model, which is treated in this article. The same result about convergence of Glauber dynamics in one space dimension had previously been proven in [2, 5], via a coupling with a simpler model, the voter model. In a subsequent paper [18], Tsatsoulis and Weber show the exponential convergence to equilibrium for the dynamical $\Phi ^4_2$ model. In [3], the authors show the convergence of the 2d Ising–Kac model on $\mathbb {Z}^2$ to $\Phi ^4_2$ by proving the convergence of the discrete Schwinger functions. In particular they were the first to explain, to the best of our knowledge, the small shift of the critical temperature for the Ising–Kac model with the renormalisation constants of the Wick powers. That result (see [3, Theorem 2]) is however restricted to temperatures satisfying a condition allowing to use Aizenman’s correlation inequalities, which corresponds to large negative values of A in (1.5).

Our main result resembles the one obtained in [3], with some differences. We will work on a periodic lattice instead of $\mathbb {Z}^2$, which we think of as a discretisation of a 2D torus. This restriction is mainly due to our techniques for bounding the solutions globally in time and a posteriori doesn’t appear to be strictly necessary since the limiting dynamic can be defined also on the whole 2D plane (see [13]). Moreover, as our main proof exploits the dynamical version of the model and not the correlation inequalities, we do not have the restriction on the temperature present in [3, Theorem 2], so that we cover arbitrary values $A \in \mathbb {R}$ in (1.5). A correlation inequality, the GHS inequality, is then employed in a subsequent corollary to partially extend the result to the case of arbitrary external magnetization b. Corollary 2.3 is the only place where we use a correlation inequality.

As remarked by one of the referees, one advantage of the strategy described in the present article is that it is robust enough to also imply tightness of the Gibbs measure of the Kac-Blume-Capel, using the result about the respective Glauber dynamic provided in [17]. In order to be able to identify the limiting law with the $\Phi ^6_2$ random field, thus generalising Theorem 2.4, one then also needs a suitable convergence to equilibrium result for the limiting $\Phi ^6_2$ dynamical model. While the result of [18] does not seem to generalised immediately (see Remark 6.2 there), this can be obtained by combining the strong Feller property obtained in [8, Sect. 5.3] with the fact that one has good control over the return times to bounded/compact sets, as well as the fact that the $\Phi ^6_2$ measure has full support, being equivalent to the free field.

The structure of the present article is as follows: our main result is Theorem 2.1 showing tightness of the fluctuations of local averages of the magnetic field in a distributional space. The proof of the theorem is based on the analysis of the dynamical $\Phi ^4_2$ model in [18, Sect. 3] and makes no use of correlation inequalities (not explicitly at least), avoids the restriction of [3, Eq. (1.8)] and exploits the regularisation provided by the time evolution of the Glauber dynamic. As a consequence of Theorem 2.1, we obtain in Corollaries 2.2 and 2.3 tightness in $\mathcal {S}'(\mathbb {T}^2)$ for the fluctuation fields (1.4).

In Theorem 2.4 we characterise the limit of each subsequence to be an invariant measure for the dynamical $\Phi ^4_2$ model constructed in [4]. Since it was shown in [4] that (1.5) is such a measure and in [18] that this invariant measure is unique, the result follows. For the proof, we make use of the uniform convergence to the invariant measure and the convergence of the Glauber dynamic in [12].

1.1 Notations

We shall consider spins arranged on a periodic lattice that we will think as embedded into a two-dimensional torus $\mathbb {T}^2 = [-\,1,1]^2$. Let $\varepsilon = N^{-\,1}$ and $\Lambda _\varepsilon {\mathop {=}\limits ^{\tiny \text{ def }}}\varepsilon \Lambda _N \subset \mathbb {T}^2$ the discretisation induced on $\mathbb {T}^2$. For $f,g:\Lambda _\varepsilon \rightarrow \mathbb {C}$ we define

$$\begin{aligned} \left\| f \right\| _{L^p(\Lambda _\varepsilon )}^p {\mathop {=}\limits ^{\tiny \text{ def }}}\sum _{x \in \Lambda _\varepsilon } \varepsilon ^2 |f(x)|^p,\qquad \left\langle f,g \right\rangle _{\Lambda _\varepsilon } {\mathop {=}\limits ^{\tiny \text{ def }}}\sum _{x \in \Lambda _\varepsilon } \varepsilon ^2 f(x) \overline{ g(x)}, \end{aligned}$$

respectively the discrete $L^p$ norm and the scalar product. We will use the discrete convolution

$$\begin{aligned} (f *g)(x){\mathop {=}\limits ^{\tiny \text{ def }}}\sum _{y \in \Lambda _\varepsilon }\varepsilon ^2 f(x-y)g(y),\qquad \text { for } x \in \Lambda _\varepsilon , \end{aligned}$$

and, when there is no possibility of confusion, we will drop the set $\Lambda _\varepsilon$ from the above definitions. We will make an extensive use of the Fourier transform

$$\begin{aligned} \widehat{f}(w) {\mathop {=}\limits ^{\tiny \text{ def }}}\left\langle f, e_w \right\rangle _{\Lambda _\varepsilon },\qquad e_w(x) = e^{i \pi w \cdot x}\qquad \text { for } w \in \Lambda _N. \end{aligned}$$

It will sometimes be convenient to also set $e_w = 0$ for $w \in \mathbb {Z}^2{\setminus } \Lambda _N$. With this notation, the Fourier inversion formula reads

$$\begin{aligned} f(x) = \frac{1}{4} \sum _{w \in \mathbb {Z}^2} \widehat{f}(w) e_w(x) \qquad \text { for } x \in \Lambda _\varepsilon . \end{aligned}$$

(1.6)

We shall use the same notation $\mathsf {Ext}(f)$ as in [12] to denote the extension of f to the continuous torus $\mathbb {T}^2$ via (1.6) applied to $x \in \mathbb {T}^2$. We recall furthermore the fact that the operator $\mathsf {Ext}$ doesn’t commute with the operation of taking the product. (Of course we could have used extensions that do commute, but $\mathsf {Ext}$ behaves nicely with respect to the scale of Besov spaces.) We will measure the regularity of a function $g:\mathbb {T}^2 \rightarrow \mathbb {R}$ (or $g:\Lambda _\varepsilon \rightarrow \mathbb {R}$) with the Besov norm, defined for $\nu \in \mathbb {R}$, and $p,q \in [1,\infty ]$ as

$$\begin{aligned} \left\| g \right\| _{\mathcal {B}^{\nu }_{p,q}} = {\left\{ \begin{array}{ll} \left( \sum _{k \ge -1} 2^{\nu k q} \left\| \delta _k g \right\| _{L^p(\mathbb {T}^2)}^q \right) ^{\frac{1}{q}} &{} \text { if }\,\,q < \infty \\ \sup _{k \ge -1} 2^{\nu k} \left\| \delta _k g \right\| _{L^p(\mathbb {T}^2)}&{} \text { if }\,\,q = \infty \end{array}\right. } \end{aligned}$$

(1.7)

[see (4.1) below for the definition of the Paley–Littlewood projection $\delta _k$] and we will denote by $\mathcal {B}^{\nu }_{p,q}$ the completion of the set of smooth test functions over the torus equipped with the corresponding Besov norm. We shall denote by $\mathcal {C}^{\nu }$ the (separable) Besov space $\mathcal {B}^{\nu }_{\infty ,\infty }$. In particular, the parameter $\nu \in \mathbb {R}$ represents the regularity of a function and the space $\mathcal {B}^{\nu }_{p,q}$ contains distributions if $\nu < 0$. It will be useful to consider, for $g : \Lambda _\varepsilon \rightarrow \mathbb {R}$, a discrete version of the Besov norm, that we shall denote by $\left\| g \right\| _{\mathcal {B}^{\nu }_{p,q}(\Lambda _\varepsilon )}$ (resp. $\left\| g \right\| _{\mathcal {C}^{\nu }(\Lambda _\varepsilon )}$),

$$\begin{aligned} \left\| f \right\| _{\mathcal {B}^{\nu }_{p,q}(\Lambda _\varepsilon ^d)} {\mathop {=}\limits ^{\tiny \text{ def }}}{\left\{ \begin{array}{ll} \left( \sum _{k \ge -1} 2^{\nu k q} \left\| \delta _k \mathsf {Ext}(f) \right\| _{L^p(\Lambda _\varepsilon ^d)}^q \right) ^{\frac{1}{q}} &{} \text { if }\,\,q < \infty \\ \sup _{k \ge -1} 2^{\nu k} \left\| \delta _k \mathsf {Ext}(f) \right\| _{L^p(\Lambda _\varepsilon ^d)}&{} \text { if }\,\,q = \infty \end{array}\right. } \end{aligned}$$

see Sect. 4 for a more precise description and for some useful properties of this norm used in the article.

2 Definitions and Statements of the Theorem

Assume for the moment that $b=0$, which is also the case studied in [12] and consider for $x \in \Lambda _N$

$$\begin{aligned} h_{\gamma }(x) {\mathop {=}\limits ^{\tiny \text{ def }}}\sum _{z \in \Lambda _N} \kappa _\gamma (x-z) \sigma _z, \end{aligned}$$

(2.1)

where the kernel is the same as in (1.2).

Following [2, 12], we define the magnetisation fluctuation field over the lattice $\Lambda _\varepsilon$ as $X_{\gamma }(z) = \gamma ^{-1} h_{\gamma }(\varepsilon ^{-1}z)$. We will consider a dynamic of Glauber type on $\Sigma _N$ in order to gain insight into the properties of the fluctuations. In order for this dynamic to converge to a non-trivial limit, we will enforce the relation between the scalings $\varepsilon$ and $\gamma$ given by (3.10).

The dynamic can be described informally as follows. Each site $x \in \Lambda _N$ is assigned an independent exponential clock with rate 1. When the clock rings, the corresponding spin changes sign with probability

$$\begin{aligned} c_{\gamma }(z,\sigma ) = \frac{1}{2}\left( 1 - \sigma _z \tanh \left( \beta h_{\gamma }(z)\right) \right) , \end{aligned}$$

(2.2)

and remains unchanged otherwise. More formally, the generator of this dynamic is given by

$$\begin{aligned} \mathscr {L}_{\gamma } f(\sigma ) = \sum _{z \in \Lambda _N} c_{\gamma }(z,\sigma )\left( f(\sigma ^{\{z\}})-f(\sigma ) \right) , \end{aligned}$$

(2.3)

for $f: \Sigma _N\rightarrow \mathbb {R}$, where

$$\begin{aligned} \sigma ^{\{z\}}_y = {\left\{ \begin{array}{ll} -\sigma _z &{} \text { if } y = z,\\ \sigma _y &{} \text { if } y \ne z. \end{array}\right. } \end{aligned}$$

The probabilities $c_{\gamma }(z,\sigma )$ are chosen precisely in such a way that $\mathbf {P}^{\gamma }_{\beta ,0}$ is invariant for this Markov process. We shall use the notations $\sigma _x(s)$ and $h_{\gamma }(s,x)$ to refer to the process at (microscopic) space $x \in \Lambda _N$ and time $s \in \mathbb {R}_+$. We will use the notation $\mathbb {P}_{\beta ,0}^{\gamma }$ (resp. $\mathbb {E}_{\beta ,0}^{\gamma }$) to refer to the probability (resp. expectation) of the process started with an initial condition drawn from $\mathbf {P}^{\gamma }_{\beta ,0}$.

In order to rewrite the process in macroscopic coordinates, we speed up the generator $\mathscr {L}_{\gamma }$ by a factor $\alpha ^{-1}$ and we will abuse the notation writing

$$\begin{aligned} X_{\gamma }(s,x) {\mathop {=}\limits ^{\tiny \text{ def }}}\delta ^{-1}h_{\gamma }(\alpha ^{-1}s,\varepsilon ^{-1}x), \end{aligned}$$

(2.4)

in (macroscopic) space $x \in \Lambda _\varepsilon$ and time $s \in \mathbb {R}_+$. In [12, Theorem 3.2] it is proven that, if the parameters $\delta , \alpha , \varepsilon$ and the inverse temperature $\beta$ are chosen such that

$$\begin{aligned} \delta = \gamma , \qquad \alpha = \gamma ^2 , \qquad \varepsilon = \gamma ^2 , \quad \beta - 1 = \alpha \left( \mathfrak {c}_\gamma + A\right) , \end{aligned}$$

(2.5)

where $\mathfrak {c}_\gamma$ is described in (2.8) below, and if the sequence of initial conditions satisfies $X_{\gamma }(0) \rightarrow X_\gamma ^0$ in $\mathcal {C}^{-\nu }$, then the law of $X_{\gamma }$ on $\mathcal {D}(\mathbb {R}_+,\mathcal {C}^{-\nu })$, converges in distribution to the solution of the stochastic quantisation equation

$$\begin{aligned} \partial _t X = \Delta X - \frac{1}{3} X^{:3:} + A X + \sqrt{2} \xi ,\qquad X(0,\cdot ) = X^0 \in \mathcal {C}^{-\nu } \end{aligned}$$

(2.6)

where $X_{\gamma }(\cdot ,0) \rightarrow X_0$ in $\mathcal {C}^{-\nu }$ and $\xi$ denotes space-time white noise. The expression $X^{:3:}$ stands for a renormalised power defined as in [4], where the relevant notion of “solution” to (2.6) is also given. The solution theory of (2.6) will be briefly summarised in Sect. 2.1. The use of the renormalised powers is necessary since the solution belongs to a distributional space.

For $x \in \Lambda _\varepsilon$, let $K_\gamma (x) = \varepsilon ^{-2} \kappa _\gamma (\varepsilon ^{-1} x)$, the macroscopic version of the kernel $K_\gamma$, and define the discrete Laplacian $\Delta _{\gamma }f = \varepsilon ^{-2}\gamma ^2 (K_\gamma *f - f)$. Under the Glauber dynamic, the process $X_{\gamma }$ satisfies on $[0,T] \times \Lambda _\varepsilon$

$$\begin{aligned} X_{\gamma }(t,x)&= X_{\gamma }(0,x) \nonumber \\&\quad + \int _0^t \Delta _{\gamma }X_{\gamma }(s,x) - \frac{1}{3} \left( X_{\gamma }^3(s,x) -\mathfrak {c}_\gamma X_{\gamma }(s,x) \right) \nonumber \\&\quad + \, A X_{\gamma }(s,x) + \mathcal {O}(\gamma ^2 X_{\gamma }^5(s,x))\ ds + M_\gamma (t,x) \end{aligned}$$

(2.7)

where $M_\gamma (t,x)$ is a martingale and $\mathfrak {c}_\gamma$ is the logarithmically diverging constant

$$\begin{aligned} \mathfrak {c}_\gamma = \frac{1}{4}\sum _{\omega \in \Lambda _N \setminus \{0\}} \frac{|\hat{K_\gamma }(\omega )|^2}{\varepsilon ^{-2}\gamma ^2(1 - \hat{K_\gamma }(\omega ))}. \end{aligned}$$

(2.8)

The next theorem is the main result of the paper. Recall the definition of the Besov norm given in Sect. 1.1. We think of $X_{\gamma }(z)$ as being a random function on $\mathbb {T}^2$, having been extended to $\mathbb {T}^2$ with the $\mathsf {Ext}$ operator.

Theorem 2.1

Assume $b=0$. Then for all positive $\nu > 0$ and for all $q > 0$

$$\begin{aligned} \limsup _{\gamma \rightarrow 0} \mathbf {E}^{\gamma }_{\beta ,0} \left[ \left\| X_{\gamma } \right\| _{\mathcal {C}^{-\nu }}^q \right] < \infty . \end{aligned}$$

In particular, the laws of $X_{\gamma }$ form a tight set of probability measures on $\mathcal {C}^{-\nu }$.

From the above theorem it is possible to deduce

Corollary 2.2

Assume $b=0$. Then the law of the field $\left( \gamma ^{-1}\sigma _{\lfloor \varepsilon ^{-1} x\rfloor }\right) _{x \in \mathbb {T}^2}$ is tight in $\mathcal {S}'(\mathbb {T}^2)$ under $\mathbf {P}^{\gamma }_{\beta ,0}$.

Proof

We will actually prove the tightness in the stronger norm of $H^{-k}(\mathbb {T}^2)$, for k sufficiently big. Let $\varphi \in \mathcal {S}(\mathbb {T}^2)$ and consider

$$\begin{aligned} \left\langle \gamma ^{-1}\sigma _{\lfloor \varepsilon ^{-1} \cdot \rfloor }, \varphi \right\rangle _{\mathbb {T}^2} = \sum _{x \in \Lambda _N} \varepsilon ^2\left( \gamma ^{-1}\sigma _{x} \right) \bar{\varphi }(\varepsilon x) \end{aligned}$$

where $\bar{\varphi }(\varepsilon x) = \epsilon ^{-2} \int _{|y|_{\infty }\le 2^{-1}\varepsilon } \varphi (\varepsilon x+y)\ dy$. Using the differentialbility of $\varphi$, we replace $\bar{\varphi }$ with $\kappa _\gamma *\bar{\varphi }$ at the cost of

$$\begin{aligned} \varepsilon ^2 \gamma ^{-2} \sup _{i_1,i_2\in \{1,2\}}\left\| \partial _{i_1}\partial _{i_2}\varphi \right\| _{L^{\infty }(\mathbb {T}^2)}, \end{aligned}$$

and this is $\mathcal {O}(\gamma ^2)$, as $\gamma \rightarrow 0$, if k is sufficiently big. Therefore (recall the form of the extension (1.6) of $X_{\gamma }$ to the continuous torus)

$$\begin{aligned} \left\langle \gamma ^{-1}\sigma _{\lfloor \varepsilon ^{-1} \cdot \rfloor }, \varphi \right\rangle _{\mathbb {T}^2} = \left\langle X_{\gamma }, \varphi \right\rangle _{\mathbb {T}^2} + \mathcal {O}(\gamma ) \end{aligned}$$

the corollary follows from Theorem 2.1. $\square$

As remarked in the proof, the topology with respect to which the convergence in Corollary 2.2 is proved is not the optimal norm. Indeed we expect the result to hold also with respect to the norm of $\mathcal {C}^{-\nu }$. In the proof of Corollary 2.2 we didn’t only show the tightness of the sequence of random variable, but we also proved that the limit of $\left\langle \gamma ^{-1}\sigma _{\lfloor \varepsilon ^{-1} \cdot \rfloor }, \varphi \right\rangle _{\mathbb {T}^2}$ coincide with $\lim _{\gamma \rightarrow 0}\left\langle X_{\gamma }, \varphi \right\rangle _{\mathbb {T}^2}$ for all $\varphi$ sufficiently smooth.

We now show how to extend the previous result to the case $b \ne 0$. It is clear that, by symmetry it is sufficient to assume $b \ge 0$. In the case of ferromagnetic pair potential $\kappa _\gamma \ge 0$ with positive external magnetisation $b \ge 0$, one has

$$\begin{aligned} \mathbf {E}^{\gamma }_{\beta ,b} \left[ \sigma _x ; \sigma _y \right] \le \mathbf {E}^{\gamma }_{\beta ,0} \left[ \sigma _x ; \sigma _y \right] \end{aligned}$$

(2.9)

where $\mathbf {E}^{\gamma }_{\beta ,b} \left[ \sigma _x ; \sigma _y \right]$ is the covariance between the spins. This follows from the fact that $\frac{d}{db}\mathbf {E}^{\gamma }_{\beta ,b} \left[ \sigma _x ; \sigma _y \right] \le 0$, which is an immediate consequence of the GHS inequality (see for instance [10] for a proof), valid for $\kappa _\gamma \ge 0$ and $b \ge 0$.

Corollary 2.3

Consider any map $\gamma \mapsto b_\gamma \ge 0$ and denote by $m_{\gamma }(b) = \mathbf {E}^{\gamma }_{\beta ,b} [\sigma _x]$ the mean of the spin $\sigma _x$, which is independent of $x \in \Lambda _N$. Then the law of the field

$$\begin{aligned} \tilde{X}_\gamma (x) = \gamma ^{-1}\bigl (\sigma _{\lfloor \varepsilon ^{-1} x\rfloor } - m_{\gamma }(b_\gamma )\bigr ),\qquad x \in \mathbb {T}^2, \end{aligned}$$

(2.10)

is tight in $\mathcal {S}'(\mathbb {T}^2)$ under $\mathbf {P}^{\gamma }_{\beta ,b_\gamma }$.

Proof

Fixing a test function $\phi$ and replacing $\bar{\varphi }$ with $\kappa _\gamma *\bar{\varphi }$ as in Corollary 2.2, we have

$$\begin{aligned} \left\langle \tilde{X}_\gamma , \varphi \right\rangle = \sum _{x \in \Lambda _N} \varepsilon ^2\gamma ^{-1}\big (\sigma _{x}-m_{\gamma }(b_\gamma ) \big )(\kappa _\gamma *\bar{\varphi })(\varepsilon x) + \mathsf {Err}, \end{aligned}$$

where $\mathsf {Err}$ converges to 0 in probability as $\gamma \rightarrow 0$. Decompose $\varphi = \varphi ^+ - \varphi ^-$ into its positive and negative part. For each of them, using the correlation inequality (2.9), we have that

$$\begin{aligned} \mathbf {E}^{\gamma }_{\beta ,b_\gamma } \Bigg |\varepsilon ^2 \sum _{x \in \Lambda _N} \frac{\sigma _{x}-m_{\gamma }(b_\gamma ) }{\gamma }(\kappa _\gamma *\overline{\varphi }^{\pm })(\varepsilon x) \Bigg |^2 \le \mathbf {E}^{\gamma }_{\beta ,0} \Bigg | \varepsilon ^2\sum _{x \in \Lambda _N} (\gamma ^{-1} \sigma _{x})\ (\kappa _\gamma *\overline{\varphi }^{\pm })(\varepsilon x) \Bigg |^2. \end{aligned}$$

Using Theorem 2.1 we see that, for $\nu \in (0,1)$, this quantity is bounded uniformly by a fixed multiple of $\left\| \varphi ^{\pm } \right\| _{\mathcal {B}^{\nu }_{1,1}}^2\mathbf {E}^{\gamma }_{\beta ,0} \left[ \left\| X_{\gamma } \right\| _{\mathcal {C}^{-\nu }}^2 \right]$, up to an error of order $\mathcal {O}(\gamma )$. In order to conclude, we observe that

$$\begin{aligned} \left\| \varphi ^{\pm } \right\| _{\mathcal {B}^{\nu }_{1,1}} \lesssim \left\| \varphi ^{\pm } \right\| _{L^1} + \left\| \varphi ^{\pm } \right\| _{\text {Lip}}^{\nu }\left\| \varphi ^{\pm } \right\| _{L^1}^{1-\nu } \lesssim \left\| \varphi \right\| _{L^1} + \left\| \nabla \varphi \right\| _{L^{\infty }} \end{aligned}$$

where the first inequality is (4.6), generalised to Lipschitz functions. $\square$

The next theorem shows that in the symmetric case $b=0$, the limit of these measures is given by the $\Phi ^4_2$ measures, as already suggested in [12].

Theorem 2.4

Assume $b=0$. Then any limiting law of the sequence $\{X_{\gamma }\}_{\gamma }$ is invariant for the dynamic (2.6), and hence, by [18] and [4, Remark 4.3], coincides with the $\Phi ^4(\mathbb {T}^2)$ measure.

Proof

In order to compare the law of a (discrete) random field $X_{\gamma }$ with fields on the torus $\mathbb {T}^2$, we will use the extension operator $\mathsf {Ext}$ defined after (1.6). For the sake of precision we will explicitly write $\mathsf {Ext}(X_{\gamma }(t))$ where the process $X_{\gamma }$ has been extended to the whole torus.

We will use the Glauber dynamic and the solution of the stochastic quantisation equation (2.6) introduced in the previous section: the idea is to exploit the exponential convergence to the invariant measure of the solution of the SPDE (2.6) proved in [18] and the convergence of the Glauber dynamic of the Kac–Ising model in [12].

By [12, Theorem 3.2], we know that if for $0< \kappa <\nu$ small enough the sequence of initial conditions $\mathsf {Ext}(X_\gamma ^0)$ is bounded in $\mathcal {C}^{-\nu +\kappa }$ and converges to a limit $X^0$ in $\mathcal {C}^{-\nu }$ as $\gamma \rightarrow 0$, one has

$$\begin{aligned} \mathsf {Ext}\left( X_{\gamma }\right) {\mathop {\longrightarrow }\limits ^{\mathcal {L}}} X \qquad \text { in }\mathcal {D}\left( [0,T];\mathcal {C}^{-\nu }\right) , \end{aligned}$$

(2.11)

where X solves (2.6) starting from $X^0$. In the above equation we took into account the fact that $X_{\gamma }$ is defined on the discrete lattice and therefore has to be extended with the operator $\mathsf {Ext}$ to be comparable with X.

We first want to show that (2.11) holds true when instead of a deterministic sequence $\mathsf {Ext}X_\gamma ^0\rightarrow X^0$ in $\mathcal {C}^{-\nu }$, we have the convergence in law of the initial conditions $\mathcal {L}(\mathsf {Ext}X_\gamma ^0) \rightarrow \mathcal {L}(X^0)$ in the topology of $\mathcal {C}^{-\nu }$. In order to do this call $\mathfrak {L}_{\gamma }$ (resp. $\mathfrak {L}_{0}$) the laws at time zero of the processes $\mathsf {Ext}X_{\gamma }$ (resp. X) and assume that $\mathfrak {L}_{\gamma } \rightarrow \mathfrak {L}_{0}$. Consider then a bounded continuous function $G :\mathcal {D}\left( [0,T];\mathcal {C}^{-\nu }\right) \rightarrow \mathbb {R}$: we want to show that

$$\begin{aligned} \lim _{\gamma \rightarrow 0} \left| \mathbb {E}\big [ G(\mathsf {Ext}(X_{\gamma })) \big | X_\gamma ^0\sim \mathfrak {L}_{\gamma } \big ] - \mathbb {E}\big [ G(X) \big | X^0\sim \mathfrak {L}_0 \big ] \right| = 0. \end{aligned}$$

Conditioning over the initial conditions we can define

$$\begin{aligned} f_{G}^{\gamma }(X_\gamma ^0)&:= \mathbb {E}\left[ G(\mathsf {Ext}(X_{\gamma }))\Big | X_{\gamma }(0) = X_\gamma ^0\right] \\ f_G(X^0)&:= \mathbb {E}\left[ G(X)\Big | X(0) = X^0\right] . \end{aligned}$$

The result [12, Theorem 3.2] implies that $f_{G}^{\gamma }(X_\gamma ^0) \rightarrow f_{G}(X^0)$ whenever $\mathsf {Ext}X_\gamma ^0\rightarrow X^0$ in $\mathcal {C}^{-\nu }$. Since $\mathcal {C}^{-\nu }$ is separable, we can apply the Skorokhod’s representation theorem to deduce that there is a probability space $(\tilde{\mathbb {P}},\tilde{\mathcal {F}},\tilde{\Omega })$ where all the processes $\mathsf {Ext}(X^0_{\gamma })$ and $X^0$ can be realised and the sequence $\mathsf {Ext}(X^0_{\gamma })(\tilde{\omega })$ converge to $X^0(\tilde{\omega })$ in $\mathcal {C}^{-\nu }$ for $\tilde{\mathbb {P}}$-a.e. $\tilde{\omega } \in \tilde{\Omega }$.

An application of the dominated convergence theorem then shows that, as $\gamma \rightarrow 0$

$$\begin{aligned}&\left| \mathbb {E}\big [ G(\mathsf {Ext}(X_{\gamma })) \big | X_\gamma ^0\sim \mathfrak {L}_{\gamma } \big ] - \mathbb {E}\big [ G(X) \big | X^0\sim \mathfrak {L}_0 \big ] \right| \nonumber \\&\quad \le \int \left| f_{G}^{\gamma }(X_\gamma ^0(\tilde{\omega })) - f_{G}(X^0(\tilde{\omega })) \right| \tilde{\mathbb {P}}(d\tilde{\omega }) \rightarrow 0, \end{aligned}$$

(2.12)

so that we can assume (2.11) to hold even when the initial datum is convergent in law.

By Theorem 2.1 we know that, if at time 0 the configuration $\sigma (0) \in \Sigma _N$ is distributed according to $\mathbf {P}^{\gamma }_{\beta ,0}$, then the law of $X_\gamma ^0(x) = \gamma ^{-1}\kappa _\gamma *\sigma _{\left\lfloor \varepsilon ^{-1}x \right\rfloor }(0)$ is tight, and therefore there exists a subsequence $\gamma _{k}$ for $k \ge 0$ and a measure $\mu ^*$ on $\mathcal {C}^{-\nu }$ such that the law of $\mathsf {Ext}X^0_{\gamma _k}$ converges to $\mu ^*$. In the following calculations we will tacitly assume $\gamma \rightarrow 0$ along the sequence $\gamma _k$ to avoid the subscript. We will show that, if $\mu$ if the unique invariant measure of (2.6) then $\mu ^* = \mu$.

Let $F:\mathcal {C}^{-\nu } \rightarrow \mathbb {R}$ be a bounded and continuous function, then, by the invariance of the Gibbs measure under the Glauber dynamic, for $t \ge 0$

$$\begin{aligned} \mathbf {E}^{\gamma }_{\beta ,0} \left[ F(\mathsf {Ext}X_\gamma ^0)\right] = \mathbb {E}_{\beta ,0}^{\gamma } \left[ F(\mathsf {Ext}X_{\gamma }(t))\right] . \end{aligned}$$

Recall that the evaluation map, that associates to a process in $\mathcal {D}\left( [0,T];\mathcal {C}^{-\nu }\right)$ its value at a given time, is not continuous with respect to the Skorokhod topology, however the integral map $G : u \mapsto \int _0^T F(u(s))\ ds$ is continuous in its argument in virtue of the the continuity and boundedness of F. Hence for any fixed T we have

$$\begin{aligned} \mathbf {E}^{\gamma }_{\beta ,0} \left[ F(\mathsf {Ext}X_\gamma ^0)\right] = \mathbb {E}^{\gamma }_{\beta ,0}\Bigg [ T^{-1}\int _0^T F\left( \mathsf {Ext}X_{\gamma }(s) \right) \ ds \Bigg ] \end{aligned}$$

and

$$\begin{aligned} \lim _{\gamma \rightarrow 0}\left| \mathbb {E}^{\gamma }_{\beta ,0}\Bigg [ \int _0^T F\left( \mathsf {Ext}X_{\gamma }(s) \right) \ ds \Bigg ] - \mathbb {E}\Bigg [ \int _0^T F\left( X(s) \right) \ ds\Bigg | X(0) \sim \mu ^*\Bigg ] \right| = 0. \end{aligned}$$

By the uniform convergence to equilibrium of the stochastic quantisation equation [18, Corollary 6.6] there exist constants $c,C > 0$

$$\begin{aligned} \left| \mathbb {E}[F\big ( X(s)\big )\big | X(0) \sim \mu ^*] - \mu [F]\right| \le C \left|F \right|_{\infty } e^{-c s}. \end{aligned}$$

From the above inequality it follows that

$$\begin{aligned} \left| T^{-1}\int _0^T \mathbb {E}[F\big ( X(s)\big )\big | X(0) \sim \mu ^*] -\mu [F] ds\right| \lesssim T^{-1} \left|F \right|_{\infty } \end{aligned}$$

and letting T be large enough the last difference can be made arbitrarily small. From the above estimates we can see that, for arbitrary $T>0$,

$$\begin{aligned} \limsup _{\gamma \rightarrow 0}\left| \mathbf {E}^{\gamma }_{\beta ,0} \left[ F(\mathsf {Ext}X_\gamma ^0)\right] - \mu [F] \right| \le C \left|F \right|_{\infty } T^{-1} \end{aligned}$$

and the result follows. $\square$

Remark 2.5

For $b_\gamma = b$ constant, we actually expect the limiting points to vanish under the scaling (2.10). On the other hand, for $b_\gamma = b\gamma$, one can follow an argument virtually identical to the one given in this article to show that the limit is given by the law of the $\Phi ^4_2$ measure with external magnetic field b.

2.1 Solution of the Limiting Equation

Before the proof of the main theorem, let us briefly explain the construction of the solution in [4] to the following SPDE

$$\begin{aligned} d X = \Big ( \Delta X + \sum _{j=1}^n a_{2j-1} X^{2j-1}\Big )dt + \sqrt{2}dW. \end{aligned}$$

As in (2.6), the powers in the above SPDE have to be renormalised in order to find a nontrivial solution. The precise way the process is renormalised follows [12, 17]. Consider at first Z(t) the solution of the stochastic heat equation

$$\begin{aligned} d Z = \Delta Z dt+ \sqrt{2}dW\,\qquad Z(\cdot ,0) = 0, \end{aligned}$$

and therefore in two dimension Z belongs to $\mathcal {C}([0,T];\mathcal {C}^{-\nu })$ a.s. for any $\nu > 0$. Consider the Galerkin approximation

$$\begin{aligned} d Z_{\varepsilon } = \Delta Z_{\varepsilon } dt+ \sqrt{2}dW_{\varepsilon }\,\qquad Z_{\varepsilon }(\cdot ,0) = 0, \end{aligned}$$

(2.13)

where $dW_{\varepsilon }$ is the $L^2(\mathbb {T}^2)$-projection of the space time white noise on the span of the Fourier modes generated by $\{e_{\omega }\}_{|\omega | \le N}$.

From the above SPDE we see that $Z_{\varepsilon }$ has a representation in terms of the stochastic convolution with the heat kernel $P_t = e^{t \Delta }$

$$\begin{aligned} Z_{\varepsilon }(t,x) = \sqrt{2} \int _0^t P_{t-s}d\hat{W}(s,x) . \end{aligned}$$

Define the renormalisation constant

$$\begin{aligned} \mathfrak {c}_{\varepsilon } := \lim _{t \rightarrow \infty }\mathbb {E}\left[ Z_{\varepsilon }^2(t,0) - \frac{t}{2}\right] = \sum _{\omega \in \Lambda _N\setminus \{0\}} \frac{1 }{4\pi ^2|\omega |^2}. \end{aligned}$$

In order to renormalise the process $Z_{\varepsilon }(t)$ at finite time, it is more convenient to use its time dependent version

$$\begin{aligned} \mathfrak {c}_{\varepsilon }(t) {\mathop {=}\limits ^{\tiny \text{ def }}}\mathbb {E}\left[ Z_{\varepsilon }^2(t,0)\right] = \frac{1}{2}\sum _{\omega \in \Lambda _N} \int _0^t e^{-2(t-s)\pi ^2|\omega |^2}ds = \frac{t}{2} + \sum _{\omega \in \Lambda _N\setminus \{0\}} \frac{1 - e^{-2t\pi ^2|\omega |^2}}{4\pi ^2|\omega |^2} \end{aligned}$$

We therefore define the renormalised powers of the process $Z_{\varepsilon }$ as

$$\begin{aligned} Z_{\varepsilon }^{:n:}(t) {\mathop {=}\limits ^{\tiny \text{ def }}}H_n(Z_{\varepsilon }(t),\mathfrak {c}_{\varepsilon }(t)). \end{aligned}$$

By [4, Lemma 3.2], the process $Z_{\varepsilon }^{:n:}(t)$ is Cauchy in $L^p\left( \mathcal {C}([0,T];\mathcal {C}^{-\nu }),\mathbb {P}\right)$ for every $p \ge 1$ and we will be referring to its limit as $Z^{:n:}$. To be precise, the result in [4, Lemma 3.2] is proven for a fixed time, but the extension to the whole process is immediate.

By the variation of constants formula, the solution to (2.13) started from the initial condition $X^0_{\varepsilon } {\mathop {=}\limits ^{\tiny \text{ def }}}\Pi _{\varepsilon }X^0$ is given by $\tilde{Z}_{\varepsilon } (t)= e^{\Delta t} X^0_{\varepsilon } + Z_{\epsilon }(t)$. To extend the definition of the renormalised powers to the process $\tilde{Z}_{\varepsilon }(t)$ one uses the following property of the Hermite polynomial

$$\begin{aligned} H_n(a+b,c) = \sum _{j=0}^n \left( {\begin{array}{c}n\\ j\end{array}}\right) b^{n-j} H_j(a,c) \end{aligned}$$

and let

$$\begin{aligned} \tilde{Z}_{\varepsilon }^{:n:}(t) = \sum _{j=0}^n \left( {\begin{array}{c}n\\ j\end{array}}\right) \left( e^{\Delta t} X^0_{\varepsilon } \right) ^{n-j} Z_{\varepsilon }^{:j:}(t). \end{aligned}$$

The above random variable is well defined because $e^{\Delta t} X^0_{\varepsilon }$ is a smooth function and the product with $Z_{\varepsilon }^{:j:}(t)$ is in $\mathcal {C}^{-\nu }$ for any $t>0$ (see for instance [13, Corollary 3.2] or Theorems 2.82 and 2.85 in [1] for a proof).

We then set $X_{\varepsilon }(t)$ the Galerkin approximation of X(t) solving

$$\begin{aligned} {\left\{ \begin{array}{ll} d X_{\varepsilon }(t) &{}= \left( \Delta X_{\varepsilon }(t) + \sum _{j=1}^n a_{2j-1} H_{n}(X_{\varepsilon }(t),\mathfrak {c}_{\varepsilon })\right) dt + \sqrt{2}dW_{\varepsilon }(t)\\ X_{\varepsilon }(0) &{}= \Pi _{\varepsilon }X^0\in \mathcal {C}^{-\nu } \end{array}\right. } \end{aligned}$$

(2.14)

Since $H_n$ is a polynomial in both variables, it is possible to replace $\mathfrak {c}_{\varepsilon }$ in the above formula with $\mathfrak {c}_{\varepsilon }(t)$ provided one compensates it in the coefficient of the polynomial.

$$\begin{aligned} \sum _{j=1}^n a_{2j-1} H_{n}(X_{\varepsilon }(t),\mathfrak {c}_{\varepsilon }) = \sum _{j=1}^n a_{2j-1}(t,\varepsilon ) H_{n}(X_{\varepsilon }(t),\mathfrak {c}_{\varepsilon }(t)) \end{aligned}$$

with new coefficients $a_{2j-1}(t)$ depending polynomially only on the old coefficients and on the difference $\mathfrak {c}_{\varepsilon }-\mathfrak {c}_{\varepsilon }(t)$. From the definitions of $\mathfrak {c}_{\varepsilon }$ and $\mathfrak {c}_{\varepsilon }(t)$ one can see that their difference is diverging logarithmically as $t \rightarrow 0$ and therefore each power of $a_{2j-1}(t)$ is integrable in [0, T]. Hence we can rewrite (2.14) as

$$\begin{aligned} {\left\{ \begin{array}{ll} d X_{\varepsilon }(t) &{}= \left( \Delta X_{\varepsilon }(t) + \sum _{j=1}^n a_{2j-1}(t,\varepsilon ) H_{n}(X_{\varepsilon }(t),\mathfrak {c}_{\varepsilon }(t))\right) dt + \sqrt{2}dW_{\varepsilon }(t)\\ X_{\varepsilon }(0) &{}= \Pi _{\varepsilon }X^0\in \mathcal {C}^{-\nu } \end{array}\right. } \end{aligned}$$

(2.15)

We decompose $X_{\varepsilon }(t)=\tilde{Z}_{\varepsilon }(t)+V_{\varepsilon }(t)$ where, for a.e. realisation of $Z_{\varepsilon }$, the process $V_{\varepsilon }$ solves the PDE

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t V_{\varepsilon }(t) &{}= \Delta V_{\varepsilon }(t) + \sum _{j=1}^n a_{2j-1}(t,\varepsilon ) H_{n}\left( \tilde{Z}_{\varepsilon }(t)+V_{\varepsilon }(t),\mathfrak {c}_{\varepsilon }(t)\right) \\ V_{\varepsilon }(0) &{}=0 \end{array}\right. } \end{aligned}$$

(2.16)

where

$$\begin{aligned} H_n\left( \tilde{Z}_{\varepsilon }(t)+V_{\varepsilon }(t),\mathfrak {c}_{\varepsilon }(t)\right) = \sum _{j=0}^n \left( {\begin{array}{c}n\\ j\end{array}}\right) V^{n-j}_{\varepsilon }(t) \tilde{Z}^{:j:}_{\varepsilon }(t). \end{aligned}$$

The last product is again well-posed thanks to the fact that $V_{\varepsilon }(t) \in \mathcal {C}^{2-\nu -\kappa }$ for any $\kappa >0$, from the regularizing properties of the parabolic Eq. (2.16).

As proven in [4], the processes $V_{\varepsilon }$ converge in $\mathcal {C}([0,T];\mathcal {C}^{2-\nu })$ to the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} d V &{}= \Delta V(t) + \sum _{j=1}^n a_{2j-1}(t) \sum _{j=0}^n \left( {\begin{array}{c}n\\ j\end{array}}\right) V^{n-j}(t) \tilde{Z}^{:j:}(t)\\ V(0) &{}=0 \end{array}\right. } \end{aligned}$$

(2.17)

where $\tilde{Z}^{:j:}(t) {\mathop {=}\limits ^{\tiny \text{ def }}}\lim _{\varepsilon \rightarrow 0} P_tX^0+ Z^{:j:}_{\varepsilon }(t)$. For all $\kappa >0$, the solution of the above PDE is unique and $V \in \mathcal {C}([0,T];\mathcal {C}^{2-\nu -\kappa })$ and only depends on the realisation of the process Z via the tuple $(\tilde{Z},\dots ,\tilde{Z}^{:2n-1:}) \in L^{\infty }([0,T];\mathcal {C}^{-\nu })^n$. We summarise it with the following proposition, essentially proven in [4].

Theorem 2.6

For all $\nu > 0$ and $T>0$ there exists a locally Lipschitz continuous function

$$\begin{aligned} \mathcal {S}_T :L^{\infty }([0,T];\mathcal {C}^{-\nu })^n \rightarrow \mathcal {C}([0,T];\mathcal {C}^{2-\nu -\kappa }) \end{aligned}$$

that associates to $(\tilde{Z},\tilde{Z}^{:2:}\dots ,\tilde{Z}^{:2n-1:})$ the solution of (2.17).

Using the definitions above, we now outline the skeleton of the proof in [12]. First of all we want to remark that we made the decision of absorbing the initial conditions in the process $\tilde{Z}_{\varepsilon }$, instead we could have started (2.16) from $\Pi _\varepsilon X_\gamma ^0$ and defined a similar solution map $\mathcal {S}_{T}^{X^0}$. Consider $Z_{\gamma }$ the solution to the linearised part of (2.7) satisfying

$$\begin{aligned} Z_{\gamma }(t,x) = \int _0^t\Delta _{\gamma }Z_{\gamma }(s,x) ds + M_\gamma (t,x), \end{aligned}$$

(2.18)

which is an approximation to the stochastic heat equation. In [4] and [12] the authors provided a useful definition of $Z_{\gamma }^{:n:}$ the renormalized powers of $Z_{\gamma }$ that we will not introduce here. In this article will only use the fact that for any $T>0$, $q > 0$, $\nu > 0$, $j \ge 0$ and $\lambda >0$,

$$\begin{aligned} \limsup _{\gamma \rightarrow 0}\mathbb {E}\left[ \sup _{s \in [0,T]}s^{\lambda }\left\| H_j\left( Z_{\gamma }(s,\cdot ),\mathfrak {c}_\gamma \right) \right\| _{\mathcal {C}^{-\nu }}^q \right] < \infty , \end{aligned}$$

(2.19)

which follows from [12, Eq. (3.15), Propositions 5.3 and 5.4].

In [12, Sect. 6] it is proven that processes $\left( Z_\gamma ,Z_\gamma ^{:2:}, \dots , Z_\gamma ^{:2n-1:} \right)$ jointly converge in law to $\left( Z,Z^{:2:},\dots ,Z^{:2n-1:}\right)$. Using the decomposition $X_{\gamma }= Z_{\gamma }+ V_{\gamma }$, it is possible to see that $V_{\gamma }$ satisfies an equation similar to (2.16), with initial condition $X_\gamma ^0$ and

$$\begin{aligned} \left\| V_{\gamma }- \mathcal {S}_T^{X_\gamma ^0}\left( Z_\gamma ,Z_\gamma ^{:2:}, \dots , Z_\gamma ^{:2n-1:} \right) \right\| _{L^{\infty }([0,T];\mathcal {C}^{-\nu })} \rightarrow 0. \end{aligned}$$

Therefore, by the continuity of $\mathcal {S}_T$, we have that, as $\gamma \rightarrow 0$,

$$\begin{aligned} X_{\gamma }\sim Z_{\gamma }+ \mathcal {S}_T^{X_\gamma ^0}\left( Z_\gamma ,Z_\gamma ^{:2:}, \dots , Z_\gamma ^{:2n-1:} \right) {\mathop {\longrightarrow }\limits ^{\mathcal {L}}} Z + \mathcal {S}_T^{X^0}\left( Z,Z^{:2:}, \dots , Z^{:2n-1:} \right) = X, \end{aligned}$$

as required.

3 Proof of Theorem 2.1

We are now going to prove the statements used in Sect. 2 and in particular Theorem 2.1. We first obtain a very suboptimal bound on $X_{\gamma }$ which can be used as a starting point for the derivation of sharper bounds.

Proposition 3.1

Let $p \ge 2$ an even integer, and $\lambda \in [0,1]$ then there exists $C(p,\lambda ) > 0$ such that

$$\begin{aligned} \mathbb {E}\big [\left\| X_{\gamma }(t,\cdot ) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ] \le C \left( \mathbb {E}\big [\left\| X_{\gamma }(0,\cdot ) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ]^{1-\lambda } t^{-\frac{p}{2}\lambda } \right) \vee \gamma ^{-\frac{p}{2}}. \end{aligned}$$

In particular, if we start the process from the invariant measure, we obtain that there exists $C = C(p)>0$ such that for all $t \ge 0$

$$\begin{aligned} \mathbf {E}^{\gamma }_{\beta ,0} \big [\left\| X_{\gamma } \right\| _{L^p(\Lambda _\varepsilon )}^p\big ] = \mathbb {E}_{\beta ,0}^{\gamma }\big [\left\| X_{\gamma }(t,\cdot ) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ] \le C \gamma ^{-\frac{p}{2}} \end{aligned}$$

(3.1)

Proof

In the following proof we will denote with C a generic constant whose value depends on p and might change from line to line. Recall the action of the generator of the Glauber dynamic (2.3):

$$\begin{aligned} \mathscr {L}_{\gamma } h_{\gamma }^p(t,x)&= \sum _{z \in \Lambda _N} c_{\gamma }(z,\sigma (t)) \Big (\left( h_{\gamma }(t,x)-2 \sigma _z(t)\kappa _\gamma (z-x) \right) ^p - h_{\gamma }^p(t,x) \Big )\\&\quad \le p\big (- h_{\gamma }+ \kappa _\gamma *\tanh (\beta h_{\gamma })\big )(t,x) h_{\gamma }^{p-1}(t,x) + C \big (|h_{\gamma }(t,x)| + \gamma ^2\big )^{p-2} \gamma ^2. \end{aligned}$$

The second inequality is a consequence of the fact that $\left\| \kappa _\gamma \right\| _{\infty } \lesssim \gamma ^2$ and $\left\| \kappa _\gamma \right\| _{L^1} \lesssim 1$. We can take the average over $x \in \Lambda _N$ to obtain

$$\begin{aligned} \mathscr {L}_{\gamma } \left\| h_{\gamma }(t) \right\| _{L^p(\Lambda _N)}^p\le & {} p \left\langle h_{\gamma }^{p-1}(t), \kappa _\gamma *\tanh (\beta h_{\gamma }(t)) \right\rangle _{ \Lambda _N } - p\left\| h_{\gamma }(t) \right\| _{L^p(\Lambda _N)}^{p} \\&+ C \gamma ^2 \left\| h_{\gamma }(t) \right\| _{L^{p-2}(\Lambda _N)}^{p-2} + C \gamma ^{2p-2}. \end{aligned}$$

We use the fact that p is even and the hyperbolic tangent is monotone to bound

$$\begin{aligned}&\left\langle h_{\gamma }^{p-1}(t), \kappa _\gamma *\tanh (\beta h_{\gamma }(t)) \right\rangle _{\Lambda _N} = \left\langle h_{\gamma }^{p-1}(t),\tanh (\beta h_{\gamma }(t)) \right\rangle _{\Lambda _N} \\&\quad + \frac{1}{2} \sum _{x,y \in \Lambda _N} \kappa _\gamma (x-y)\big (h_{\gamma }^{p-1}(t,x) - h_{\gamma }^{p-1}(t,y)\big )\big (\tanh (\beta h_{\gamma }(t,y)) - \tanh (\beta h_{\gamma }(t,x))\big )\\&\quad \le \left\langle h_{\gamma }^{p-1}(t),\tanh (\beta h_{\gamma }(t)) \right\rangle _{\Lambda _N}. \end{aligned}$$

Moreover, it is easy to see that there exists a constant $c_0>0$ such that

$$\begin{aligned} \frac{\tanh (\beta h)}{h} \le \beta - c_0 h^2\qquad \text { for } h \in [1,1]. \end{aligned}$$

Since $|h_{\gamma }(t,x)| \le 1$ and $\beta =1+ \gamma ^2 (\mathfrak {c}_{\gamma }+A)$, we can bound $\mathscr {L}_{\gamma } \left\| h_{\gamma }(t) \right\| _{L^p(\Lambda _N)}^p$ by

$$\begin{aligned}&p[\beta -1]\left\| h_{\gamma }(t) \right\| _{L^p(\Lambda _N)}^p - c_0 p \left\| h_{\gamma }(t) \right\| _{L^{p+2}(\Lambda _N)}^{p+2} + C \gamma ^2 \left\| h_{\gamma }(t) \right\| _{L^{p-2}(\Lambda _N)}^{p-2} +C \gamma ^{2p-2}\\&\quad \le C ( \gamma ^2 \mathfrak {c}_{\gamma })^{\frac{p+2}{2}} - \frac{c_0}{2} p \left\| h_{\gamma }(t) \right\| _{L^{p}(\Lambda _N)}^{p+2} + C\gamma ^{\frac{p+2}{2}} +C \gamma ^{2p-2}\\&\quad \le - \frac{c_0}{2} p \left\| h_{\gamma }(t) \right\| _{L^{p}(\Lambda _N)}^{p+2} + C \gamma ^{\frac{p+2}{2}}, \end{aligned}$$

where we used the fact that $|A|\le \mathfrak {c}_\gamma \le \gamma ^{-1}$ for $\gamma$ small enough and the generalised Young inequality in the last line. Therefore, taking the expectation

$$\begin{aligned}&\mathbb {E}\big [ \left\| h_{\gamma }(\alpha ^{-1} t) \right\| _{L^p(\Lambda _N)}^p \big ] \\&\quad \le \mathbb {E}\big [ \left\| h_{\gamma }(0) \right\| _{L^p(\Lambda _N)}^p \big ] + \alpha ^{-1}\int _0^{ t} - \frac{c_0}{2} p \mathbb {E}\big [ \left\| h_{\gamma }(\alpha ^{-1} s) \right\| _{L^{p}(\Lambda _N)}^{p+2} \big ] + C \gamma ^{\frac{p+2}{2}} ds \end{aligned}$$

and multiplying both sides by $\gamma ^{-p}$ and applying Jensen’s inequality we obtain

$$\begin{aligned}&\mathbb {E}\big [\left\| X_{\gamma }(t) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ] = \mathbb {E}\big [\left\| X_{\gamma }(0) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ] + \int _0^t \mathbb {E}\big [\mathscr {L}_{\gamma }\left\| X_{\gamma }(s) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ]\ ds\\&\quad \le \mathbb {E}\big [\left\| X_{\gamma }(0) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ] - \frac{c_0}{2} p \gamma ^2 \int _0^t \mathbb {E}\big [\left\| X_{\gamma }(s) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ]^{\frac{p+2}{p}}\ ds + C \gamma ^{\frac{2-p}{2}}. \end{aligned}$$

From the comparison test in Lemma 4.10 we have that

$$\begin{aligned} \mathbb {E}\big [\left\| X_{\gamma }(t) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ] \lesssim \frac{\mathbb {E}\big [\left\| X_{\gamma }(0) \right\| _{L^p}^p\big ] }{\big ( 1 + C t\mathbb {E}\big [\left\| X_{\gamma }(0) \right\| _{L^p(\Lambda _\varepsilon )}^p\big ]^{\frac{2}{p}}\big )^{\frac{p}{2}}} \vee \gamma ^{-\frac{p}{2}}, \end{aligned}$$

and the result follows. $\square$

Remark 3.2

Despite its simplicity, Proposition 3.1 has the advantage of making the proof of [12, Theorem 6.1] simpler, avoiding the need for the stopping time $\tau _{\gamma ,\mathfrak {m}}$ and providing sufficient control over [12, Eq. (6.7)].

Proposition 3.3

Recall the definitions given in Sect. 2 of the processes $X_{\gamma }$, $Z_{\gamma }$ and $V_{\gamma }:= X_{\gamma }- Z_{\gamma }$. We want to remark that $V_{\gamma }$ is not similar to of $V_{\varepsilon }$, because of the initial condition [see (2.17)]. Let $p \ge 2$ an even integer. Then there exist $\nu _0 > 0$, $\lambda _{j,i} > 0$ for $i=1,2$ and $j=0,1,2$ such that for all $0< \nu < \nu _0$ and $0 \le s \le t \le T$

$$\begin{aligned}&\left\| V_{\gamma }(t,\cdot ) \right\| _{L^p(\Lambda _\varepsilon )}^p - \left\| V_{\gamma }(s,\cdot ) \right\| _{L^p(\Lambda _\varepsilon )}^p \nonumber \\&\quad +\, C_1 \int _s^t \left\| V_{\gamma }(r,\cdot ) \right\| _{L^p(\Lambda _\varepsilon )}^{p+2} dr + C_1 \int _s^t \left\langle V_{\gamma }^{p-1}(r),(-\Delta _{\gamma })V_{\gamma }(r) \right\rangle _{\Lambda _\varepsilon } dr \nonumber \\&\quad \le C_2 \int _s^t\sum _{j=0}^3 \sum _{i=1,2}\left\| H_j(Z_\gamma (r,\cdot ),\mathfrak {c}_\gamma ) \right\| _{\mathcal {C}^{-\nu }(\Lambda _\varepsilon )}^{\lambda _{j,i}}\ dr + \int _s^t\mathsf {Err}(r)\ dr \end{aligned}$$

(3.2)

where, for every $q>0$

$$\begin{aligned} \sup _{0 \le r \le T}\mathbb {E}_{\beta ,0}^{\gamma }\left[ \mathsf {Err}^q(r) \right] ^{\frac{1}{q}} \lesssim C_3(p,q,T) \gamma ^{\frac{p-2}{6}-2\nu \frac{(p-2)}{3}}. \end{aligned}$$

(3.3)

Proof

The proof follows the argument provided in [18, Proposition 3.7] for the dynamical $\Phi ^4_2$ model, but applied to the model of [12], with the important difference that in our case all the operators are discrete operators. Without loss of generality, we will prove (3.2) starting at time $s=0$ from $V_{\gamma }^0= X_\gamma ^0$.

In the following calculations, since there is no possibility of confusion, we will use $L^p$ instead of $L^p(\Lambda _\varepsilon )$, and $\left\langle \cdot ,\cdot \right\rangle$ instead of $\left\langle \cdot ,\cdot \right\rangle _{\Lambda _\varepsilon }$. From (2.7) and (2.18) we see that $V_{\gamma }(t,x)$ satisfies, for $x \in \Lambda _\varepsilon , t \ge 0$

$$\begin{aligned}&V_{\gamma }(t,x) \\&\quad = V_{\gamma }^0(x) + \int _0^t \Delta _{\gamma }V_{\gamma }(s,x) ds + \int _0^t \gamma ^{-2}K_\gamma *\left( \gamma ^{-1}\tanh (\beta \gamma X_{\gamma }(s,x)) - X_{\gamma }(s,x) \right) ds \end{aligned}$$

and in particular $V_{\gamma }(t,x)$ is continuous and weakly differentiable in time, for all $\gamma >0$. Recall that $\beta = 1 + \gamma ^2 (\mathfrak {c}_{\gamma }+ A)$ and expand the hyperbolic tangent up to third order

$$\begin{aligned} \tanh (\beta \gamma X_{\gamma }(s))= \gamma X_{\gamma }(s) + \gamma ^3 (\mathfrak {c}_{\gamma }+ A) X_{\gamma }(s) - \frac{\gamma ^3}{3} X_{\gamma }^3(s) \\ + \,\gamma ^3(\beta - 1)\mathcal {O}\left( X_{\gamma }^3(s) \right) + \mathcal {O}\left( \gamma ^5 X_{\gamma }^5(s) \right) . \end{aligned}$$

With the above formula the derivative of the discrete $L^p$ norm of $V_{\gamma }$ is calculated

$$\begin{aligned} \left\| V_{\gamma }(t) \right\| _{L^p}^p = \left\| V_{\gamma }^0 \right\| _{L^p}^p + p \int _0^t \left\langle V_{\gamma }^{p-1} , \Delta _{\gamma }V_{\gamma } \right\rangle (s)\ ds + \frac{1}{3} D(s) + B(s)\ ds \end{aligned}$$

(3.4)

where

$$\begin{aligned} D(s) = - \left\langle K_\gamma *V_{\gamma }^{p-1}(s) , X_{\gamma }^3(s) - 3(\mathfrak {c}_{\gamma }+ A )X_{\gamma }(s) \right\rangle \end{aligned}$$

and B(s) is produced by the remainder of the Taylor expansion of the hyperbolic tangent

$$\begin{aligned} B(s) \le C \gamma ^2 \left\langle |V_{\gamma }^{p-1}|(s),\mathfrak {c}_{\gamma }|X_{\gamma }|^3(s) + |X_{\gamma }|^5(s) \right\rangle . \end{aligned}$$

(3.5)

where we used the fact that $|A| \le \mathfrak {c}_{\gamma }$ for $\gamma$ small enough.We will first replace D(s) with

$$\begin{aligned} D_1(s):= & {} -\left\langle V_{\gamma }^{p-1}(s) ,X_{\gamma }^3(s) - 3\mathfrak {c}_{\gamma }X_{\gamma }(s) \right\rangle + 3A \left\langle V_{\gamma }^{p-1}(s) , X_{\gamma }(s) \right\rangle \nonumber \\\le & {} -\left\| V_{\gamma }^{p+2}(s) \right\| _{L^1} + 3\left| \left\langle V_{\gamma }^{p+1}(s),Z_{\gamma }(s) \right\rangle \right| + 3\left| \left\langle V_{\gamma }^{p}(s),H_2(Z_{\gamma }(s),\mathfrak {c}_\gamma ) \right\rangle \right| \nonumber \\&\quad + \left| \left\langle V_{\gamma }^{p-1}(s),H_3(Z_{\gamma }(s),\mathfrak {c}_\gamma ) \right\rangle \right| + 3A \left\| V_{\gamma }^{p}(s) \right\| _{L^1} + 3A\left| \left\langle V_{\gamma }^{p-1}(s),Z_{\gamma }(s) \right\rangle \right| \nonumber \\ \end{aligned}$$

(3.6)

Let

$$\begin{aligned} L_s {\mathop {=}\limits ^{\tiny \text{ def }}}\left\| V_{\gamma }(s) \right\| _{L^{p+2}}^{p+2},\qquad K_s {\mathop {=}\limits ^{\tiny \text{ def }}}\left\langle V_{\gamma }^{p-1}(s) , \Delta _{\gamma }V_{\gamma }(s) \right\rangle . \end{aligned}$$

Those terms are the good terms of (3.4), and the idea is now to bound all the other errors $|D(s)-D_1(s)|$ with expression containing $L_s$ and $K_s$. In the following calculations we assume $\gamma$ to be small enough such that $|A| \le \mathfrak {c}_{\gamma }$. The cost of replacing D(s) with $D_1(s)$ is given by

$$\begin{aligned} |D(s) - D_1(s)|&\le \sum _{x,y \in \Lambda _\varepsilon } \varepsilon ^4 K_\gamma (x-y) \left| V_{\gamma }^{p-1}(s,y)- V_{\gamma }^{p-1}(s,x) \right| \\&\quad \times \big |\big (X_{\gamma }^3(s,y) -X_{\gamma }^3(s,x) \big ) - 3(\mathfrak {c}_{\gamma }+ A) \left( X_{\gamma }(s,y) - X_{\gamma }(s,x)\right) \big |\\&\le 3 \sum _{x,y \in \Lambda _\varepsilon } \varepsilon ^4 K_\gamma (x-y) \left| V_{\gamma }^{p-1}(s,y)- V_{\gamma }^{p-1}(s,x) \right| \\&\quad \times \big (\big | V_{\gamma }(s,y)- V_{\gamma }(s,x) \big | + \left| Z_{\gamma }(s,y)- Z_{\gamma }(s,x) \right| \big )\left( 2 \mathfrak {c}_\gamma + X_{\gamma }^2(s,x)\right) . \end{aligned}$$

Denote with

$$\begin{aligned} D_2= & {} 3 \sum _{x,y \in \Lambda _\varepsilon } \varepsilon ^4 K_\gamma (x-y) \left| V_{\gamma }^{p-1}(s,y)- V_{\gamma }^{p-1}(s,x) \right| \\&\quad \times \left| V_{\gamma }(s,y)- V_{\gamma }(s,x) \right| \left( 2 \mathfrak {c}_\gamma + X_{\gamma }^2(s,x)\right) \\ D_3= & {} 3 \sum _{x,y \in \Lambda _\varepsilon } \varepsilon ^4 K_\gamma (x-y) \left| V_{\gamma }^{p-1}(s,y)- V_{\gamma }^{p-1}(s,x) \right| \\&\quad \times \left| Z_{\gamma }(s,y)- Z_{\gamma }(s,x) \right| \left( 2 \mathfrak {c}_\gamma + X_{\gamma }^2(s,x)\right) . \end{aligned}$$

We will now bound $D_3$ with a small multiple of $L_s$ and $K_s$ plus an error in (3.3), the term $D_2$ can be bounded in a similar way.

By $a^n - b^n = (a-b)(a^{n-1} + \dots + b^{n-1})$ and the generalized Young inequality

$$\begin{aligned} a^{p-1} - \, b^{p-1}= & {} (a-b)(a^{p-2} + \dots + b^{p-2})\\\le & {} |a^{p-1} - b^{p-1}|\frac{|a-b|}{2 \lambda } + (|a|^{p-1} + |b|^{p-1}) \lambda 2^{p-2} \end{aligned}$$

Therefore, applying the previous inequality to each summands of $D_3$ and choosing $\lambda = c_1^{-1}(\gamma ^{-1} \varepsilon )^2\left| Z_{\gamma }(s,y)- Z_{\gamma }(s,x) \right| \left( 2 \mathfrak {c}_\gamma + X_{\gamma }^2(s,x)\right)$ we have that

$$\begin{aligned} D_3\le & {} c_1 K_s + C c_1^{-1}(\varepsilon ^2 \gamma ^{-2})\sum _{x \in \Lambda _\varepsilon } \varepsilon ^2 |V_{\gamma }^{p-1}(s,x)|| Z_{\gamma }(s,x)|\left( \mathfrak {c}_\gamma + X_{\gamma }^2(s,x)\right) \nonumber \\\le & {} c_1 K_s + c_1 L_s +C c_1^{-1} \left\| \varepsilon ^2 \gamma ^{-2}| Z_{\gamma }( s)|\left( \mathfrak {c}_\gamma + X_{\gamma }^2( s)\right) \right\| _{L^{\frac{p-2}{3}}}^{(p-2)/3} \end{aligned}$$

(3.7)

where $c_1>0$ can be chosen to be for instance $c_1 = 1/8$. The last term will be part of the error (3.3). Recall that $\varepsilon = \gamma ^2$ and the last term of (3.7) is bounded in expectation using Proposition 3.1, Lemma 4.2 and (2.19)

$$\begin{aligned}&\mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\| \varepsilon ^2 \gamma ^{-2}| Z_{\gamma }( s)|\left( \mathfrak {c}_\gamma + X_{\gamma }^2( s)\right) \right\| _{L^{\frac{p-2}{3}}}^{(p-2)/3}\right] \nonumber \\&\quad \le \mathbb {E}_{\beta ,0}^{\gamma } \left[ \left\| Z_{\gamma }( s) \right\| _{\mathcal {C}^{-\nu }(\mathbb {T}^2)}^{2(p-2)/3} \right] ^{1/2} \left( (\gamma ^2 \mathfrak {c}_\gamma )^{\frac{2(p-2)}{3}} + \mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\| \gamma X_{\gamma }( s) \right\| _{L^{2(p-2)/3}}^{2(p-2)/3} \right] \right) ^{1/2}\nonumber \\&\quad \le C(T) \gamma ^{\frac{p-2}{6}-2\nu \frac{(p-2)}{3}} \end{aligned}$$

(3.8)

which is negligible if $\nu$ is small enough and $p > 2$. It is immediate to generalize (3.8) to any power, as in (3.3).

We will then bound the term B(t) in (3.5) with Proposition 3.1. Using Young’s inequality we have that

$$\begin{aligned} B(s)\le & {} C \gamma ^2 \left\langle |V_{\gamma }^{p-1}|(s),\mathfrak {c}_{\gamma }+ \left| Z_{\gamma }^2(s)+2V_{\gamma }(s)Z_{\gamma }(s) + V_{\gamma }^2(s)\right| |X_{\gamma }|^3(s) \right\rangle \\\le & {} \frac{1}{24} \left\| V_{\gamma }(s) \right\| _{L^{p+2}}^{p+2} + C \mathfrak {c}_{\gamma }^{\frac{p+2}{3}} \big \Vert (\gamma ^{\frac{2}{3}}X_{\gamma }(s))^{p+2} \big \Vert _{L^1} \\&\quad +\, C \big \Vert Z_{\gamma }^{\frac{2p+4}{3}}(\gamma ^{2}X_{\gamma }^3(s))^{\frac{p+2}{3}} \big \Vert _{L^1} + \big \Vert Z_{\gamma }^{\frac{p+2}{2}}(\gamma ^{2}X_{\gamma }^3(s))^{\frac{p+2}{2}} \big \Vert _{L^1}+ \left\| (\gamma ^{2}X_{\gamma }^3(s))^{p+2} \right\| _{L^1} . \end{aligned}$$

The constant 1 / 24 has been arbitrarily chosen in order to control B(s) with a small multiple of $L_s$ plus a quantity that will be part of the error in (3.3) and can be bounded in expectation, as we did in (3.8), by $C(T) \gamma ^{\frac{p+2}{6} - 2\nu \frac{2p+4}{3}}$, which is negligible for $\nu$ small enough.

We are now in the setting of [18, Eq. (3.13)], namely the discrete process $V_{\gamma }$ satisfies

$$\begin{aligned}&\left\| V_{\gamma }(t) \right\| _{L^p}^p - p \int _0^t \frac{5}{6} K_s + \frac{5}{24} L_s \ ds \nonumber \\&\quad \le \left\| V_{\gamma }^0 \right\| _{L^p}^p + \frac{p}{3} \int _0^t \sum _{j=0}^2 \left( {\begin{array}{c}3\\ j\end{array}}\right) \left\langle V_{\gamma }^{p-1+j}(s) , H_{3-j}(Z_{\gamma }(s),\mathfrak {c}_\gamma ) \right\rangle \ ds\nonumber \\&\qquad + \,\,A \int _0^t \left| \left\langle V_{\gamma }^{p-1}(s) , (V_{\gamma }(s) + Z_{\gamma }(s)) \right\rangle \right| \ ds + \int _0^t \mathsf {Err}(s) ds, \end{aligned}$$

(3.9)

where $\mathbb {E}_{\beta ,0}^{\gamma }[|\mathsf {Err}(s)|^q]^{\frac{1}{q}} \le C(T,p,q) \gamma ^{\frac{p-2}{6}-2\nu \frac{(p-2)}{3}}$ for any positive q. We will now show that, for $\nu$ small enough and $j = 0,1,2$, there exist $\lambda _{j,1},\lambda _{j,1} > 0$

$$\begin{aligned}&\left\langle V_{\gamma }^{p-1+j} , H_{3-j}(Z_{\gamma }(s),\mathfrak {c}_\gamma ) \right\rangle \nonumber \\&\quad \lesssim \left( L_s^{\frac{p-1+j}{p+2} - \nu \frac{p }{p+2}} K^{\nu }_s + L_s^{\frac{p-1+j}{p+2}} \right) \left\| H_{3-j}(Z_{\gamma }(s),\mathfrak {c}_\gamma ) \right\| _{\mathcal {C}^{-\nu }(\Lambda _\varepsilon )}\nonumber \\&\quad \le \frac{1}{7} K_s + \frac{1}{30} L_s + C \sum _{i=1,2} \left\| H_{3-j}(Z_{\gamma }(s),\mathfrak {c}_\gamma ) \right\| _{\mathcal {C}^{-\nu }(\Lambda _\varepsilon )}^{\lambda _i} \end{aligned}$$

(3.10)

where the last line follows from the Young inequality for $\nu$ sufficiently small. In a similar way,

$$\begin{aligned} A\left| \left\langle V_{\gamma }^{p-1}(s) , (V_{\gamma }(s) +Z_{\gamma }(s)) \right\rangle \right| \le \frac{1}{7} K_s + \frac{1}{30} L_s + C(A)\left( 1 + \left\| Z_{\gamma }(s) \right\| _{\mathcal {C}^{-\nu }(\Lambda _\varepsilon )}^{\lambda _i}\right) \quad \quad \end{aligned}$$

(3.11)

Recall that all the norms appearing the proof so far are norms on the discrete lattice. The same proof of [18, Prop. 3.7] can be used to prove (3.10) and (3.11), provided the same inequalities hold in the discrete setting.

We are going to prove (3.10), (3.11) being essentially the same. Using the duality for discrete Besov spaces proved in Proposition 4.6,

$$\begin{aligned} \left\langle V_{\gamma }^{p-1+j}( s), H_{3-j}(Z_{\gamma }(s),\mathfrak {c}_\gamma ) \right\rangle _{\Lambda _\varepsilon } \le \left\| V_{\gamma }^{p-1+j}( s) \right\| _{\mathcal {B}_{1,1}^{\nu }(\Lambda _\varepsilon )} \left\| H_{3-j}(Z_{\gamma }(s),\mathfrak {c}_\gamma ) \right\| _{\mathcal {C}^{-\nu }(\Lambda _\varepsilon )}. \end{aligned}$$

We then control $\big \Vert V_{\gamma }^{p-1+j}( s) \big \Vert _{\mathcal {B}_{1,1}^{\nu }(\Lambda _\varepsilon )}$ with Lemma 4.7. From (4.7) applied to $f(x) = V_{\gamma }^{p-1+j}(s,x)$

$$\begin{aligned} \left\| f \right\| _{\mathcal {B}_{1,1}^{\nu }(\Lambda _\varepsilon )} \lesssim \left\| f \right\| _{L^1(\Lambda _\varepsilon )}^{1-2\nu } \left( \sum _{x,y \in \Lambda _\varepsilon } \varepsilon ^4 K_\gamma (x-y)\varepsilon ^{-1}\gamma |f(x)-f(y)| \right) ^{2\nu }+ \left\| f \right\| _{L^1(\Lambda _\varepsilon )}. \end{aligned}$$

We will now estimate the term inside the brackets. For p even and $j \in \mathbb {N}$, we have

$$\begin{aligned} \left| a^{p-1+j}-b^{p-1+j}\right| ^{\frac{p-1 }{p-1+j}} \le \left| a^{p-1}-b^{p-1}\right| \end{aligned}$$

the above equation follows easily from the Minkowski inequality if one assumes a and b to have the same sign. If the a and b have different signs, the inequality follows by the fact that p is an even integer and hence the right-hand-side is equal to $|a|^{p-1} + |b|^{p-1}$. Therefore from the generalized Young inequality for $\lambda >0$

$$\begin{aligned}&\left| a^{p-1+j}-b^{p-1+j}\right| \le \left| a^{p-1}-b^{p-1}\right| \left| a^{p-1}-b^{p-1}\right| ^{\frac{j}{p-1}} \\&\qquad \le \lambda \left| a^{p-1}-b^{p-1}\right| |a -b | + \frac{C}{ \lambda } \left( |a|^{p-2 + 2j}+|b|^{p-2 + 2j}\right) , \end{aligned}$$

we have for every $\lambda >0$

$$\begin{aligned}&\sum _{x,y \in \Lambda _\varepsilon } \varepsilon ^2 K_\gamma (x-y)\varepsilon ^{-1}\gamma \left| V_{\gamma }^{p-1+j}(s,x)-V_{\gamma }^{p-1+j}(s,y)\right| \\&\qquad \lesssim \lambda \left\langle V_{\gamma }^{p-1}(s),\Delta _{\gamma }V_{\gamma }(s) \right\rangle + \frac{1}{\lambda }\left\| V_{\gamma }^{p+2+2j}(s) \right\| _{L^1} \end{aligned}$$

and optimizing in $\lambda$ we get (3.10). Finally we can combine (3.9) and (3.10) to conclude the proof.

We remark that the right-hand-side of (3.10) is slightly different from [18] since we have to use $\Delta _{\gamma }$, the discrete (long range) Laplacian, which is a good approximation of the continuous Laplacian only on low frequencies. $\square$

We now turn to the proof of Theorem 2.1, which follows the lines of [18, Corollary 3.10].

Proof of Theorem 2.1

By the monotonicity of $L^q$ norms it is sufficient to prove the statement of Theorem 2.1 for q large enough. In the following proof C will denote a constant possibly changing from line to line. The Gibbs measure $\mathbf {P}^{\gamma }_{\beta ,0}$ is an invariant measure for the Glauber dynamic. Let $T \ge 0$ and consider

$$\begin{aligned} \mathbf {E}^{\gamma }_{\beta ,0} \big [\left\| X_{\gamma } \right\| _{\mathcal {C}^{-\nu }}^q\big ] =\frac{2}{T} \int _{T/2}^T \mathbb {E}_{\beta ,0}^{\gamma }\big [\left\| X_{\gamma }(s) \right\| _{\mathcal {C}^{-\nu }}^q\big ] ds. \end{aligned}$$

(3.12)

From the definition of $V_{\gamma }$ we can write

$$\begin{aligned} \left\| X_{\gamma }(s) \right\| _{\mathcal {C}^{-\nu }} \le \left\| Z_{\gamma }(s) \right\| _{\mathcal {C}^{-\nu }} + \left\| V_{\gamma }(s) \right\| _{\mathcal {C}^{-\nu }} \end{aligned}$$

By (2.19) proven in [12, Proposition 5.4], we have that

$$\begin{aligned} \mathbb {E}_{\beta ,0}^{\gamma }\left[ \sup _{s \in [T/4,T]}\left\| H_j(Z_{\gamma }(s),\mathfrak {c}_\gamma ) \right\| _{\mathcal {C}^{-\nu }}^q\right] \le C(T,q,j) \end{aligned}$$

(3.13)

where the proportionality constant may depend on T and q. From the definition of the discrete Besov norm it follows that (3.13) holds true also when we replace the Besov norm with the discrete Besov Norm. By Proposition 4.4 and Lemma 4.2, for any $q > d/\nu$ and $\kappa >0$ there exists $C(p,\kappa )$

$$\begin{aligned}&\left\| V_{\gamma }(s) \right\| _{\mathcal {C}^{-\nu }} \le \left\| \mathsf {Ext}V_{\gamma }(s) \right\| _{L^q(\mathbb {T}^2)} \nonumber \\&\quad \lesssim \left\| V_{\gamma }(s) \right\| _{L^{q}(\Lambda _\varepsilon )} + \varepsilon ^{- \kappa } \left\| V_{\gamma }(s) \right\| _{L^{2q-2}(\Lambda _\varepsilon )}^{1 - \frac{1}{q}} \Bigg \{ \sum _{\begin{array}{c} |x-y| = \varepsilon \\ x,y \in \Lambda _\varepsilon \end{array}} \varepsilon ^2 (V_{\gamma }(s,y)-V_{\gamma }(s,x))^2 \Bigg \}^{\frac{1}{2q}}\nonumber \\ \end{aligned}$$

(3.14)

where the proportionality constant depends on q and $\kappa$.

In Proposition 3.3, using (3.13) and (3.3) we obtain that

$$\begin{aligned}&\mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\| V_{\gamma }(t) \right\| _{L^q(\Lambda _\varepsilon )}^q\right] +\, C_1 \int _s^t \mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\| V_{\gamma }(r) \right\| _{L^q(\Lambda _\varepsilon )}^q\right] ^{\frac{q+2}{p}} dr \nonumber \\&\qquad + C_1\int _s^t \mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\langle V_{\gamma }^{q-1}(r),(-\Delta _{\gamma })V_{\gamma }(r) \right\rangle _{\Lambda _\varepsilon } \right] dr \nonumber \\&\qquad \le \mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\| V_{\gamma }(s) \right\| _{L^q(\Lambda _\varepsilon )}^q \right] + C(q,T) \end{aligned}$$

(3.15)

From Lemma 4.10, applied to $\mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\| V_{\gamma }(t) \right\| _{L^q(\Lambda _\varepsilon )}^q\right]$ we have that there exists C(q, T) such that for all $T/4 \le s \le t \le T$ we have

$$\begin{aligned} \mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\| V_{\gamma }(t) \right\| _{L^q(\Lambda _\varepsilon )}^q\right] \lesssim C(q,T) \left( |t-s|^{-\frac{q}{2}} \vee 1 \right) . \end{aligned}$$

Let us choose $s=T/4$ and $t \in [T/2,T]$: from the above inequality we have that

$$\begin{aligned} \mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\| V_{\gamma }(t) \right\| _{L^p(\Lambda _\varepsilon )}^p\right] \le C(p,T). \end{aligned}$$

(3.16)

At this point we only need to provide a bound for

$$\begin{aligned} \sum _{\begin{array}{c} |x-y| = \varepsilon \\ x,y \in \Lambda _\varepsilon \end{array}} \varepsilon ^2 (V_{\gamma }(s,y)-V_{\gamma }(s,x))^2 \lesssim \varepsilon ^2 \sum _{\omega \in \Lambda _N} |\omega |^2 |\hat{V}_{\gamma }(s)|^2. \end{aligned}$$

By (4.9), the operator $\Delta _{\gamma }$ approximates the discrete Laplacian only for low frequencies $|\omega |\le \gamma ^{-1}$

$$\begin{aligned} | \widehat{\Delta _{\gamma }V_{\gamma }}(s)(\omega )| = \gamma ^{-2}(1 -\hat{K}_{\gamma }(\omega ))| \hat{V}_{\gamma }(s)(\omega )| \ge c |\omega |^2| \hat{V}_{\gamma }(s)(\omega )|. \end{aligned}$$

On the other hand, for high frequencies $\gamma ^{-1} \le |\omega | \le \gamma ^{-2}$, we have

$$\begin{aligned} | \widehat{\Delta _{\gamma }V_{\gamma }}(s)(\omega )| \ge \gamma ^{-2}(1 -\hat{K}_{\gamma }(\omega ))| \hat{V}_{\gamma }(s)(\omega )| \gtrsim \gamma ^{-2}| \hat{V}_{\gamma }(s)(\omega )|, \end{aligned}$$

hence for all $\omega \in \Lambda _N$,

$$\begin{aligned} |\omega |^2 |\hat{V}_{\gamma }(s)|^2 \le \gamma ^{-2}(|\omega |^2 \wedge \gamma ^{-2}) | \hat{V}_{\gamma }(s)(\omega )|^2 \le \gamma ^{-4}(1 -\hat{K}_{\gamma }(\omega ))| \hat{V}_{\gamma }(s)(\omega )|^2 \end{aligned}$$

and therefore

$$\begin{aligned} \sum _{\omega \in \Lambda _N} |\omega |^2 |\hat{V}_{\gamma }(s)|^2 \le \gamma ^{-2} \left\langle V_{\gamma }(s),(-\Delta _{\gamma })V_{\gamma }(s) \right\rangle _{\Lambda _\varepsilon }. \end{aligned}$$

Equation (3.15) holds for any positive even integer $q \ge 2$ and $T/4 \le s \le t \le T$, if we choose $s=T/2$, $t=T$ and $q=2$ we can conclude that

$$\begin{aligned}&\int _{T/2}^T\sum _{|x-y|=\varepsilon } \epsilon ^2 (V_{\gamma }(s,y)-V_{\gamma }(s,x))^2 ds\nonumber \\&\qquad \le \varepsilon ^2 \gamma ^{-2} \int _{T/2}^T \mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\langle V_{\gamma }(r),(-\Delta _{\gamma })V_{\gamma }(r) \right\rangle _{\Lambda _\varepsilon } \right] dr \le C(T) . \end{aligned}$$

(3.17)

It is sufficient now to control the right-hand-side of (3.12) with (3.14). Again choose $s=T/2$, $t=T$ in (3.15): by (3.13), (3.16) and (3.17)

$$\begin{aligned}&\mathbf {E}^{\gamma }_{\beta ,0} \big [\left\| X_{\gamma } \right\| _{\mathcal {C}^{-\nu }}^q\big ] \le \frac{2}{T} \int _{T/2}^T \mathbb {E}_{\beta ,0}^{\gamma }\big [\left\| Z_{\gamma }(s) \right\| _{\mathcal {C}^{-\nu }}^q\big ] + \mathbb {E}_{\beta ,0}^{\gamma }\big [\left\| V_{\gamma }(s) \right\| _{\mathcal {C}^{-\nu }}^q\big ] ds \\&\quad \le C(T,q,\kappa ) \int _{T/2}^T \mathbb {E}_{\beta ,0}^{\gamma }\big [\left\| Z_{\gamma }(s) \right\| _{\mathcal {C}^{-\nu }}^q\big ] + \mathbb {E}_{\beta ,0}^{\gamma }\left\| V_{\gamma }(s) \right\| _{L^q}^q ds \\&\qquad + \,\,C(T,q,\kappa )\ \mathbb {E}_{\beta ,0}^{\gamma } \int _{T/2}^T \left\| V_{\gamma }(s) \right\| _{L^{2q-1}}^{q-1} \varepsilon ^{-q \kappa } \left\{ \sum _{\begin{array}{c} |x-y| = \varepsilon \\ x,y \in \Lambda _\varepsilon \end{array}} \varepsilon ^2 (V_{\gamma }(s,y)-V_{\gamma }(s,x))^2 \right\} ^{\frac{1}{2}} ds \\&\quad \le C(T,q,\kappa )\left( 1 + \varepsilon ^{-q \kappa } \varepsilon \gamma ^{-1} \left\{ \int _{T/2}^T \mathbb {E}_{\beta ,0}^{\gamma }\left[ \left\langle V_{\gamma }(r),(-\Delta _{\gamma })V_{\gamma }(r) \right\rangle _{\Lambda _\varepsilon } \right] dr\right\} ^{1/2} \right) , \end{aligned}$$

where in the last line we applied the Cauchy-Schwarz inequality. The claim follows by choosing $\kappa >0$ small enough. $\square$

4 Bounds for Discrete Besov Spaces

In this section we collect and prove some results in the context of discrete Besov spaces which are difficult to find. Let us first define the Besov norm on the discrete torus as follows. The definitions and proofs are based upon [12, 13] and [1]. In [1, Proposition 2.10] it is proven the existence of continuous functions $\tilde{\chi },\chi : \mathbb {R}^d \rightarrow \mathbb {R}$ such that

$$\begin{aligned} \mathrm {supp}(\tilde{\chi }) \subseteq B_0(4/3),\qquad \mathrm {supp}(\chi ) \subseteq B_0(8/3) \setminus B_0(3/4) \end{aligned}$$

and such that, setting

$$\begin{aligned} \chi _{-1} {\mathop {=}\limits ^{\tiny \text{ def }}}\tilde{\chi }, \qquad \chi _k(\cdot ) {\mathop {=}\limits ^{\tiny \text{ def }}}\chi (2^{-k}\ \cdot ) \qquad \text { for } (k \ge 0) \end{aligned}$$

one has $\tilde{\chi }(r) + \sum _{k=0}^{\infty } \chi (2^{-k}\ r) = 1$ for all $r \in \mathbb {R}^d$.

For $g : \mathbb {T}^d \rightarrow \mathbb {R}$ define the projection onto the k-th Paley-Littlewood block as

$$\begin{aligned} \delta _k g(x) = 2^{-d}\sum _{\omega \in \mathbb {Z}^d} \chi _k(\omega )\ \widehat{g}(\omega ) e_{\omega }(x) \end{aligned}$$

(4.1)

for $x \in \mathbb {T}^d$ and $k \ge -1$. [The factor $2^{-d}$ is such that $\sum _k \delta _k g = g$, see also (1.6).] Recall that the continuous Besov norm given in (1.7) is then defined in terms of these projections.

We now define a version of the Besov norm for functions defined in the discrete lattice. This is obtained by not only extending the function with the extension operator of Sect. 1.1, but also performing the $L^p$ norm in (1.7) on the discrete space $\Lambda _\varepsilon ^d$ instead of $\mathbb {T}^d$. Let $f : \Lambda _\varepsilon \rightarrow \mathbb {R}$, for $\nu \in \mathbb {R}$, $p,q \in [1,\infty ]$, with $\left\| \cdot \right\| _{\mathcal {B}^{\nu }_{p,q}(\Lambda _\varepsilon ^d)}$ we define

$$\begin{aligned} \left\| f \right\| _{\mathcal {B}^{\nu }_{p,q}(\Lambda _\varepsilon ^d)} {\mathop {=}\limits ^{\tiny \text{ def }}}{\left\{ \begin{array}{ll} \left( \sum _{k \ge -1} 2^{\nu k q} \left\| \delta _k \mathsf {Ext}(f) \right\| _{L^p(\Lambda _\varepsilon ^d)}^q \right) ^{\frac{1}{q}} &{} \text { if }\,\,q < \infty \\ \sup _{k \ge -1} 2^{\nu k} \left\| \delta _k \mathsf {Ext}(f) \right\| _{L^p(\Lambda _\varepsilon ^d)}&{} \text { if }\,\,q = \infty \end{array}\right. } \end{aligned}$$

(4.2)

It is clear, from the definitions of $\mathsf {Ext}(g)$, (4.1) and (4.3), that for $x \in \Lambda _\varepsilon ^d$

$$\begin{aligned} \delta _k\mathsf {Ext}(f)(x) = 2^{-d}\sum _{\omega \in \Lambda _N^d} \chi _k(\omega )\ \widehat{f}(\omega ) e_{\omega }(x) = \eta ^N_{k} *f(x) \qquad \text { for }\,\, x \in \Lambda _\varepsilon \end{aligned}$$

where $\eta ^N_{k}(x)$ is defined, for $k \ge -1$ and $x \in \mathbb {T}^d$, by

$$\begin{aligned} \eta ^N_{k}(x) {\mathop {=}\limits ^{\tiny \text{ def }}}2^{-d} \sum _{\omega \in \Lambda _N^d } \chi _k( \omega ) e_{\omega }(x), \end{aligned}$$

(4.3)

where we abused the notations omitting N from the definition of $\eta ^N_{k}$.

Remark 4.1

We could have easily avoided the definition of a discrete version of the Besov norm. There is only one point where such definition is really needed, and this is in Proposition 3.3 and Lemma 4.7, where we need to control the Besov norm with a combination of discrete $L^p$ norms.

The next lemma is a minor generalisation of [12, Lemma B.6].

Lemma 4.2

For $p \in [1,\infty ]$ and $\kappa > 0$, there exists a constant C such that for all $f: \Lambda _\varepsilon \rightarrow \mathbb {R}$,

$$\begin{aligned} \left\| \mathsf {Ext}(f) \right\| _{L^p(\mathbb {T}^2)}\le & {} C \log ^2(\varepsilon ^{-1}) \left\| f \right\| _{L^p(\Lambda _\varepsilon )} \end{aligned}$$

(4.4)

$$\begin{aligned} \left\| \mathsf {Ext}(f) \right\| _{L^p(\mathbb {T}^2)}&\le C\left\| f \right\| _{L^p(\Lambda _\varepsilon )} \nonumber \\&\quad +\, C \varepsilon ^{ - \kappa } \left\| f \right\| _{L^{2p-2}(\Lambda _\varepsilon )}^{1 - \frac{1}{p}} \left\{ \sum _{\begin{array}{c} |x-y| = \varepsilon \\ x,y \in \Lambda _\varepsilon \end{array}}\varepsilon ^2 (f(y)-f(x))^2 \right\} ^{\frac{1}{2p}}. \end{aligned}$$

(4.5)

The same lemma holds true in any dimension, with a factor $C(d) \log ^d(\varepsilon ^{-1})$ in (4.4).

Proof

We first show (4.4). Recall that from the definition of the extension operator $\mathsf {Ext}f(x) = f(x)$ for $x \in \Lambda _\varepsilon ^2$, and

$$\begin{aligned} \mathsf {Ext}(f)(x)= \sum _{z \in \Lambda _\varepsilon } \varepsilon ^2 f(z) \prod _{j=1,2}\frac{\sin \left( \pi \varepsilon ^{-1} (x_j - z_j)\right) }{2\sin \left( \frac{\pi }{2}(x_j - z_j)\right) } \qquad x \in \mathbb {T}^2. \end{aligned}$$

Using the inequality $\sin (2\varepsilon ^{-1}a)/\sin (a) \lesssim \varepsilon ^{-1} \wedge |a|^{-1}$ we can bound

$$\begin{aligned} |\mathsf {Ext}(f)(x)| \lesssim \sum _{z \in \Lambda _\varepsilon } \varepsilon ^2| f(z) |\prod _{j=1,2} \varepsilon ^{-1} \wedge |z_i-x_i|^{-1}. \end{aligned}$$

For $x \in \mathbb {T}^2$, denote with $[x]_{\varepsilon }$ the closest point to x in $\Lambda _\varepsilon$. We can then rewrite the above inequality as

$$\begin{aligned} |\mathsf {Ext}(f)(x)| \lesssim \sum _{z \in \Lambda _\varepsilon } \varepsilon ^2 |f(z+[x]_{\varepsilon }) |\prod _{j=1,2} \varepsilon ^{-1} \wedge |z_i+ [x]_{\varepsilon ,i}-x_i|^{-1}. \end{aligned}$$

we observe now that if $|z_i| \le \varepsilon$, then $|z_i+ [x]_{\varepsilon ,i}-x_i|^{-1} \gtrsim \varepsilon ^{-1}$, while if $|z_i| > \varepsilon$ we have that $|z_i+ [x]_{\varepsilon ,i}-x_i|^{-1} \lesssim |z_i|^{-1} \lesssim \varepsilon ^{-1}$, hence

$$\begin{aligned} |\mathsf {Ext}(f)(x)| \lesssim \sum _{z \in \Lambda _\varepsilon } \varepsilon ^2 |f(z+[x]_{\varepsilon })| \prod _{j=1,2} \varepsilon ^{-1} \wedge |z_i|^{-1}, \end{aligned}$$

and taking the $L^p(\mathbb {T}^2,dx)$ norm yields

$$\begin{aligned} \Big (\int _{\mathbb {T}^2}|f( [x]_{\varepsilon })|^p dx\Big )^{\frac{1}{p}} \Big ( 1 + 2 \sum _{1 \le k \le \varepsilon ^{-1}} k^{-1}\Big )^2 \lesssim \left\| f \right\| _{L^p(\Lambda _\varepsilon )} \log ^2(\varepsilon ^{-1}), \end{aligned}$$

as claimed. The inequality (4.5) is a consequence of Hölder’s inequality

$$\begin{aligned} \left\| \mathsf {Ext}(f) \right\| _{L^p(\mathbb {T}^2)}^p\lesssim & {} \left\| f \right\| _{L^p(\Lambda _\varepsilon )}^p + \int _{|y|\le \varepsilon /2} |\mathsf {Ext}f(x+y) - f(x)|^p d^2y\\\le & {} \left\| f \right\| _{L^p(\Lambda _\varepsilon )}^p+ \left\| \mathsf {Ext}f \right\| _{L^{2p-2}(\mathbb {T}^2)}^{p-1} \left( \sum _{x \in \Lambda _\varepsilon } \int _{|y|\le \varepsilon /2} |f(x+y) - f(x)|^2 d^2y\right) ^{\frac{1}{2}}\\\le & {} \left\| f \right\| _{L^p(\Lambda _\varepsilon )}^p+ \left\| \mathsf {Ext}f \right\| _{L^{2p-2}(\mathbb {T}^2)}^{p-1} \left( \sum _{x \in \Lambda _\varepsilon } \int _{|y|\le \varepsilon /2} |f(x+y) - f(x)|^2 d^2y\right) ^{\frac{1}{2}}\\\le & {} \left\| f \right\| _{L^p(\Lambda _\varepsilon )}^p+ \varepsilon \left\| \mathsf {Ext}f \right\| _{L^{2p-2}(\mathbb {T}^2)}^{p-1} \left\| \mathsf {Ext}f \right\| _{\dot{H}^1(\mathbb {T}^2)} \end{aligned}$$

where we denoted by $\left\| \mathsf {Ext}f \right\| _{\dot{H}^1(\mathbb {T}^2)}$ the homogeneous Sobolev seminorm. From the definition of the extension operator it is easy to see that

$$\begin{aligned} \left\| \mathsf {Ext}f \right\| _{\dot{H}^1(\mathbb {T}^2)}^2 = \sum _{\omega \in \Lambda _N} |\omega |^2|\hat{f}(\omega )|^2 \lesssim \sum _{\begin{array}{c} |x-y| = \varepsilon \\ x,y \in \Lambda _\varepsilon \end{array}} \varepsilon ^2 \frac{(f(y)-f(x))^2}{\varepsilon ^2} , \end{aligned}$$

and an application of (4.4) yields (4.5). $\square$

Let $d \in \mathbb {N}^+$ and $\Lambda _\varepsilon ^d$ a discretisation of the d-dimensional torus $\mathbb {T}^d=[-1,1]^d$.

Lemma 4.3

Let $\chi : \mathbb {R}^d \rightarrow \mathbb {R}$ be a smooth function with compact support. For every $p \in [0,\infty ]$ we have $\sup _{\lambda \in (0,1)} \lambda ^{d\left( 1-\frac{1}{p}\right) } \left\| \sum _{w \in \Lambda _N} \chi (\lambda w) e_w \right\| _{L^p(\Lambda _\varepsilon ^d)}< \infty .$

The above result is proven in [12, Lemma B.1] for the continuous $L^{(\mathbb {T}^d)}$ norm, the generalisation to $\Lambda _\varepsilon ^d$ follows trivially from the same argument. See also [13] for a proof in the case of a continuous Fourier transform. We quote in the next proposition a useful embedding between Besov and $L^p$ spaces proven, for instance, in [1, Proposition 2.39].

Proposition 4.4

For any $\nu > 0$, and $p \ge \frac{d}{\nu }$ there exists $C>0$

$$\begin{aligned} \left\| f \right\| _{\mathcal {B}^{-\nu }_{\infty ,\infty }(\mathbb {T}^d)} \le C \left\| f \right\| _{L^p(\mathbb {T}^d)} \end{aligned}$$

We also mention some classical estimates for Besov norms restated in case of discrete Besov spaces. The proofs are omitted since they follow closely their continuous counterparts.

Proposition 4.5

(Product estimates for discrete Besov spaces) Let $\beta< 0 < \alpha$ and $p,q \in [1,\infty ]$. There exists $C>0$ such that, uniformly over $\varepsilon \in (0,1]$,

$$\begin{aligned} \left\| f \right\| _{\mathcal {B}^{\beta }_{p,q}(\Lambda _\varepsilon )}\le C\left\| f \right\| _{\mathcal {B}^{\alpha }_{p,q}(\Lambda _\varepsilon )}\left\| g \right\| _{\mathcal {B}^{\beta }_{p,q}(\Lambda _\varepsilon )}. \end{aligned}$$

Proposition 4.6

(Duality for discrete Besov spaces) Let $\alpha \in \mathbb {R}$, $p,q,p',q'\ge 1$ with $\frac{1}{p}+\frac{1}{p'} = \frac{1}{q}+\frac{1}{q'}=1$. There exists $C>0$ such that, uniformly over $\varepsilon \in (0,1]$,

$$\begin{aligned} \left\langle f,g \right\rangle _{\Lambda _\varepsilon } \le C \left\| f \right\| _{\mathcal {B}^{\alpha }_{p,q}(\Lambda _\varepsilon )} \left\| g \right\| _{\mathcal {B}^{-\alpha }_{p',q'}(\Lambda _\varepsilon )}. \end{aligned}$$

The next proposition is the main technical tool of the paper, and it allows to control the discrete Besov norm with the same discrete Laplacian of the dynamic. Recall that, in the case of continuous Besov spaces, for a differentiable function f and $\nu \in (0,1)$, one has [13, Proposition 3.8]

$$\begin{aligned} \left\| f \right\| _{\mathcal {B}^{\nu }_{1,1}} \lesssim \left\| f \right\| _{L^1} + \left\| f \right\| _{L^1}^{1-\nu }\left\| \nabla f \right\| _{L^1}^{\nu }. \end{aligned}$$

(4.6)

We have the following analogue of this result.

Lemma 4.7

For $f: \Lambda _\varepsilon \rightarrow \mathbb {R}$ and $\nu \in (0,1/2)$

$$\begin{aligned} \left\| f \right\| _{\mathcal {B}_{1,1}^{\nu }(\Lambda _\varepsilon )} \lesssim \left\| f \right\| _{L^1(\Lambda _\varepsilon )}^{1-2\nu } \left( \sum \limits _{x,y \in \Lambda _\varepsilon } \varepsilon ^4 K_\gamma (x-y)\varepsilon ^{-1}\gamma |f(x)-f(y)| \right) ^{2\nu }+ \left\| f \right\| _{L^1(\Lambda _\varepsilon )}\nonumber \\ \end{aligned}$$

(4.7)

where the constant is independent of $\varepsilon$ or f.

Remark 4.8

Here we write $x-y$ for the shortest element in the corresponding equivalence class (viewing $\Lambda _\varepsilon$ as a quotient of $\mathbb {Z}_\varepsilon ^2$ with $\mathbb {Z}_\varepsilon = \varepsilon \mathbb {Z}$). In cases where this might be ambiguous one has $K_\gamma (x-y)=0$ for all possible interpretations anyway. Another equivalent interpretation is that one of the variables runs over $\Lambda _\varepsilon$ and the other one runs over all of $\mathbb {Z}_\varepsilon$, f being identified with its periodic continuation.

Compare (4.7) with (4.6). The factor 2 in front of $\nu$ depends on the scale at which $\Delta _{\gamma }$ changes its behaviour, and this is not the best result that is possible to obtain.

Proof

Rewrite the definition of $\left\| f \right\| _{\mathcal {B}^{\nu }_{1,1}(\Lambda _\varepsilon )}$ in (4.2) as

$$\begin{aligned} \left\| f \right\| _{\mathcal {B}^{\nu }_{1,1}(\Lambda _\varepsilon )} = \sum _{k \ge -1} 2^{\nu k} \left\| \eta ^N_k *f \right\| _{L^1(\Lambda _\varepsilon )} \end{aligned}$$

(4.8)

where $\eta ^N_k$ are the projections on the Paley-Littlewood blocks defined in (4.3). In the discrete case the summation over k extends up to a multiple of $\log (\varepsilon ^{-1})$. In the proof, since there is no possibility of confusion, we will use $L^p$ instead of $L^p(\Lambda _\varepsilon )$. We will divide the sum into

$$\begin{aligned} \sum _{-1 \le k \le L} 2^{\nu k} \left\| \eta ^N_k *f \right\| _{L^1} + \sum _{L < k \le -\log _2(\varepsilon )} 2^{\nu k} \left\| \eta ^N_k *f \right\| _{L^1} \end{aligned}$$

where L will be chosen later. We bound the first part with

$$\begin{aligned} \sum _{-1 \le k \le L} 2^{\nu k} \left\| \eta ^N_k *f \right\| _{L^1} \le \sum _{-1 \le k \le L} 2^{\nu k} \sup _{k'\le L} \left\| \eta ^N_{k'} \right\| _{L^1} \left\| f \right\| _{L^1} \lesssim 2^{\nu L} \left\| f \right\| _{L^1}. \end{aligned}$$

(4.9)

In order to control the second summation we will now prove, for $k \ge 0$, the inequality

$$\begin{aligned} \left\| \eta ^N_k *f \right\| _{L^1} \lesssim \left( 2^{-k} \vee \varepsilon \gamma ^{-1} \right) \sum _{x,y \in \Lambda _\varepsilon } \varepsilon ^4 K_\gamma (x-y) \frac{|f(y) -f(x)|}{\varepsilon \gamma ^{-1}}. \end{aligned}$$

(4.10)

If $k \ge 0$ the projection kernel $\eta ^N_k$ has mean zero and therefore

$$\begin{aligned} \left\| \eta ^N_k *f \right\| _{L^1} = \sum _{x \in \Lambda _\varepsilon } \varepsilon ^2 \left| \sum _{y \in \Lambda _\varepsilon } \varepsilon ^2 \eta ^N_k(-y) \Big (f(x+y) - f(x)\Big )\right| . \end{aligned}$$

At this point the treatment differs from the proof of [13, Proposition 3.8], because of the particular form of the Laplacian. The definition of $K_\gamma$ (in particular the continuity of $\mathfrak {K}$) implies that there exists $b_0 > 0$ such that

$$\begin{aligned} \inf _{ |w|\le b_0 \varepsilon \gamma ^{-1}} \sum _{ z \in \mathbb {Z}_\varepsilon }\varepsilon ^2 \big (K_\gamma (z) \wedge K_\gamma (w-z)\big ) \ge 1/2. \end{aligned}$$

If $|y| \le b_0 \varepsilon \gamma ^{-1}$, then

$$\begin{aligned}&\Big |f(x+y) - f(x)\Big | \le 2 \Big ( \sum _{ z \in \mathbb {Z}_\varepsilon }\varepsilon ^2 \big (K_\gamma (z) \wedge K_\gamma (y-z)\big ) \Big ) \Big |f(x+y) - f(x)\Big |\\&\quad \le 2\varepsilon ^2 \sum _{ z \in \mathbb {Z}_\varepsilon } K_\gamma (y-z) \Big |f(x+y) - f(x+z)\Big | + K_\gamma (z) \Big |f(x+z) - f(x)\Big |. \end{aligned}$$

If $|y| \ge b_0 \varepsilon \gamma ^{-1}$ on the other hand, then there exists a path $\{y_0,y_1,\dots ,y_n\}$ in $\mathbb {Z}_\varepsilon$ of length n proportional to $|y|\gamma \varepsilon ^{-1}$ connecting $y_0 = 0$ with $y_n = y$ and such that $|y_{j+1}-y_j| \le b_0 \varepsilon \gamma ^{-1}$ for $j= 0,\dots ,n-1$. We can then apply the above inequality to every step of the path. Combining these bounds, we obtain

$$\begin{aligned} \left\| \eta ^N_k *f \right\| _{L^1} \lesssim \sum _{y \in \Lambda _\varepsilon } \varepsilon ^2 |\eta ^N_k(-y)| \{ |y|\gamma \varepsilon ^{-1} \vee 1 \} \sum _{x \in \Lambda _\varepsilon , z \in \mathbb {Z}_\varepsilon } \varepsilon ^4 K_\gamma (z ) |f(x+z) -f(x)|,\nonumber \\ \end{aligned}$$

(4.11)

and (4.10) follows from the fact that

$$\begin{aligned} \sum _{y \in \Lambda _\varepsilon } \varepsilon ^2 |\eta ^N_k( y)| \{ |y|\varepsilon ^{-1} \gamma \vee \gamma \} \lesssim \left\| \eta ^N_k \right\| _{L^1} + \varepsilon ^{-1} \gamma 2^{-k} \sum _{y \in \Lambda _\varepsilon }2^k|y||\eta ^N_k(y)| \lesssim 1 \vee \varepsilon ^{-1} \gamma 2^{-k}. \end{aligned}$$

Summing over k yields

$$\begin{aligned}&\sum _{L< k \le \log _2(\varepsilon ^{-1})} 2^{\nu k} \left\| \eta ^N_k *f \right\| _{L^1} \\&\qquad \qquad \qquad \lesssim \sum _{L < k \le \log _2(\varepsilon ^{-1})} 2^{\nu k}\{ \varepsilon \gamma ^{-1} \vee 2^{-k}\} \sum _{x \in \Lambda _\varepsilon , z\in \mathbb {Z}_\varepsilon } {\varepsilon ^4 K_\gamma (z ) \over \varepsilon \gamma ^{-1}} |f(x+z) -f(x)|. \end{aligned}$$

At this point we use the fact that $\varepsilon \gamma ^{-1} \vee 2^{-k} \le 2^{-\frac{k}{2}}$ for $k \le -\log _2(\varepsilon ) = -2\log _2(\varepsilon \gamma ^{-1})$ and, recalling (4.8) and (4.9), we obtain

$$\begin{aligned} \left\| f \right\| _{\mathcal {B}^{\nu }_{1,1}(\Lambda _\varepsilon )} \lesssim 2^{\nu L} \left\| f \right\| _{L^1} + 2^{\left( \nu -\frac{1}{2}\right) L}\sum _{x \in \Lambda _\varepsilon , z\in \mathbb {Z}_\varepsilon } {\varepsilon ^3\gamma K_\gamma (z )} |f(x+z) -f(x)|. \end{aligned}$$

The claim now follows by optimising this expression over L. (The second term in (4.7) comes from the fact that we had to impose $L > 1$.) $\square$

The next proposition quantifies the decay in the Fourier space of the kernel $K_\gamma$ used in the article. The proof is given in [12, Lemma 8.2]

Proposition 4.9

(Estimates on the kernel) For $\omega \in \Lambda _N$ and $\gamma$ small enough the following inequalities hold

• There exists a positive constant $C>0$ such that,

  $$\begin{aligned} |\hat{K}_{\gamma }(\omega )|\le 1 \wedge \frac{\gamma ^{-2}}{|\omega |^2} \end{aligned}$$

• There exists a positive constant $c>0$ such that, for $|\omega | \ge \gamma ^{-1}$

  $$\begin{aligned} 1 - \hat{K}_{\gamma }(\omega ) \ge c \left( |\gamma \omega |^2\wedge 1\right) \end{aligned}$$

For completeness, we state the following simple but crucial comparison test which can be found in this specific form in [18].

Lemma 4.10

(Comparison test) Let $\lambda > 1$ and $f:[0,T] \rightarrow \mathbb {R}^+$ differentiable satisfying for $t \in [0,T]$

$$\begin{aligned} f'(t) + 2 c_1 \left( f(t) \right) ^{\lambda }\le c_2. \end{aligned}$$

Then for $t \in [0,T]$

$$\begin{aligned} f(t) \le \frac{f(0)}{\left( 1 + c_1 (\lambda -1) t f(0)^{\lambda -1}\right) ^{\frac{1}{\lambda -1}}} \vee \left( \frac{c_2}{c_1}\right) ^{\frac{1}{\lambda }} \le \left( c_1(\lambda -1) t\right) ^{-\frac{1}{\lambda -1}} \vee \left( \frac{c_2}{c_1}\right) ^{\frac{1}{\lambda }}. \end{aligned}$$