Four-Dimensional Weakly Self-avoiding Walk with Contact Self-attraction

Abstract

We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on $\mathbb {Z}^4$, for sufficiently small attraction. We prove that the susceptibility and correlation length of order p (for any $p>0$) have logarithmic corrections to mean field scaling, and that the critical two-point function is asymptotic to a multiple of $|x|^{-2}$. This shows that small contact self-attraction results in the same critical behaviour as no contact self-attraction; a collapse transition is predicted for larger self-attraction. The proof uses a supersymmetric representation of the two-point function, and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self-avoiding walk, without self-attraction.

1 The Model and Main Result

The self-avoiding walk is a basic model for a linear polymer chain in a good solution. The repulsive self-avoidance constraint models the excluded volume effect of the polymer. In a poor solution, the polymer tends to avoid contact with the solution by instead making contact with itself. This is modelled by a self-attraction favouring nearest-neighbour contacts. The self-avoiding walk is already a notoriously difficult problem, and the combination of these two competing tendencies creates additional difficulties and an interesting phase diagram.

In this paper, we consider a continuous-time version of the weakly self-avoiding walk with nearest-neighbour contact self-attraction on $\mathbb {Z}^4$. When both the self-avoidance and self-attraction are sufficiently weak, we prove that the susceptibility and finite-order correlation length have logarithmic corrections to mean field scaling with exponents $\frac{1}{4}$ and $\frac{1}{8}$ for the logarithm, respectively, and that the critical two-point function is asymptotic to a multiple of $|x|^{-2}$ as $|x| \rightarrow \infty$.

1.1 Definition of the Model

For $d>0$, let X denote the continuous-time simple random walk on ${{\mathbb {Z}}}^{d}$. That is, X is the stochastic process with right-continuous sample paths that takes its steps at the times of the events of a rate-2d Poisson process. A step is independent both of the Poisson process and of all other steps, and is taken uniformly at random to one of the 2d nearest neighbours of the current position. The field of local times $L_T = (L_T^x)_{x\in \mathbb {Z}^d}$ of X, up to time $T \ge 0$, is defined by

$$\begin{aligned} L_T^x = \int _0^T \mathbbm {1}_{X_t = x} \; dt . \end{aligned}$$

(1.1)

The self-intersection local time and self-contact local time of X up to time T are the random variables defined, respectively, by

$$\begin{aligned} I_T&= \sum _{x \in \mathbb {Z}^d} (L_T^x)^2 = \int _0^T ds \int _0^T dt \; \mathbbm {1}_{X_{s}=X_{t}} ,\end{aligned}$$

(1.2)

$$\begin{aligned} C_T&= \sum _{x \in \mathbb {Z}^d}\sum _{e\in \mathcal {U}} L_T^xL_T^{x+e} = \int _0^T ds \int _0^T dt \; \mathbbm {1}_{X_{s} \sim X_{t}} , \end{aligned}$$

(1.3)

where $\mathcal {U}$ is the set of unit vectors in ${{\mathbb {Z}}}^{d}$ and $y\sim x$ indicates that x and y are nearest neighbours.

Given $\beta > 0$ and $\gamma \in \mathbb {R}$, we define

$$\begin{aligned} U_{\beta ,\gamma }(f) = \beta \sum _{x\in {{\mathbb {Z}}}^{d}} f_x^2 - \frac{\gamma }{2d} \sum _{x\in {{\mathbb {Z}}}^{d}} \sum _{e\in \mathcal {U}} f_x f_{x+e} \end{aligned}$$

(1.4)

for $f:{{\mathbb {Z}}}^{d}\rightarrow \mathbb {R}$ with $f_x = 0$ for all but finitely many x. The potential that associates an energy to X in terms of its field of local times is given by

$$\begin{aligned} U_{\beta ,\gamma ,T} = U_{\beta ,\gamma }(L_T) = \beta I_T - \frac{\gamma }{2d} C_T . \end{aligned}$$

(1.5)

The energy $U_{\beta ,\gamma ,T}$ increases with the self-intersection local time, corresponding to weak self-avoidance. For $\gamma >0$, the energy decreases when the self-contact local time increases, corresponding to a contact self-attraction. For $\gamma <0$, the contact term is repulsive. We are primarily interested in the case of positive $\gamma$, but our results hold also for small negative $\gamma$.

Figure 1 shows a sample path X and indicates one self-intersection and four self-contacts. Although $I_T$ also receives contributions from the time the walk spends at each vertex, and $C_T$ receives a contribution from each step, these contributions have the same distribution for all walks taking the same number of steps. The depicted intersections and contacts are the meaningful ones.

Fig. 1

Polymer with one self-intersection and four self-contacts shown

Let $a,b \in {{\mathbb {Z}}}^{d}$, and let $E_a$ denote the expectation for the process X started at $X(0)=a$. We define

$$\begin{aligned} c_T = E_a\left( e^{-U_{\beta ,\gamma ,T}}\right) , \quad c_T(a,b) = E_a\left( e^{-U_{\beta ,\gamma ,T}}\mathbbm {1}_{X_T = b}\right) . \end{aligned}$$

(1.6)

By translation-invariance, $c_T$ does not depend on a. For $\nu \in \mathbb {R}$, the two-point function is defined by

$$\begin{aligned} G_{\beta ,\gamma ,\nu }(a,b)&= \int _0^\infty c_T(a,b) e^{-\nu T} \; dT, \end{aligned}$$

(1.7)

and the susceptibility is defined by

$$\begin{aligned} \chi (\beta , \gamma , \nu ) = \int _0^\infty c_T e^{-\nu T} \; dT = \sum _{x\in {{\mathbb {Z}}}^{d}} G_{\beta ,\gamma ,\nu }(0, x) . \end{aligned}$$

(1.8)

For $p>0$, we define the correlation length of order p by

$$\begin{aligned} \xi _p(\beta ,\gamma ,\nu ) = \left( \frac{1}{\chi (\beta , \gamma , \nu )} \sum _{x\in {{\mathbb {Z}}}^{d}} |x|^p G_{\beta ,\gamma ,\nu }(0, x) \right) ^{1/p}. \end{aligned}$$

(1.9)

In (1.7)–(1.9), self-intersections are suppressed by the factor $\exp [-\beta I_T]$, whereas nearest-neighbour contacts are encouraged by the factor $\exp [\frac{\gamma }{2d}C_T]$ when $\gamma > 0$.

1.2 The Critical Point

The right-hand sides of (1.7)–(1.8) are positive or $+\infty$, and monotone decreasing in $\nu$ by definition. We define the critical point

$$\begin{aligned} \nu _c(\beta , \gamma ) = \inf \{ \nu \in \mathbb {R}: \chi (\beta , \gamma , \nu ) < \infty \} . \end{aligned}$$

(1.10)

For $\gamma =0$, an elementary argument shows that $\nu _c(\beta ,0) > -\infty$ for all dimensions, and furthermore that $\nu _c(\beta , 0) \in [ -2 \beta (-\Delta _{{{\mathbb {Z}}}^{d}}^{-1})_{0,0}, 0]$ for dimensions $d>2$; see [3, Lemma A.1]. Here, $\Delta _{{{\mathbb {Z}}}^{d}}$ is the Laplacian on ${{\mathbb {Z}}}^{d}$, i.e., the ${{\mathbb {Z}}}^{d}\times {{\mathbb {Z}}}^{d}$ matrix with entries

$$\begin{aligned} (\Delta _{{{\mathbb {Z}}}^{d}})_{x, y} = \mathbbm {1}_{x\sim y} - 2 d \mathbbm {1}_{x=y}. \end{aligned}$$

(1.11)

An equivalent definition is as follows: given a unit vector $e \in {{\mathbb {Z}}}^{d}$, the discrete gradient is defined by $\nabla ^e f_x = f_{x+e}-f_x$, and the Laplacian is $\Delta _{{{\mathbb {Z}}}^{d}} f_{x} = \sum _{e \in \mathcal {U}} \nabla ^e f_x = -\frac{1}{2}\sum _{e \in \mathcal {U}}\nabla ^{-e} \nabla ^{e} f_x$.

To estimate the critical point when $\gamma \ne 0$, we also define

$$\begin{aligned} |\nabla f_x|^2&= \sum _{e\in \mathcal {U}} |\nabla ^e f_x|^2, \quad |\nabla f|^2 = \sum _{x\in {{\mathbb {Z}}}^{d}} |\nabla f_x|^2. \end{aligned}$$

(1.12)

From the definition, we see that

$$\begin{aligned} \sum _{x\in {{\mathbb {Z}}}^{d}} f_x \Delta _{{{\mathbb {Z}}}^{d}} f_x = -\frac{1}{2} |\nabla f|^2. \end{aligned}$$

(1.13)

It follows that

$$\begin{aligned} \sum _{x\in {{\mathbb {Z}}}^{d}} \sum _{e\in \mathcal {U}} f_x f_{x+e} = 2 d \sum _{x\in {{\mathbb {Z}}}^{d}} f_x^2 + \sum _{x\in {{\mathbb {Z}}}^{d}} f_x \Delta _{{{\mathbb {Z}}}^{d}} f_x = 2 d \sum _{x\in {{\mathbb {Z}}}^{d}} f_x^2 - \frac{1}{2} \sum _{x\in {{\mathbb {Z}}}^{d}} |\nabla f_x|^2\nonumber \\ \end{aligned}$$

(1.14)

and so we get the useful representation:

$$\begin{aligned} U_{\beta ,\gamma }(f) = (\beta - \gamma ) \sum _{x\in {{\mathbb {Z}}}^{d}} f_x^2 + \frac{\gamma }{4d} \sum _{x\in {{\mathbb {Z}}}^{d}} \sum _{e\in \mathcal {U}} |\nabla ^e f_x|^2. \end{aligned}$$

(1.15)

In particular,

$$\begin{aligned} U_{\beta ,\gamma ,T} = (\beta - \gamma ) I_T + \frac{\gamma }{4d} |\nabla L_T|^2 . \end{aligned}$$

(1.16)

A version of (1.16) can be found in [21].

Lemma 1.1

Let $d >0$. Let $\beta >0$ and $|\gamma | < \beta$. If $\gamma \ge 0$ then $\nu _c(\beta , \gamma ) \in [\nu _c(\beta , 0),\nu _c(\beta -\gamma , 0)]$. If $\gamma < 0$ then $\nu _c(\beta ,\gamma ) \in [\nu _c(\beta -\gamma ,0),\nu _c(\beta ,0)]$.

Proof

Suppose first that $\gamma \in [0,\beta )$. It follows from (1.5) and (1.16) that

$$\begin{aligned} U_{\beta -\gamma ,0,T} \le U_{\beta ,\gamma ,T} \le U_{\beta ,0,T}, \end{aligned}$$

(1.17)

which implies the desired estimates for $\nu _c(\beta ,\gamma )$.

On the other hand, if $\gamma \in (-\beta , 0)$ then the inequalities are reversed and now

$$\begin{aligned} U_{\beta ,0,T} \le U_{\beta ,\gamma ,T} \le U_{\beta -\gamma ,0,T}, \end{aligned}$$

(1.18)

which again implies the desired result. $\square$

1.3 The Main Result

Our main result is the following theorem. It shows that in dimension $d = 4$, for sufficiently small $\beta$ and $\gamma$, the two-point function (1.7) has the same asymptotic decay, to leading order, as the simple random walk two-point function. It also shows that the susceptibility and correlation length of order p exhibit logarithmic corrections to mean-field behaviour. These results were all proved for $\gamma =0$ in [2, 3, 6], and we extend them here to small nonzero $\gamma$.

We denote the Laplacian on ${{\mathbb {R}}}^{d}$ by $\Delta _{{{\mathbb {R}}}^{d}}$ and define a constant $\mathsf{c}_p$ by

$$\begin{aligned} \mathsf{c}_p^p = \int _{\mathbb {R}^4} |x|^p (-\Delta _{\mathbb {R}^4} + 1)^{-1}_{0x} \; dx. \end{aligned}$$

(1.19)

Theorem 1.2

Let $d = 4$. There exist $\beta _* > 0$ and a positive function $\gamma _* : (0, \beta _*) \rightarrow \mathbb {R}$ such that whenever $0< \beta < \beta _*$ and $|\gamma | < \gamma _*(\beta )$, there are constants $A_{\beta ,\gamma }$ and $B_{\beta ,\gamma }$ such that the following hold:

• (i) The critical two-point function decays as

  $$\begin{aligned} G_{\beta ,\gamma ,\nu _c}(0, x) = A_{\beta ,\gamma } |x|^{-2} \left( 1 + O\left( \frac{1}{\log |x|}\right) \right) \quad \text {as} |x|\rightarrow \infty , \end{aligned}$$

  (1.20)

  with $A_{\beta ,\gamma } = \frac{1}{4 \pi ^2} (1 + O(\beta ))$ as $\beta \downarrow 0$.

• (ii) The susceptibility diverges as

  $$\begin{aligned} \chi (\beta , \gamma , \nu _c + \varepsilon ) \sim B_{\beta ,\gamma } \varepsilon ^{-1} \left( \log \varepsilon ^{-1}\right) ^{1/4}, \quad \varepsilon \downarrow 0, \end{aligned}$$

  (1.21)

  with $B_{\beta ,\gamma } = \left( \frac{\beta }{2\pi ^2}\right) ^{1/4} (1 + O(\beta ))$ as $\beta \downarrow 0$.

• (iii) For any $p >0$, if $\beta _*$ is chosen small depending on p, then the correlation length of order p diverges as

  $$\begin{aligned} \xi _p(\beta , \gamma , \nu _c + \varepsilon ) \sim B_{\beta ,\gamma }^{1/2} \mathsf{c}_p \varepsilon ^{-1/2} \left( \log \varepsilon ^{-1}\right) ^{1/8}, \quad \varepsilon \downarrow 0. \end{aligned}$$

  (1.22)

Our method of proof extends the renormalisation group argument, used for $\gamma =0$ in [2, 3, 6, 27], to small nonzero $\gamma$. In Sect. 2, as a first step, we show that the two-point function can be approximated by a finite-volume one. The finite-volume two-point function has a supersymmetric integral representation [7, 9, 10], which we state in Sect. 3. These two sections do not involve the renormalisation group. The application of the renormalisation group method requires the following new ingredients: (i) in Sect. 4, we provide estimates on the contact attraction which show that it is compatible with the renormalisation group method developed in [13, 14], and also with the dynamical systems theorem proved in [5], (ii) in Sect. 5, we use the implicit function theorem to extend the identification of the critical point from $\gamma =0$ to $\gamma \ne 0$, and complete the proof of Theorem 1.2.

In fact, we demonstrate that after the introduction of $\gamma$, chosen sufficiently small depending on g, we may use the the same renormalisation group flow of the remaining coupling constants as in the case $\gamma =0$, to second order in these coupling constants. Thus, since the critical exponents are determined by this second-order flow, they are independent of small $\gamma$, and take the same values as for $\gamma =0$. The critical value $\nu _c(\beta ,\gamma )$ does, however, depend on $\gamma$.

1.4 Critical Exponents and Polymer Collapse

It has been known for decades that self-avoiding walk obeys mean-field behaviour in dimensions $d \ge 5$. In particular, a version of Theorem 1.2 for the strictly self-avoiding walk (in discrete time with $\beta =\infty$ and $\gamma =0$) in dimensions $d \ge 5$ was proved in [18, 19] using the lace expansion [15]. In its original applications, the lace expansion relied on the purely repulsive nature of the self-avoidance interaction. Models incorporating attraction require new ideas. For a particular model with self-attraction and specially chosen exponentially decaying step weights, the lace expansion was used in [28] to prove that, for $d \ge 5$, the mean-square displacement grows diffusively for small attraction. More recently [20], the lace expansion has been applied in situations where repulsion occurs only in an average sense. In a further development [17], the lace expansion has been applied to a model of strictly self-avoiding walk with a self-attraction that rewards visits to adjacent parallel edges, to prove that sufficiently weak self-attraction does not affect the critical behaviour in dimensions $d \ge 5$. The results of [17, 28] for $d \ge 5$ complement our results for $d=4$, via entirely different methods.

Fig. 2

The predicted phase diagram for $d \ge 2$

Assuming it exists, the critical exponent $\bar{\nu }$ for the mean-square displacement is defined by

$$\begin{aligned} \langle |X(T)|^2 \rangle = \frac{1}{c_T} E_0\left( |X(T)|^2 e^{-U_{\beta ,\gamma ,T}}\right) \approx T^{2\bar{\nu }} , \end{aligned}$$

(1.23)

possibly with logarithmic corrections. A general tenet of the theory of critical phenomena asserts that other natural length scales, such as the correlation length of order p, are also governed by the exponent $\bar{\nu }$. A typical argument for this, found in physics textbooks, goes as follows. It is predicted that $c_T \approx e^{\nu _c T}T^{{\bar{\gamma }} -1}$, where ${\bar{\gamma }}$ is the critical exponent for the susceptibility [for $d=4$, ${\bar{\gamma }}=1$ with a logarithmic correction, by (1.21)]. By definition,

$$\begin{aligned} \xi _2(\beta , \gamma , \nu )^2 = \frac{\int _0^\infty \langle |X(T)|^2 \rangle c_T e^{-\nu T} \; dT}{\int _0^\infty c_T e^{-\nu T} \; dT} . \end{aligned}$$

(1.24)

In (1.24), we substitute the asymptotic formula for $c_T$, as well as (1.23), to obtain

$$\begin{aligned} \xi _2(\beta , \gamma , \nu ) \approx (\nu - \nu _c)^{-\bar{\nu }} \quad \text {as}\; \nu \downarrow \nu _c, \end{aligned}$$

(1.25)

with the same exponent $\bar{\nu }$ as in (1.23).

The weakly self-avoiding walk with contact self-attraction is a model for polymer collapse. Polymer collapse corresponds to a discontinuous reduction in the exponent $\bar{\nu }$ as $\gamma$ increases. A summary of results, predictions, and references can be found in [23, Chapter 6]. See also [24, 29]. The predicted phase diagram for dimensions $d \ge 2$ is shown in Figure 2. The predicted values of the exponent at the $\theta$-transition are $\bar{\nu }_\theta = \frac{4}{7}$ for $d=2$ and $\bar{\nu }_\theta = \frac{1}{2}$ for $d \ge 3$ [23]. The phase labelled $\bar{\nu }_\mathrm{SAW}$ takes its name from the fact that in this phase the model with attraction is predicted to be in the same universality class as the self-avoiding walk. The predicted values of the exponent $\bar{\nu }_\mathrm{SAW}$ for the self-avoiding walk are respectively $\frac{3}{4}$, 0.587597(7), $\frac{1}{2}$ for $d=2,3,4$ (with a logarithmic correction for $d=4$; see [16] for $d=3$), and it has been proved that $\bar{\nu }_\mathrm{SAW}=\frac{1}{2}$ for $d \ge 5$ [15, 19]. It remains a major challenge in the mathematical theory of polymers to prove the full validity of the phase diagram in all dimensions $d\ge 2$. Very recently, the existence of a collapse transition (a singularity of the free energy) has been proven for a two-dimensional prudent self-avoiding walk with contact self-attraction [26].

For $\gamma \ge 0$, the significance of the restriction $\gamma <\beta$ has been noted for a closely related discrete-time model, for which it is proved that for $\gamma > \beta$ the walk is in a compact phase in the sense that $\bar{\nu }= 0$, whereas for $\gamma < \beta$ it is the case that $\bar{\nu }\ge 1/d$ [21]. In the compact phase, the discrete-time model obeys the analogue of $c_T \approx e^{kT^2}$ with $k>0$, so $\chi (\beta ,\gamma ,\nu )=\infty$ for all $\nu \in \mathbb {R}$ and $\nu _c = +\infty$. For the 1-dimensional case, the behaviour for the transition line $\gamma =\beta$ has been studied in [22].

The axis $\gamma =0$ corresponds to the weakly self-avoiding walk which is well understood in dimensions $d \ge 5$ [15, 19], and in dimension 4 [2, 3, 6]. Theorem 1.2 extends the results of [2, 3, 6] for dimension $d = 4$ to the region bounded by the dashed line. Our results show that for $d=4$ there is no polymer collapse for small contact self-attraction, in the sense that the critical behaviour remains the same with small contact attraction as with no contact attraction. In particular, Theorem 1.2(iii) shows that, in the sense of (1.25), when $\gamma$ is small, $\bar{\nu }= \frac{1}{2}$ holds with a logarithmic correction.

2 Finite-Volume Approximation

The first step in the proof of Theorem 1.2 is an approximation of $G_{\beta ,\gamma ,\nu }(a,b)$ and $\chi (\beta , \gamma , \nu )$ by finite-volume analogues of these quantities. This is the content of Proposition 2.2.

Before proving the proposition, we require some preliminaries. Let $P^n$ be the projection of ${{\mathbb {Z}}}^{d}$ onto the discrete torus of side n, which we denote $\mathbb {Z}_n^d$. Then $P^n$ has a natural action on the path space $({{\mathbb {Z}}}^{d})^{[0,\infty )}$. We let $X^n = P^n(X)$ be the projection of X and note that $X^n$ is a simple random walk on $\mathbb {Z}^d_n$.

We call $h = (h_x)_{x\in {{\mathbb {Z}}}^{d}}$ a field of path functionals if $h_x : ({{\mathbb {Z}}}^{d})^{[0,\infty )} \rightarrow \mathbb {R}$ is a function on continuous-time paths for each $x \in {{\mathbb {Z}}}^{d}$; a simple example is given by the local time functional. We assume that the random field $h(X) = (h_x(X))_{x\in {{\mathbb {Z}}}^{d}}$ has finite support almost surely, i.e., with probability 1, $h_x(X) = 0$ for all but finitely many x. Denote by $h(X^n)$ the corresponding random field for $X^n$, i.e., for $x \in \mathbb {Z}_n^d$,

$$\begin{aligned} h_x(X^n) = \sum _{y\in {{\mathbb {Z}}}^{d}} h_{x+ny}(X). \end{aligned}$$

(2.1)

Given a positive integer k, we define $Q_k \subset \mathbb {Z}^d$ by $Q_k = \{y \in \mathbb {Z}^d : 0 \le y_i < k, \; i=1,\ldots ,d\}$. Then, for integers $n,k \ge 1$,

$$\begin{aligned} \sum _{y \in Q_k} h_{x+ny}(X^{kn}) = \sum _{y \in Q_k} \sum _{z\in {{\mathbb {Z}}}^{d}} h_{x+ny+knz}(X) = \sum _{y\in {{\mathbb {Z}}}^{d}} h_{x+ny}(X) = h_x(X^n), \end{aligned}$$

(2.2)

and it follows by summation over $x \in \mathbb {Z}^d_n$ that

$$\begin{aligned} \sum _{x\in \mathbb {Z}^d_{kn}} h_x(X^{kn}) = \sum _{x\in \mathbb {Z}^d_n} h_x(X^n). \end{aligned}$$

(2.3)

Lemma 2.1

Let $n,k \ge 1$ and let f and g be nonnegative fields of path functionals with finite support almost surely. Then

$$\begin{aligned} \sum _{x\in \mathbb {Z}^d_{kn}} f_x(X^{kn}) g_x(X^{kn}) \le \sum _{x\in \mathbb {Z}^d_n} f_x(X^n) g_x(X^n). \end{aligned}$$

(2.4)

Proof

By (2.3) and (2.2),

$$\begin{aligned} \sum _{x\in \mathbb {Z}_{kn}^d} f_x(X^{kn}) g_x(X^{kn}) = \sum _{x\in \mathbb {Z}_n^d} \sum _{y \in Q_k} f_{x+ny}(X^{kn}) g_{x+ny}(X^{kn}). \end{aligned}$$

(2.5)

By nonnegativity and two more applications of (2.2),

$$\begin{aligned} \sum _{x\in \mathbb {Z}_n^d} \sum _{y \in Q_k} f_{x+ny}(X^{kn}) g_{x+ny}(X^{kn})&\le \sum _{x\in \mathbb {Z}_n^d} \left( \sum _{y \in Q_k} f_{x+ny}(X^{kn})\right) \left( \sum _{y \in Q_k} g_{x+ny}(X^{kn})\right) \nonumber \\&= \sum _{x\in \mathbb {Z}_n^d} f_x(X^n) g_x(X^n). \end{aligned}$$

(2.6)

This completes the proof. $\square$

Fix $L \ge 2$ and $N \ge 1$. We write $\Lambda _N$ for the torus $\mathbb {Z}^d_n$ with $n=L^N$. Thus, $X^{L^N}$ is the simple random walk on $\Lambda _N$. For $F_T = F_T(X)$ any one of the functions $L_T^x,I_T,C_T$ of X defined in (1.1)–(1.3), we write $F_{N,T} = F_T(X^{L^N})$. For instance, with $n=L^N$,

$$\begin{aligned} L^x_{N,T} = \int _0^T \mathbbm {1}_{X^{n}_t=\;x} \; dt, \quad I_{N,T} = \sum _{x \in \Lambda _N}(L_{N,T}^x)^2 . \end{aligned}$$

(2.7)

We apply Lemma 2.1 with $k = L$ and $n = L^N$ for three choices of f, g:

$$\begin{aligned} I_{N+1,T}&\le I_{N,T}\quad&\left( f_x=g_x=L_T^x\right) ,\end{aligned}$$

(2.8)

$$\begin{aligned} C_{N+1,T}&\le C_{N,T} \quad&\left( f_x=\textstyle {\sum _{e\in \mathcal {U}}L_T^{x+e}},\; g_x=L_T^x\right) ,\end{aligned}$$

(2.9)

$$\begin{aligned} \sum _{x\in \Lambda _{N+1}} |\nabla ^e L^x_{N+1,T}|^2&\le \sum _{x\in \Lambda _N} |\nabla ^e L^x_{N,T}|^2 \quad&\left( f_x = g_x = \left| \nabla ^e L_T^x\right| \right) . \end{aligned}$$

(2.10)

Summation of (2.10) over $e\in \mathcal {U}$ also gives

$$\begin{aligned} \sum _{x\in \Lambda _{N+1}} |\nabla L^x_{N+1,T}|^2 \le \sum _{x\in \Lambda _N} |\nabla L^x_{N,T}|^2. \end{aligned}$$

(2.11)

We identify the vertices of $\Lambda _N$ with nested subsets of ${{\mathbb {Z}}}^{d}$, centred at the origin (approximately if L is even), with $\Lambda _{N+1}$ paved by $L^d$ translates of $\Lambda _N$. We can thus define $\partial \Lambda _N$ to be the inner vertex boundary of $\Lambda _N$. We denote the expectation of $X^{L^N}$ started from $a \in \Lambda _N$ by $E^{\Lambda _N}_a$ and define

$$\begin{aligned} c_{N,T}(a, b)&= E^{\Lambda _N}_a \left( e^{-U_{\beta ,\gamma ,T}} \mathbbm {1}_{X(T)=b} \right) \quad (a, b \in \Lambda _N), \end{aligned}$$

(2.12)

$$\begin{aligned} c_{N,T}&= E^{\Lambda _N}_0 \left( e^{-U_{\beta ,\gamma ,T}} \right) . \end{aligned}$$

(2.13)

The finite-volume two-point function and susceptibility are defined by

$$\begin{aligned} G_{N,\beta ,\gamma ,\nu }(a,b)&= \int _0^\infty c_{N,T}(a, b) e^{-\nu T} \; dT, \end{aligned}$$

(2.14)

$$\begin{aligned} \chi _N(\beta , \gamma , \nu )&= \int _0^\infty c_{N,T} e^{-\nu T} \; dT . \end{aligned}$$

(2.15)

Proposition 2.2

Let $d >0$, $\beta >0$ and $\gamma < \beta$. For all $\nu \in \mathbb {R}$,

$$\begin{aligned} \lim _{N \rightarrow \infty } G_{N,\beta ,\gamma ,\nu }(a,b) = G_{\beta ,\gamma ,\nu }(a,b) \end{aligned}$$

(2.16)

and

$$\begin{aligned} \lim _{N\rightarrow \infty }\chi _N(\beta ,\gamma ,\nu )= \chi (\beta ,\gamma ,\nu ). \end{aligned}$$

(2.17)

Proof

Fix $a, b \in {{\mathbb {Z}}}^{d}$, and consider N sufficiently large that a, b can be identified with points in $\Lambda _N$. By (1.16), (2.8) and (2.11) (if $0 \le \gamma <\beta$), or by (1.5), (2.8) and (2.9) (if $\gamma < 0$),

$$\begin{aligned} c_{N,T}(a, b) \le c_{N+1,T}(a, b). \end{aligned}$$

(2.18)

Thus, (2.16) follows by monotone convergence, once we show that

$$\begin{aligned} \lim _{N\rightarrow \infty } c_{N,T}(a, b) = c_T(a, b). \end{aligned}$$

(2.19)

This follows as in [2, (2.8)]. That is, first we define

$$\begin{aligned} c_{N,T}^*(a, b)&= E^{\Lambda _N}_a \left( e^{-U_{\beta ,\gamma ,T}} \mathbbm {1}_{X(T)=b} \mathbbm {1}_{\{X([0, T]) \cap \partial \Lambda _N \ne \varnothing \}} \right) \end{aligned}$$

(2.20)

$$\begin{aligned} c_T^*(a, b)&= E_a \left( e^{-U_{\beta ,\gamma ,T}} \mathbbm {1}_{X(T)=b} \mathbbm {1}_{\{X([0, T]) \cap \partial \Lambda _N \ne \varnothing \}} \right) . \end{aligned}$$

(2.21)

Since walks which do not reach $\partial \Lambda _N$ make equal contributions to both $c_T(a,b)$ and $c_{N,T}(a,b)$, we have

$$\begin{aligned} c_T(a, b) - c_T^*(a, b) = c_{N,T}(a, b) - c_{N,T}^*(a, b). \end{aligned}$$

(2.22)

Thus,

$$\begin{aligned} |c_T(a, b) - c_{N,T}(a, b)| = |c_T^*(a, b) - c_{N,T}^*(a, b)| \le c_T^*(a, b) + c_{N,T}^*(a, b). \end{aligned}$$

(2.23)

Let $P^{\Lambda _N}_a$ and $P_a$ be the measures associated with $E^{\Lambda _N}_a$ and $E_a$, respectively. With $Y_t$ a rate-2d Poisson process with measure $\mathsf{P}$,

$$\begin{aligned} c_T^*(a, b) + c_{N,T}^*(a, b)&\le P_a (X([0, T]) \cap \partial \Lambda _N \ne \varnothing ) + P^{\Lambda _N}_a (X([0, T]) \cap \partial \Lambda _N \ne \varnothing ) \nonumber \\&\le 2 \mathsf{P} (Y_T \ge \text {diam}(\Lambda _N)) \rightarrow 0 \end{aligned}$$

(2.24)

as $N\rightarrow \infty$. This completes the proof of (2.16).

Finally, by monotone convergence of $G_N$ to G, for $\nu \in \mathbb {R}$,

$$\begin{aligned} \lim _{N\rightarrow \infty } \chi _N(g, \gamma , \nu ) = \sum _{b\in {{\mathbb {Z}}}^{d}} \lim _{N\rightarrow \infty } G_{N,g,\gamma ,\nu }(a, b) \mathbbm {1}_{b\in \Lambda _N} = \chi (g, \gamma , \nu ), \end{aligned}$$

(2.25)

which proves (2.17). $\square$

3 Integral Representation and Progressive Integration

In this section, we reformulate the model in terms of a perturbation of a supersymmetric Gaussian integral, in order to prepare for the application of the renormalisation group. The integral representation, which is a special case of a result from [9], makes use of the Grassmann integral. We begin by recalling the definition of the Grassmann integral in Sect. 3.1 and state the integral representation in Sect. 3.2. In Sect. 3.3, we split the integral into a Gaussian part and a perturbation. The basic idea underlying the renormalisation group is the progressive evaluation of this Gaussian integral via a multi-scale decomposition of its covariance, which we introduce in Sect. 3.4.

3.1 Boson and Fermion Fields

We fix N and write $\Lambda = \Lambda _N$. Given complex variables $\phi _x, {\bar{\phi }}_x$ (the boson field) for $x \in \Lambda$, we define the differentials (the fermion field)

$$\begin{aligned} \psi _x = \frac{1}{\sqrt{2\pi i}} d\phi _x, \quad {\bar{\psi }}_x = \frac{1}{\sqrt{2\pi i}} d{\bar{\phi }}_x, \end{aligned}$$

(3.1)

where we fix a choice of complex square root. The fermion fields are multiplied with each other via the anti-commutative wedge product, though we suppress this in our notation.

A differential form that is the product of a function of $(\phi , {\bar{\phi }})$ with p differentials is said to have degree p. A sum of forms of even degree is said to be even. We introduce a copy $\bar{\Lambda }$ of $\Lambda$ and we denote the copy of $X \subset \Lambda$ by ${\bar{X}} \subset \bar{\Lambda }$. We also denote the copy of $x \in \Lambda$ by ${\bar{x}} \in \bar{\Lambda }$ and define $\phi _{{\bar{x}}} = {\bar{\phi }}_x$ and $\psi _{{\bar{x}}} = {\bar{\psi }}_x$. Then any differential form F can be written

$$\begin{aligned} F = \sum _{{\vec {y}}} F_{{\vec {y}}} (\phi , {\bar{\phi }}) \psi ^{{\vec {y}}} \end{aligned}$$

(3.2)

where the sum is over finite sequences ${\vec {y}}$ over $\Lambda \sqcup \bar{\Lambda }$, and $\psi ^{{\vec {y}}} = \psi _{y_1} \ldots \psi _{y_p}$ when ${\vec {y}} = (y_1, \ldots , y_p)$. When ${\vec {y}} = \varnothing$ is the empty sequence, $F_\varnothing$ denotes the 0-degree (bosonic) part of F.

In order to apply the results of [2, 3, 6], we require smoothness of the coefficients $F_{{\vec {y}}}$ of F. For Theorem 1.2(i, ii), we need these coefficients to be $C^{10}$, and for Theorem 1.2(iii) we require a p-dependent number of derivatives for the analysis of $\xi _p$, as discussed in [6]. We let $\mathcal {N}$ be the algebra of even forms with sufficiently smooth coefficients and we let $\mathcal {N}(X) \subset \mathcal {N}$ be the sub-algebra of even forms only depending on fields in X. Thus, for $F \in \mathcal {N}(X)$, the sum in (3.2) runs over sequences ${\vec {y}}$ over $X \sqcup {\bar{X}}$. Note that $\mathcal {N}= \mathcal {N}(\Lambda )$.

Now let $F = (F_j)_{j \in J}$ be a finite collection of even forms indexed by a set J and write $F_\varnothing = (F_{\varnothing ,j})_{j \in J}$. Given a $C^\infty$ function $f : \mathbb {R}^J \rightarrow \mathbb {C}$, we define f(F) by its Taylor expansion about $F_\varnothing$:

$$\begin{aligned} f(F) = \sum _\alpha \frac{1}{\alpha !} f^{(\alpha )}(F_\varnothing ) (F - F_\varnothing )^\alpha . \end{aligned}$$

(3.3)

The summation terminates as a finite sum, since $\psi _x^2 = {\bar{\psi }}_x^2 = 0$ due to the anti-commutative product.

We define the integral $\int F$ of a differential form F in the usual way as the Riemann integral of its top-degree part (which may be regarded as a function of the boson field). In particular, given a positive-definite $\Lambda \times \Lambda$ symmetric matrix C with inverse $A = C^{-1}$, we define the Gaussian expectation (or super-expectation) of F by

$$\begin{aligned} \mathbb {E}_C F = \int e^{-S_A} F, \end{aligned}$$

(3.4)

where

$$\begin{aligned} S_A = \sum _{x\in \Lambda } \Big (\phi _x (A{\bar{\phi }})_x + \psi _x (A {\bar{\psi }})_x\Big ). \end{aligned}$$

(3.5)

Finally, for $F = f(\phi , {\bar{\phi }}) \psi ^{{\vec {y}}}$, we let

$$\begin{aligned} \theta F = f(\phi + \xi , {\bar{\phi }} + \bar{\xi }) (\psi + \eta )^{{\vec {y}}}, \end{aligned}$$

(3.6)

where $\xi$ is a new boson field, $\eta = (2\pi i)^{-1/2} d\xi$ a new fermion field, and $\bar{\xi }, \bar{\eta }$ are the corresponding conjugate fields. We extend $\theta$ to all $F \in \mathcal {N}$ by linearity and define the convolution operator $\mathbb {E}_C\theta$ by letting $\mathbb {E}_C\theta F \in \mathcal {N}$ denote the Gaussian expectation of $\theta F$ with respect to $(\xi , \bar{\xi }, \eta , \bar{\eta })$, with $\phi ,\bar{\phi },\psi ,\bar{\psi }$ held fixed.

3.2 Integral Representation of the Two-Point Function

An integral representation formula applying to general local time functionals is given in [7, 9]; see also [27, Appendix A]. We state the result we need in the proposition below.

Let $\Delta$ denote the Laplacian on $\Lambda$, i.e. $\Delta _{xy}$ is given by the right-hand side of (1.11) for $x, y \in \Lambda$. We define the differential forms:

$$\begin{aligned} \tau _x&= \phi _x {\bar{\phi }}_x + \psi _x {\bar{\psi }}_x \end{aligned}$$

(3.7)

$$\begin{aligned} \tau _{\Delta ,x}&= \frac{1}{2} \Big ( \phi _{x} (- \Delta \bar{\phi })_{x} + (- \Delta \phi )_{x} \bar{\phi }_{x} + \psi _{x} (- \Delta \bar{\psi })_{x} + (- \Delta \psi )_{x} \bar{\psi }_{x} \Big ) \end{aligned}$$

(3.8)

$$\begin{aligned} |\nabla \tau _x|^2&= \sum _{e\in \mathcal {U}} (\nabla ^e \tau )_x^2. \end{aligned}$$

(3.9)

Proposition 3.1

Let $d > 0$ and $\beta > 0$. For $\gamma < \beta$ and $\nu \in \mathbb {R}$,

$$\begin{aligned} G_{N,\beta ,\gamma ,\nu }(a, b)&= \int e^{-\sum _{x\in \Lambda } \left( U_{\beta ,\gamma }(\tau ) + \nu \tau _x + \tau _{\Delta ,x}\right) } {\bar{\phi }}_a \phi _b . \end{aligned}$$

(3.10)

Proof

The proof is identical to the proof of the $p = 1$ case of [27, Proposition 2.2] when, in the notation used in [27], we set

$$\begin{aligned} F(S) = e^{-U_{\beta ,\gamma }(S) - (\nu - 1) \sum _{x\in \Lambda } S_x} \end{aligned}$$

(3.11)

in [27, (A.13)]. $\square$

3.3 Gaussian Approximation

We divide the integral in (3.10) into a Gaussian part and a perturbation. Although the division is arbitrary here, a careful choice of the division must be made, and it is made in Theorem 5.1. We require several definitions. Let $z_0>-1$ and $m^2 >0$. We set

$$\begin{aligned} g_0 = (\beta - \gamma ) (1 + z_0)^2, \quad \nu _0 = \nu (1 + z_0) - m^2, \quad \gamma _0 = \frac{1}{4d} \gamma (1 + z_0)^2, \end{aligned}$$

(3.12)

and define

$$\begin{aligned} V^+_{0,x} = g_0\tau _x^2 + \nu _0 \tau _x + z_0 \tau _{\Delta ,x}, \quad U^+_x = |\nabla \tau _x|^2. \end{aligned}$$

(3.13)

The monomial $U^+_x$ should not be confused with the potential $U_{\beta ,\gamma }$. We define

$$\begin{aligned} Z_0 = \prod _{x\in \Lambda } e^{-(V^+_{0,x} + \gamma _0 U^+_x)}, \end{aligned}$$

(3.14)

and, with $C = (-\Delta + m^2)^{-1}$ and with the expectation given by (3.4),

$$\begin{aligned} Z_N = \mathbb {E}_C \theta Z_0. \end{aligned}$$

(3.15)

Recall that $Z_{N,\varnothing }$ denotes the 0-degree part of $Z_N$. We define a test function $\mathbbm {1}: \Lambda _N \rightarrow \mathbb {R}$ by $\mathbbm {1}_x=1$ for all x, and write $D^2 Z_{N,\varnothing }(0, 0; \mathbbm {1}, \mathbbm {1})$ for the directional derivative of $Z_{N,\varnothing }$ at $(\phi , {\bar{\phi }}) = (0, 0)$, with both directions equal to $\mathbbm {1}$. That is,

$$\begin{aligned} D^2 Z_{N,\varnothing }(0, 0; \mathbbm {1}, \mathbbm {1}) = \frac{\partial ^2}{\partial s\partial t} Z_{N,\varnothing }(s \mathbbm {1}, t\mathbbm {1})\big |_{s=t=0}. \end{aligned}$$

(3.16)

Proposition 3.2

Let $d > 0$, $\gamma , \nu \in \mathbb {R}$, $\beta >0$ and $\gamma <\beta$. If the relations (3.12) hold, then

$$\begin{aligned} G_{N,\beta ,\gamma ,\nu }(a,b) = (1+z_0) \mathbb {E}_C (Z_0 {\bar{\phi }}_a \phi _b), \end{aligned}$$

(3.17)

and

$$\begin{aligned} \chi _N\left( \beta ,\gamma ,\nu \right) = (1+z_0){\hat{\chi }}_N(m^2, g_0, \gamma _0, \nu _0, z_0) , \end{aligned}$$

(3.18)

with

$$\begin{aligned} {\hat{\chi }}_N(m^2, g_0, \gamma _0, \nu _0, z_0) = \frac{1}{m^2} + \frac{1}{m^4} \frac{1}{|\Lambda |} D^2 Z_{N,\varnothing }(0, 0; \mathbbm {1}, \mathbbm {1}). \end{aligned}$$

(3.19)

Proof

We make the change of variables $\varphi _x \mapsto (1 + z_0)^{1/2} \varphi _x$ (with $\varphi = \phi , {\bar{\phi }}, \psi , {\bar{\psi }}$) in (3.10), and obtain

$$\begin{aligned} G_{N,\beta ,\gamma ,\nu }(a, b)&= (1+z_0) \int e^{-\sum _{x\in \Lambda } \left( g_0 \tau _x^2 + \gamma _0 |\nabla \tau _x|^2 + \nu (1+z_0) \tau _x + (1+z_0)\tau _{\Delta ,x}\right) } {\bar{\phi }}_a \phi _b . \end{aligned}$$

(3.20)

Then, for any $m^2 \in \mathbb {R}$, we have

$$\begin{aligned} G_{N,\beta ,\gamma ,\nu }(a, b) = (1 + z_0) \int e^{-\sum _{x\in \Lambda } (\tau _{\Delta ,x} + m^2 \tau _x)} Z_0 {\bar{\phi }}_a \phi _b \end{aligned}$$

(3.21)

($m^2$ simply cancels with $\nu _0$ on the right-hand side). We use this with $m^2>0$, so that the inverse matrix $C=(-\Delta +m^2)^{-1}$ exists. By symmetry of the matrix $\Delta$, (3.5) gives

$$\begin{aligned} S_{(-\Delta +m^2)} = \sum _{x\in \Lambda } \left( \tau _{\Delta ,x} + m^2 \tau _x \right) . \end{aligned}$$

(3.22)

Then (3.17) follows from (3.21)–(3.22) and (3.4). Summation over $b\in \Lambda _N$ gives the formula $\chi _N(\beta ,\gamma ,\nu ) = (1+z_0)\sum _{x\in \Lambda } \mathbb {E}_C (Z_0\bar{\phi }_0\phi _x)$. Then (3.18), with (3.19), follows by an elementary computation as in [3, Section 4.1]. $\square$

3.4 Progressive Integration

The identity (3.17) splits the two-point function into a Gaussian part and a perturbation $Z_0$. The Gaussian part is parametrised by $(m^2, z_0)$, although the dependence on $z_0$ has been shifted out of the integral. We analyse the integral (3.17) using the renormalisation group method developed in [4, 11,12,13,14], which is itself inspired by [30]. This method is based on a decomposition

$$\begin{aligned} C = C_1 + \cdots + C_{N-1} + C_{N,N}, \end{aligned}$$

(3.23)

of the covariance C used to define $Z_N$ in (3.15), where $C_1, \ldots , C_{N-1}, C_{N,N}$ are covariances. For simplicity, we write $C_N = C_{N,N}$. A finite-range decomposition of this sort was constructed in [1, 8]. Specifically, we use the decomposition of [1].

The covariance decomposition allows us to evaluate $Z_N$ progressively by defining inductively

$$\begin{aligned} Z_{j+1} = \mathbb {E}_{C_{j+1}}\theta Z_j \quad (j < N). \end{aligned}$$

(3.24)

It is a basic fact that a sum of two independent Gaussian random variables with covariances $C'$ and $C''$ is itself Gaussian with covariance $C' + C''$. By [11, Proposition 2.6], this property extends to the Gaussian super-expectation in the sense that

$$\begin{aligned} \mathbb {E}_C\theta = \mathbb {E}_{C_N}\theta \circ \ldots \circ \mathbb {E}_{C_1}\theta . \end{aligned}$$

(3.25)

Thus, the definition of $Z_{j+1}$ in (3.24) agrees with (3.15) when ${j+1} = N$.

From the perspective of the renormalisation group, we view the map $Z_j \mapsto Z_{j+1}$ as defining a dynamical system. The evaluation of $Z_N$ can be accomplished by studying this system’s dependence on its initial condition, as we discuss in the next section.

4 Initial Coordinates for the Renormalisation Group

Following the approach of [3], we represent $Z_j$ by a pair of coordinates $I_j$ and $K_j$ that capture the relevant (expanding), marginal, and irrelevant (contracting) parts of $Z_j$. We begin in Sect. 4.1 by defining coordinates $(I_0, K_0)$ for $Z_0$. Norms used to control the evolution of these coordinates are introduced in Sect. 4.2, and it is shown in Sects. 4.3, 4.4 that $K_0$ satisfies norm estimates that permit the results of [5, 14] to be applied. The initial coordinate $K_0$ depends on the coupling constants $(g_0, \gamma _0, \nu _0, z_0)$ of (3.12), and regularity of $K_0$ as a function of these variables is established in Sect. 4.5.

4.1 Initial Coordinates for the Renormalisation Group

We now divide $Z_0$ into coordinates $I_0$ and $K_0$. The division depends on the sign of $\gamma$.

4.1.1 Coordinates for Positive $\gamma$

Assume that $\gamma \ge 0$. For $X \subset \Lambda$, we define

$$\begin{aligned} I_0^+(X) = \prod _{x\in X} e^{-V^+_{0,x}}, \quad \quad K_0^+(X) = \prod _{x \in X} I_{0,x}^+ (e^{-\gamma _0 U^{+}_{x}} - 1). \end{aligned}$$

(4.1)

Here, $I^+_{0,x} = I^+_0(\{x\})$, and we usually denote evaluation at a singleton by a subscript. By definition and binomial expansion,

$$\begin{aligned} Z_0 = \prod _{x\in \Lambda } \left( I^+_{0,x} + K^+_{0,x} \right) = \sum _{X\subset \Lambda } I_0^+(\Lambda \setminus X) K_0^+(X) . \end{aligned}$$

(4.2)

This polymer gas representation of $Z_0$ extends a much simpler representation used to study the weakly self-avoiding walk previously, e.g., in [2, 3]. In particular, when $\gamma _0 = 0$,

$$\begin{aligned} K_0^+(X) = \mathbbm {1}_\varnothing (X) = {\left\{ \begin{array}{ll} 1 &{} X = \varnothing \\ 0 &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

(4.3)

and (4.2) agrees with [3, (5.27)]. Thus the effect of nonzero $\gamma _0$ is incorporated entirely into the non-trivial $K_0^+$ of (4.1), rather than (4.3).

Then $(V^+_0, K_0^+)$ can be viewed as the initial condition of the dynamical system (3.24). This initial condition is not uniquely defined as a function of $(\beta , \gamma , \nu )$. Rather, the constraints (3.12) leave us with the freedom to choose $\nu _0$ and $z_0$ as we please. The key to the success of the renormalisation group method is the identification of critical values $\nu _0^c, z_0^c$ that lie on a stable manifold for the Gaussian fixed point $(V_0, K_0) = 0$. The existence of the stable manifold, which is a highly non-trivial fact, is obtained using the main result of [5]. This result allows for the possibility that $K_0^+$ is non-zero as long as $\Vert K^+_0\Vert = O(g_0^3)$ in an appropriate norm. We take advantage of this additional generality in order to prove Theorem 1.2.

4.1.2 Coordinates for Negative $\gamma$

Assume that $\gamma <0$. Define

$$\begin{aligned} V^-_{0,x} = V^+_{0,x} + 4 d \gamma _0 \tau _x^2, \quad U^-_x = 2 \sum _{e\in \mathcal {U}} \tau _x \tau _{x+e}. \end{aligned}$$

(4.4)

By the identity

$$\begin{aligned} \sum _{x\in \Lambda } \Big ( g_0 \tau _x^2 + \gamma _0 \sum _{e\in \mathcal {U}} (\nabla ^e \tau _x)^2 \Big ) = \sum _{x\in \Lambda } \Big ( (g_0 + 4d \gamma _0) \tau _x^2 - 2 \gamma _0 \sum _{e\in \mathcal {U}} \tau _x \tau _{x+e} \Big ), \end{aligned}$$

(4.5)

we can write

$$\begin{aligned} Z_0 = \prod _{x\in \Lambda } (I^-_{0,x} + K^-_{0,x}) = \sum _{X\subset \Lambda } I^-_0(\Lambda \setminus X) K^-_0(X), \end{aligned}$$

(4.6)

with

$$\begin{aligned} I^-_0(X) = \prod _{x\in X} e^{-V^-_{0,x}}, \quad \quad K^-_0(X) = \prod _{x\in X} I^-_{0,x} (e^{\gamma _0 U^-_x} - 1). \end{aligned}$$

(4.7)

Thus, we can parametrise $Z_0$ via either pair $(I^\pm _0, K^\pm _0)$. We use $(I^+_0, K^+_0)$ when $\gamma _0 \ge 0$ and $(I^-_0, K^-_0)$ when $\gamma _0 < 0$. With this convention,

$$\begin{aligned} K^\pm _0(X) = \prod _{x\in X} I^\pm _{0,x} (e^{-|\gamma _0| U^\pm _x} - 1) \quad (\mathrm{use} + \mathrm{for}\,\,\gamma _0 \ge 0, \mathrm{use} - \mathrm{for} \,\,\gamma _0<0). \end{aligned}$$

(4.8)

4.2 Norms

In this section, we recall some definitions and basic facts concerning norms, from [11]. For now, we only consider the case of scale $j = 0$.

Recall the notation introduced in Sect. 3.1. A test function g is defined to be a function $({\vec {x}}, {\vec {y}}) \mapsto g_{{\vec {x}},{\vec {y}}}$, where ${\vec {x}}$ and ${\vec {y}}$ are finite sequences of elements in $\Lambda \sqcup \bar{\Lambda }$. When ${\vec {x}}$ or ${\vec {y}}$ is the empty sequence $\varnothing$, we drop it from the notation as long as this causes no confusion; e.g., we may write $g_{{\vec {x}}} = g_{{\vec {x}},\varnothing }$. The length of a sequence ${\vec {x}}$ is denoted $|{\vec {x}}|$. Gradients of test functions are defined component-wise. Thus, if ${\vec {x}} = (x_1, \ldots , x_m)$ and $\alpha = (\alpha _1, \ldots , \alpha _m)$ with each $\alpha _i \in \mathbb {N}_0^\mathcal {U}$, and similarly for ${\vec {y}}=(y_1,\ldots ,y_n)$ and $\beta =(\beta _1,\ldots ,\beta _n)$, then

$$\begin{aligned} \nabla ^{\alpha ,\beta }_{{\vec {x}},{\vec {y}}} g_{{\vec {x}},{\vec {y}}} = \nabla ^{\alpha _1}_{x_1} \ldots \nabla ^{\alpha _m}_{x_m} \nabla ^{\beta _1}_{y_1} \ldots \nabla ^{\beta _n}_{y_n} g_{x_1,\ldots ,x_m,y_1,\ldots ,y_n}. \end{aligned}$$

(4.9)

Let $\mathfrak {h}_0 > 0$ be a parameter, which we set below. We fix positive constants $p_\Phi \ge 4$ and $p_\mathcal {N}$ and assume that all test functions vanish when $|{\vec {x}}| +|{\vec {y}}| > p_\mathcal {N}$. For Theorem 1.2(i, ii), any choice of $p_\mathcal {N}\ge 10$ is sufficient, whereas for Theorem 1.2(iii) it is necessary to choose $p_\mathcal {N}$ large depending on p [6]. The $\Phi = \Phi (\mathfrak {h}_0)$ norm on such test functions is defined by

$$\begin{aligned} \Vert g\Vert _\Phi = \sup _{{\vec {x}}, {\vec {y}}} \mathfrak {h}_0^{-(|{\vec {x}}| +|{\vec {y}}|)} \!\!\!\!\!\!\!\!\sup _{\alpha ,\beta : |\alpha |_1+|\beta |_1 \le p_\Phi } |\nabla ^{\alpha ,\beta } g_{{\vec {x}}, {\vec {y}}}|, \end{aligned}$$

(4.10)

where $|\alpha |_1$ denotes the total order of the differential operator $\nabla ^\alpha$. Thus, for any test function g and for sequences ${\vec {x}}, {\vec {y}}$ with $|{\vec {x}}| +|{\vec {y}}| \le p_\mathcal {N}$ and corresponding $\alpha , \beta$ with $|\alpha |_1 + |\beta |_1 \le p_\Phi$,

$$\begin{aligned} |\nabla ^{\alpha ,\beta } g_{{\vec {x}},{\vec {y}}}| \le \mathfrak {h}_0^{|{\vec {x}}| + |{\vec {y}}|} \Vert g\Vert _\Phi . \end{aligned}$$

(4.11)

For any $F \in \mathcal {N}$, there exist unique functions $F_{{\vec {y}}}$ of $(\phi , {\bar{\phi }})$ that are anti-symmetric under permutations of ${\vec {y}}$, such that

$$\begin{aligned} F = \sum _{{\vec {y}}} \frac{1}{|{\vec {y}}|!} F_{{\vec {y}}}(\phi , {\bar{\phi }}) \psi ^{{\vec {y}}}. \end{aligned}$$

(4.12)

Given a sequence $\vec {x}$ with $|\vec {x}| = m$, we define

$$\begin{aligned} F_{{\vec {x}}, {\vec {y}}} = \frac{\partial ^m F_{{\vec {y}}}}{\partial \phi _{x_1} \ldots \partial \phi _{x_m}}. \end{aligned}$$

(4.13)

We define a $\phi$-dependent pairing of elements of $\mathcal {N}$ with test functions, by

$$\begin{aligned} \langle F, g \rangle _\phi = \sum _{{\vec {x}}, {\vec {y}}} \frac{1}{|{\vec {x}}|! |{\vec {y}}|!} F_{{\vec {x}},{\vec {y}}}(\phi , {\bar{\phi }}) g_{{\vec {x}},{\vec {y}}}. \end{aligned}$$

(4.14)

Let $B(\Phi )$ denote the unit $\Phi$-ball in the space of test functions. Then the $T_\phi = T_\phi (\mathfrak {h}_0)$ semi-norm on $\mathcal {N}$ is defined by

$$\begin{aligned} \Vert F\Vert _{T_\phi } = \sup _{g\in B(\Phi )} |\langle F, g \rangle _\phi |. \end{aligned}$$

(4.15)

We need several properties of the $T_\phi$ semi-norm, whose proofs can be found in [11]. First, there is the important product property [11, Proposition 3.7]

$$\begin{aligned} \Vert F G\Vert _{T_\phi } \le \Vert F\Vert _{T_\phi } \Vert G\Vert _{T_\phi }. \end{aligned}$$

(4.16)

An immediate consequence is that $\Vert e^{-F}\Vert _{T_\phi } \le e^{\Vert F\Vert _{T_\phi }}$. This is improved in [11, Proposition 3.8], which states that (recall that $F_\varnothing$ denotes the 0-degree part of F)

$$\begin{aligned} \Vert e^{-F}\Vert _{T_\phi } \le e^{-2 \mathrm{Re} F_\varnothing (\phi ) + \Vert F\Vert _{T_\phi }}. \end{aligned}$$

(4.17)

Each of the two choices $\varphi = \phi , {\bar{\phi }}$ can be viewed as a test function supported on sequences with $|{\vec {x}}| = 1$ and $|{\vec {y}}| = 0$ and satisfying $\varphi _{{\bar{x}}} = \bar{\varphi }_x$. In particular, $\Vert \phi \Vert _\Phi$ is defined as the norm of a test function. We use [11, Proposition 3.10], which states that if $F \in \mathcal {N}$ is a polynomial in $\phi ,\bar{\phi },\psi ,\bar{\psi }$ of total degree $A \le p_\mathcal {N}$, then

$$\begin{aligned} \Vert F\Vert _{T_\phi } \le \Vert F\Vert _{T_0} (1 + \Vert \phi \Vert _\Phi )^A. \end{aligned}$$

(4.18)

We write $x^\Box = \{y: |y-x|_\infty \le 2^d-1\}$, where $|x|_\infty = \max \{|x_i| : 1 \le i \le d\}$ (this is the scale-0 version of [13, (1.37)] for a single point). The $\Phi _x \equiv \Phi (x^\square )$ norm of $\phi \in \mathbb {C}^\Lambda$ is defined by

$$\begin{aligned} \Vert \phi \Vert _{\Phi _x} = \inf \left\{ \Vert \phi - f\Vert _\Phi : f \in \mathbb {C}^\Lambda \text { such that } f_y = 0 \;\forall y \in x^\square \right\} . \end{aligned}$$

(4.19)

By taking the infimum in (4.18) over all possible re-definitions of $\phi _y$ for $y \notin x^\square$, we get

$$\begin{aligned} \Vert F\Vert _{T_\phi } \le \Vert F\Vert _{T_0} (1 + \Vert \phi \Vert _{\Phi _x})^A \end{aligned}$$

(4.20)

when $F \in \mathcal {N}(x^\square )$.

We need two choices of the parameter $\mathfrak {h}_0$ (for both choices, $\mathfrak {h}_0 \ge 1$): either $\mathfrak {h}_0 = \ell _0$, an L-dependent constant; or $\mathfrak {h}_0 = h_0 = k_0 \tilde{g}_0^{-1/4}$, where $k_0$ is a small constant and $\tilde{g}_0$ is a constant which must be chosen small depending on L. Some discussion of these constants occurs in the proof of Proposition 4.1. In [13], two regulators are defined. At scale 0, these are given by

$$\begin{aligned} G_0(x, \phi ) = e^{\Vert \phi \Vert ^2_{\Phi _x(\ell _0)}}, \qquad {\tilde{G}}_0(x, \phi ) = e^{\frac{1}{2} \Vert \phi \Vert ^2_{{\tilde{\Phi }}_x(\ell _0)}}. \end{aligned}$$

(4.21)

The ${\tilde{\Phi }}_x$ norm in the definition of ${\tilde{G}}_0$, is defined in [13, (1.40)]; it is a modification of the $\Phi _x$ norm that is invariant under shifts by linear test functions. Its specific properties do not play a direct role in this paper. Two regulator norms are defined for $F \in \mathcal {N}(x^\square )$ by

$$\begin{aligned} \Vert F\Vert _{G_0} = \sup _{\phi \in \mathbb {C}^\Lambda } \frac{\Vert F\Vert _{T_\phi (\ell _0)}}{G_0(x,\phi )} , \quad \Vert F\Vert _{\tilde{G}^\mathsf{t}_0} = \sup _{\phi \in \mathbb {C}^\Lambda } \frac{\Vert F\Vert _{T_\phi (h_0)}}{\tilde{G}^\mathsf{t}_0(x,\phi )} , \end{aligned}$$

(4.22)

where $\mathsf{t} \in (0, 1]$ is a constant power.

4.3 Bounds on $K_0$

The main estimate on $K^\pm _{0,x}$ is given by the following proposition. Consistent with [13, (1.83)], we fix a large constant $C_\mathcal {D}$ and define

$$\begin{aligned} \mathcal {D}_0 = \mathcal {D}_0(\tilde{g}_0) = \{(g,\nu ,z) \in \mathbb {R}^3 : C_{\mathcal {D}}^{-1}\tilde{g}_0< g< C_{\mathcal {D}}\tilde{g}_0, \; |\nu |,|z| < C_{\mathcal {D}}\tilde{g}_0\}. \end{aligned}$$

(4.23)

Proposition 4.1

Suppose that $V^\pm _0 \in \mathcal {D}_0$, with $\tilde{g}_0$ sufficiently small. If $|\gamma _0| \le \tilde{g}_0$, then (with constants that may depend on L)

$$\begin{aligned} \Vert K^\pm _{0,x}\Vert _{G_0} = O(|\gamma _0|), \quad \Vert K^\pm _{0,x}\Vert _{{\tilde{G}}_0} = O(|\gamma _0|/g_0), \end{aligned}$$

(4.24)

where the bounds on $K^+$ and $K^-$ hold for $\gamma _0 \ge 0$ and $\gamma _0 < 0$, respectively.

The form of the estimates (4.24) can be anticipated from the definition of $K_0^\pm$ in (4.8). The upper bound arises from the small size of $e^{-|\gamma _0|U_x^\pm }-1$. For small fields, hence small $U_x^\pm$, this is of order $|\gamma _0|$, as reflected by the $G_0$ norm estimate of (4.24). For large fields, namely fields of size $|\phi | \approx \tilde{g}_{0}^{-1/4}$, the difference $e^{-|\gamma _0|U_x^\pm }-1$ is roughly of size $|\gamma _0|\,|\phi |^4 \approx |\gamma _0|/\tilde{g}_0$. This effect is measured by the ${\tilde{G}}_0$ norm.

Before proving the proposition, we write (4.8) for a singleton as

$$\begin{aligned} K^\pm _{0,x} = I^\pm _{0,x} J^\pm _x, \end{aligned}$$

(4.25)

where, by the fundamental theorem of calculus,

$$\begin{aligned} I^\pm _{0,x}&= e^{-V^\pm _{0,x}} \end{aligned}$$

(4.26)

$$\begin{aligned} J^\pm _x&= e^{-|\gamma _0|U^\pm _x} - 1 = - \int _0^{1} |\gamma _0| U^\pm _x e^{-t |\gamma _0| U^\pm _x} \; dt. \end{aligned}$$

(4.27)

As in (4.8), the $+$ versions of (4.25)–(4.27) hold only for $\gamma _0 \ge 0$ and the − versions only for $\gamma _0 < 0$.

Let $F \in \mathcal {N}(x^\square )$ be a polynomial of degree at most $p_\mathcal {N}$. Then the stability estimates [13, (2.1)–(2.2)] imply that there exists $c_3 > 0$ and, for any $c_1 \ge 0$, there exist positive constants $C, c_2$ such that if $V_0^\pm \in \mathcal {D}_0$ then

$$\begin{aligned} \Vert I^\pm _{0,x} F\Vert _{T_\phi (\mathfrak {h}_0)} \le C \Vert F\Vert _{T_0(\mathfrak {h}_0)} {\left\{ \begin{array}{ll} e^{c_3 g_0 \left( 1 + \Vert \phi \Vert ^2_{\Phi _x(\ell _0)}\right) } &{} \mathfrak {h}_0 = \ell _0 \\ e^{-c_1 k_0^4 \Vert \phi \Vert ^2_{\Phi _x(h_0)}} e^{c_2 k_0^4 \Vert \phi \Vert ^2_{{\tilde{\Phi }}_x(\ell _0)}} &{} \mathfrak {h}_0 = h_0. \end{array}\right. } \end{aligned}$$

(4.28)

This essentially reduces our task to estimating $J^\pm _x$. The next lemma is an ingredient for this.

Lemma 4.2

There is a universal constant ${\tilde{C}}$ such that

$$\begin{aligned} \Vert U^\pm _x\Vert _{T_\phi (\mathfrak {h}_0)} \le 2 U^\pm _{\varnothing ,x} + {\tilde{C}} \mathfrak {h}_0^4 \left( 1 + \Vert \phi \Vert ^2_{\Phi _x(\mathfrak {h}_0)}\right) , \end{aligned}$$

(4.29)

where $U^\pm _\varnothing$ is the 0-degree part of $U^\pm$.

Proof

Let

$$\begin{aligned} M^+ = M^+_e = (\nabla ^e \tau _x)^2, \quad M^- = M^-_e = 2 \tau _x \tau _{x+e}, \end{aligned}$$

(4.30)

so that $U^\pm _x = \sum _{e\in \mathcal {U}} M^\pm _e$. It suffices to prove (4.29) with $U^\pm _x$ replaced by $M^\pm$ (on both sides of the equation). In addition, we can replace the $\Phi _x$ norm by the $\Phi$ norm; the bound with the $\Phi _x$ norm then follows in the same way that (4.20) is a consequence of (4.18), since $M^\pm \in \mathcal {N}(x^\Box )$.

By definition of $\tau _x$,

$$\begin{aligned} M^\pm = M^\pm _{\varnothing } + R^\pm , \end{aligned}$$

(4.31)

where

$$\begin{aligned}&M^+_{\varnothing } = \left( \nabla ^e |\phi _x|^2\right) ^2, \quad&R^+ = 2 \left( \nabla ^e |\phi _x|^2\right) \nabla ^e (\psi _x\bar{\psi }_x), \end{aligned}$$

(4.32)

$$\begin{aligned}&M^-_\varnothing = 2 |\phi _x|^2 |\phi _{x+e}|^2, \quad&R^- = 2 \left( |\phi _x|^2 \psi _{x+e}{\bar{\psi }}_{x+e} + \psi _x{\bar{\psi }}_x |\phi _{x+e}|^2 + \psi _x{\bar{\psi }}_x\psi _{x+e}{\bar{\psi }}_{x+e}\right) . \end{aligned}$$

(4.33)

Thus, $\Vert M^\pm \Vert _{T_\phi } \le \Vert M^\pm _{\varnothing }\Vert _{T_\phi } + \Vert R^\pm \Vert _{T_\phi }$. A straightforward computation shows that

$$\begin{aligned} \Vert R^\pm \Vert _{T_\phi } = O\left( \mathfrak {h}_0^4 (1 + \Vert \phi \Vert _\Phi )^2\right) . \end{aligned}$$

(4.34)

By definition of the $T_\phi$ semi-norm,

$$\begin{aligned} \Vert \nabla ^e |\phi _x|^2\Vert _{T_\phi } \le \nabla ^e |\phi _x|^2 + 2 \mathfrak {h}_0 (|\phi _x| + |\phi _{x+e}|) + 2 \mathfrak {h}_0^2. \end{aligned}$$

(4.35)

Together with (4.34), the product property, and (4.11), this implies that

$$\begin{aligned} \Vert M^+\Vert _{T_\phi } \le M^+_\varnothing + 2 |\nabla ^e |\phi _x|^2| (2 \mathfrak {h}_0 (|\phi _x| + |\phi _{x+e}|)) + O\left( \mathfrak {h}_0^4\right) \left( 1 + \Vert \phi \Vert ^2_\Phi \right) . \end{aligned}$$

(4.36)

By the inequality

$$\begin{aligned} 2|ab| \le |a|^2 + |b|^2 \end{aligned}$$

(4.37)

and another application of (4.11),

$$\begin{aligned} 2 |\nabla ^e |\phi _x|^2| (2 \mathfrak {h}_0 (|\phi _x| + |\phi _{x+e}|)) \le M^+_\varnothing + O\left( \mathfrak {h}_0^2 \Vert \phi \Vert ^2_\Phi \right) , \end{aligned}$$

(4.38)

and the bound on $M^+$ follows.

For the bound on $M^-$, we use the identity

$$\begin{aligned} \Vert \tau _x\Vert _{T_\phi } = (|\phi _x| + \mathfrak {h}_0)^2 + \mathfrak {h}_0^2 \end{aligned}$$

(4.39)

from [11, (3.27)]. By the product property and (4.11), this implies that

$$\begin{aligned} \Vert M^-\Vert _{T_\phi } \le&\,2 |\phi _x|^2 |\phi _{x+e}|^2 + 2 (|\phi _x| |\phi _{x+e}|) (2 \mathfrak {h}_0 (|\phi _{x+e}| + |\phi _x|)) \nonumber \\&+ O(\mathfrak {h}_0^4) (1 + \Vert \phi \Vert ^2_\Phi ). \end{aligned}$$

(4.40)

Another application of (4.37) and (4.11) gives

$$\begin{aligned} 2 (|\phi _x| |\phi _{x+e}|) (2 \mathfrak {h}_0 (|\phi _{x+e}| + |\phi _x|)) \le |\phi _x|^2 |\phi _{x+e}|^2 + O(\mathfrak {h}_0^2 \Vert \phi \Vert ^2_\Phi ), \end{aligned}$$

(4.41)

and the proof is complete. $\square$

An immediate consequence of Lemma 4.2, using (4.17), is that for any $s \ge 0$,

$$\begin{aligned} \Vert e^{-s U^\pm _x}\Vert _{T_\phi (\mathfrak {h}_0)} \le e^{{\tilde{C}} s \mathfrak {h}_0^4 \left( 1 + \Vert \phi \Vert ^2_{\Phi _x(\mathfrak {h}_0)}\right) }. \end{aligned}$$

(4.42)

Proof of Proposition 4.1 According to the definition of the regulator norms in (4.21)–(4.22), it suffices to prove that, under the hypothesis on $\gamma _0$,

$$\begin{aligned} \Vert K^\pm _{0,x}\Vert _{T_\phi (\mathfrak {h}_0)} = O(|\gamma _0| \mathfrak {h}_0^4) {\left\{ \begin{array}{ll} e^{\Vert \phi \Vert _{\Phi _x}^2} &{} (\mathfrak {h}_0=\ell _0) \\ e^{\frac{\mathsf{t}}{2} \Vert \phi \Vert _{{\tilde{\Phi }}}} &{} (\mathfrak {h}_0=h_0). \end{array}\right. } \end{aligned}$$

(4.43)

For $t \in [0,1]$, let ${\tilde{I}}^\pm _x(t) = e^{-t |\gamma _0| U^\pm _x}$. By (4.25), (4.27), and the product property,

$$\begin{aligned} \Vert K^\pm _{0,x}\Vert _{T_\phi (\mathfrak {h}_0)}&\le |\gamma _0| \Vert I^\pm _{0,x} U^\pm _x\Vert _{T_\phi (\mathfrak {h}_0)} \sup _{t\in [0, 1]} \Vert {\tilde{I}}^\pm _{x}(t)\Vert _{T_\phi (\mathfrak {h}_0)}. \end{aligned}$$

(4.44)

By (4.28) and Lemma 4.2, there exists $c_3 > 0$, and, for any $c_1 \ge 0$ there exists $c_2 > 0$, such that

$$\begin{aligned} \Vert I^\pm _{0,x} U^\pm _x\Vert _{T_\phi (\mathfrak {h}_0)} \le O(\mathfrak {h}_0^4) {\left\{ \begin{array}{ll} e^{c_3 g_0 \Vert \phi \Vert ^2_{\Phi _x(\ell _0)}} &{} \mathfrak {h}_0 = \ell _0 \\ e^{-c_1 k_0^4 \Vert \phi \Vert ^2_{\Phi _x(h_0)}} e^{c_2 k_0^4 \Vert \phi \Vert ^2_{{\tilde{\Phi }}_x(\ell _0)}} &{} \mathfrak {h}_0 = h_0. \end{array}\right. } \end{aligned}$$

(4.45)

The constant in $O(|\gamma _0| \mathfrak {h}_0^4)$ may depend on $c_1$, but this is unimportant. Also, by (4.42),

$$\begin{aligned} \sup _{t\in [0,1]} \Vert {\tilde{I}}_{x}^\pm (t) \Vert _{T_\phi (\mathfrak {h}_0)} \le e^{{\tilde{C}} |\gamma _0| \mathfrak {h}_0^4 \left( 1+\Vert \phi \Vert ^2_{\Phi _x(\mathfrak {h}_0)}\right) }. \end{aligned}$$

(4.46)

Thus, for $\mathfrak {h}_0=\ell _0$, the total exponent in our estimate for the right-hand side of (4.44) is

$$\begin{aligned} {\tilde{C}} |\gamma _0| \ell _0^4 +(c_3 g_0 + {\tilde{C}} |\gamma _0| \ell _0^4) \Vert \phi \Vert ^2_{\Phi _x(\ell _0)} . \end{aligned}$$

(4.47)

This gives the $\mathfrak {h}_0=\ell _0$ version of (4.43) provided that $g_0$ is small and $|\gamma _0|$ is small depending on L.

For $\mathfrak {h}_0=h_0$, the total exponent in our estimate for the right-hand side of (4.44) is

$$\begin{aligned} {\tilde{C}} |\gamma _0| k_0^4 \tilde{g}_0^{-1} + ({\tilde{C}} |\gamma _0| k_0^4 \tilde{g}_0^{-1} - c_1 k_0^4) \Vert \phi \Vert ^2_{\Phi _x(h_0)} + c_2 k_0^4 \Vert \phi \Vert ^2_{{\tilde{\Phi }}_x(\ell _0)}. \end{aligned}$$

(4.48)

This gives the $\mathfrak {h}_0=h_0$ version of (4.43) provided that $|\gamma _0| \le \tilde{g}_0$, $c_1\ge {\tilde{C}}$, and $c_2 k_0^4 \le \mathsf{t}/2$.

All the provisos are satisfied if we choose $c_1 \ge {\tilde{C}}$, $k_0$ small depending on $c_1$ and $\tilde{g}_0$ small. $\square$

Remark 4.3

By a small modification to the proof of Proposition 4.1, it can be shown that if $M_x \in \mathcal {N}(x^\square )$ is a monomial of degree $r \le p_\mathcal {N}-4$ (so that $M_xU_x^\pm$ has degree at most $p_\mathcal {N}$), then

$$\begin{aligned} \Vert M_x K^\pm _{0,x}\Vert _{\mathcal {G}_0} = O(|\gamma _0| \mathfrak {h}_0^{4+r}). \end{aligned}$$

(4.49)

4.4 Unified Bound on $K_0$

The results of [5, 14] are formulated in a sequence of spaces $\mathcal {W}_j$ that enable the combination of small-field and large-field estimates into a single norm estimate. In this section, we recast the result of Proposition 4.1 to see that $K_0^\pm$ fits into this formulation.

We restrict attention in this section to the $\mathcal {W}_0$ norm, whose definition is recalled below. This requires several preliminaries. Let $\mathcal {P}_0 = \mathcal {P}_0(\Lambda )$ denote the collection of subsets of vertices in $\Lambda$. We refer to the elements of $\mathcal {P}_0$ as polymers. We call a nonempty polymer $X\in \mathcal {P}_0$ connected if for any $x, x' \in X$, there is a sequence $x = x_0, \ldots , x_n = x' \in X$ such that $|x_{i+1} - x_i|_\infty = 1$ for $i = 0, \ldots , n - 1$. Let $\mathcal {C}_0$ denote the set of connected polymers. The small set neighbourhood $X^\Box$ of $X\in \mathcal {P}_0$ is defined by

$$\begin{aligned} X^\Box = \{y \in \Lambda : \exists x \in \Lambda \; \text {such that}\; |y-x|_\infty \le 2^d\}. \end{aligned}$$

(4.50)

We extend the definitions of the regulators $\mathcal {G}_0 = G_0, {\tilde{G}}_0^\mathsf{t}$, defined in (4.21), by setting

$$\begin{aligned} \mathcal {G}_0(X, \phi ) = \prod _{x\in X} \mathcal {G}_0(x, \phi ), \end{aligned}$$

(4.51)

and extend the definitions (4.22) to define norms, for $F \in \mathcal {N}(X^\Box )$, by

$$\begin{aligned} \Vert F\Vert _{G_0} = \sup _{\phi \in \mathbb {C}^\Lambda } \frac{\Vert F\Vert _{T_\phi (\ell _0)}}{G_0(X,\phi )} , \quad \Vert F\Vert _{\tilde{G}^\mathsf{t}_0} = \sup _{\phi \in \mathbb {C}^\Lambda } \frac{\Vert F\Vert _{T_\phi (h_0)}}{\tilde{G}^\mathsf{t}_0(X,\phi )} . \end{aligned}$$

(4.52)

It follows from the product property of the $T_\phi$ norm that these norms obey the product property

$$\begin{aligned} \Vert F_1F_2\Vert _{\mathcal {G}_0} \le \Vert F_1\Vert _{\mathcal {G}_0} \Vert F_2\Vert _{\mathcal {G}_0} \quad {\mathrm{for } \ F_i\in \mathcal {N}(X_i^\Box ) \ \mathrm{with } \ X_1 \cap X_2=\varnothing .} \end{aligned}$$

(4.53)

Given a map $K: \mathcal {P}_0 \rightarrow \mathcal {N}$ with the property that $K(X) \in \mathcal {N}(X^\Box )$ for all $X \in \mathcal {P}_0$, we define the $\mathcal {F}_0(\mathcal {G})$ norms (for $\mathcal {G}= G, {\tilde{G}}$) by

$$\begin{aligned} \Vert K\Vert _{\mathcal {F}_0(G)}&= \sup _{X\in \mathcal {C}_0} \tilde{g}_0^{-f_0(a, X)} \Vert K(X)\Vert _{G_0} \end{aligned}$$

(4.54)

$$\begin{aligned} \Vert K\Vert _{\mathcal {F}_0({\tilde{G}})}&= \sup _{X\in \mathcal {C}_0} \tilde{g}_0^{-f_0(a, X)} \Vert K(X)\Vert _{{\tilde{G}}_0^\mathsf{t}}, \end{aligned}$$

(4.55)

with

$$\begin{aligned} f_0 (a, X) = a(|X|-2^d)_+ = {\left\{ \begin{array}{ll} a (|X| - 2^d) &{} \text {if } |X| > 2^d \\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(4.56)

Here a is a small constant; its value is discussed below [14, (1.46)]. The $\mathcal {W}_0$ norm is then defined by

$$\begin{aligned} \Vert K\Vert _{\mathcal {W}_0}&= \max \Big \{ \Vert K \Vert _{\mathcal {F}_0(G)},\, \tilde{g}_0^{9/4} \Vert K \Vert _{\mathcal {F}_0(\tilde{G})} \Big \}. \end{aligned}$$

(4.57)

Since this definition depends on $\tilde{g}_0$ and the volume $\Lambda$, we sometimes write $\mathcal {W}_0 = \mathcal {W}_0(\tilde{g}_0, \Lambda )$. The following proposition uses Proposition 4.1 to obtain a bound on the $\mathcal {W}_0$ norm of the map $K_0^\pm : \mathcal {P}_0 \rightarrow \mathcal {N}$ defined by

$$\begin{aligned} K_0^\pm (X) = \prod _{x \in X} K_{0,x}^\pm \qquad (X \in \mathcal {P}_0) . \end{aligned}$$

(4.58)

Proposition 4.4

If $V_0^\pm \in \mathcal {D}_0$ with $\tilde{g}_0$ sufficiently small (depending on L), and if $|\gamma _0| \le O(\tilde{g}_0^{1+a'})$ for some $a' >a$, then $\Vert K_0^\pm \Vert _{\mathcal {W}_0} \le O(|\gamma _0|)$, where all constants may depend on L.

Proof

Let X be a connected polymer in $\mathcal {P}_0$. By the product property and Proposition 4.1,

$$\begin{aligned} \Vert K_0^\pm (X)\Vert _{\mathcal {G}_0} \le (c|\gamma _0|\mathfrak {h}_0^4)^{|X|}&= (c|\gamma _0|\mathfrak {h}_0^4)^{|X|\wedge 2^d} (c|\gamma _0|\mathfrak {h}_0^4)^{(|X|-2^d)_+}. \end{aligned}$$

(4.59)

For $\mathcal {G}_0=G_0$, we use $\mathfrak {h}_0=\ell _0$, $(c|\gamma _0|\mathfrak {h}_0^4)^{|X|\wedge 2^d}\le O(|\gamma _0|)$, and

$$\begin{aligned} (c|\gamma _0|\mathfrak {h}_0^4)^{(|X|-2^d)_+} \le (c' \tilde{g}_0)^{(1+a')(|X|-2^d)_+} \le \tilde{g}_0^{f_0(a,X)}. \end{aligned}$$

(4.60)

For $\mathcal {G}_0={\tilde{G}}_0$, we use $\mathfrak {h}_0=h_0 = O(\tilde{g}_0^{-1/4})$ and, since $a'>a$,

$$\begin{aligned} (c|\gamma _0|\mathfrak {h}_0^4)^{(|X|-2^d)_+} \le (c' \tilde{g}_0)^{a'(|X|-2^d)_+} \le \tilde{g}_0^{f_0(a,X)}. \end{aligned}$$

(4.61)

Since $|\gamma _0| \le \tilde{g}_0$, it follows from (4.59) that

$$\begin{aligned} \tilde{g}_0^{9/4} \Vert K_0^\pm \Vert _{\mathcal {F}_0(\tilde{G})} \le \tilde{g}_0^{9/4}O(|\gamma _0| \tilde{g}_0^{-1}) \le |\gamma _0|, \end{aligned}$$

(4.62)

and the proof is complete. $\square$

The above discussion is based on norms in the setting of the torus $\Lambda$. As in [14], a version on the infinite lattice ${{\mathbb {Z}}}^{d}$ is also required. This can be done in exactly the same manner, by defining the polymers $\mathcal {P}_0 = \mathcal {P}_0({{\mathbb {Z}}}^{d})$ to be the collection of subsets of ${{\mathbb {Z}}}^{d}$, with $K_0^\pm (X)$ defined for subsets of ${{\mathbb {Z}}}^{d}$ by $\prod _{x \in X} K_{0,x}^\pm$. The $\mathcal {W}_0 = \mathcal {W}_0(\tilde{g}_0, {{\mathbb {Z}}}^{d})$ norm (in infinite volume) can be defined analogously to (4.57). The hypotheses and conclusion of Proposition 4.4 remain the same in the setting of ${{\mathbb {Z}}}^{d}$.

4.5 Smoothness of $K_0$

Let $\mathcal {C}_0(\mathbb {Z}^d) \subset \mathcal {P}_0(\mathbb {Z}^d)$ be the set of connected polymers. By definition, a connected polymer is nonempty. Given $\tilde{g}_0>0$, let $\mathcal {W}^*_0(\tilde{g}_0, {{\mathbb {Z}}}^{d})$ denote the space of maps $F :\mathcal {C}_0({{\mathbb {Z}}}^{d}) \rightarrow \mathcal {N}$, with $F(X) \in \mathcal {N}(X^\Box )$ and $\Vert F\Vert _{\mathcal {W}_0(\tilde{g}_0, {{\mathbb {Z}}}^{d})} < \infty$. Addition in this space is defined by $(F_1+F_2)(X)=F_1(X)+F_2(X)$. We extend any $F :\mathcal {C}_0({{\mathbb {Z}}}^{d}) \rightarrow \mathcal {N}$ to $F :\mathcal {P}_0({{\mathbb {Z}}}^{d}) \rightarrow \mathcal {N}$ by taking $F(X) = \prod _{Y} F(Y)$ where the product is over the connected components Y of X.

Given any map $F : D \rightarrow \mathcal {W}^*_0(\tilde{g}_0, {{\mathbb {Z}}}^{d})$ for $D \subset \mathbb {R}$ an open interval, write $F_X, F^\phi _X : D \rightarrow \mathcal {N}(X^\square )$ for the maps defined by partial evaluation of F at X and at $(X, \phi )$, respectively. We say $F^\phi _X$ is $C^k$ if all of its coefficients in the decomposition (3.2) are $C^k$ as functions $D \rightarrow \mathbb {R}$.

Lemma 4.5

Let $D \subset \mathbb {R}$ be open and $F : D \rightarrow \mathcal {W}^*_0(\tilde{g}_0, {{\mathbb {Z}}}^{d})$ be a map. Suppose that $F^\phi _X$ is $C^2$ for all $X \in \mathcal {C}_0$ and $\phi \in \mathbb {C}^\Lambda$, and define $F^{(i)} : D \rightarrow \mathcal {W}^*_0(\tilde{g}_0, {{\mathbb {Z}}}^{d})$ by $(F^{(i)}(t))^\phi _X = (F^\phi _X)^{(i)}(t)$ for $i = 1, 2$, where the right-hand side denotes the (component-wise) $i^\mathrm{th}$ derivative of $F^\phi _X$. If $\Vert F^{(i)}(t)\Vert _{\mathcal {W}_0} < \infty$ for $i = 1, 2$ and $t \in D$, then $F^{(1)}$ is the derivative of F.

Proof

For $t, t + s \in D$, define $R(t, s) \in \mathcal {W}_0$ by

$$\begin{aligned} R^\phi _X(t, s) = F^\phi _X(t + s) - F^\phi _X(t) - s (F^\phi _X)'(t). \end{aligned}$$

(4.63)

By Taylor’s theorem, for any $\phi$ and X,

$$\begin{aligned} R^\phi _X(t, s) = s^2 \int _0^1 (F^\phi _X)''(t + u s) (1 - u) \; du, \end{aligned}$$

(4.64)

where the integral is taken component-wise. It follows that

$$\begin{aligned} \Vert R(t, s)\Vert _{\mathcal {W}_0} \le |s|^2 \sup _{u\in [0,1]} \Vert F''(t+us)\Vert _{\mathcal {W}_0} \le O(|s|^2), \end{aligned}$$

(4.65)

so F is differentiable and its derivative satisfies $(F')^\phi _X = (F^\phi _X)'$. Continuity of $F'$ follows similarly, since, by the fundamental theorem of calculus,

$$\begin{aligned} \Vert F'(t+s) - F'(t)\Vert _{\mathcal {W}_0} \le |s| \sup _{u\in [t,t+s]} \Vert F''(u)\Vert _{\mathcal {W}_0} \le O(|s|), \end{aligned}$$

(4.66)

which suffices. $\square$

Consider the map

$$\begin{aligned} (g_0, \gamma _0, \nu _0, z_0) \mapsto K_0 \in \mathcal {W}^*_0(\tilde{g}_0, {{\mathbb {Z}}}^{d}) \end{aligned}$$

(4.67)

defined by

$$\begin{aligned} K_0(g_0, \gamma _0, \nu _0, z_0) = {\left\{ \begin{array}{ll} K^+_0(g_0, \gamma _0, \nu _0, z_0) &{} (\gamma _0 \ge 0) \\ K^-_0(g_0, \gamma _0, \nu _0, z_0) &{} (\gamma _0 < 0), \end{array}\right. } \end{aligned}$$

(4.68)

for $(g_0, \gamma _0, \nu _0, z_0)$ satisfying the hypotheses of Proposition 4.4. The map $K_0$ is in fact analytic away from $\gamma _0 = 0$. However, we only prove the following, which is what we need later.

Proposition 4.6

Suppose that $V_0^\pm \in \mathcal {D}_0$, with $\tilde{g}_0$ sufficiently small (depending on L) and $|\gamma _0| \le O(\tilde{g}_0^{1+a'})$ for some $a' >a$. The map $K_0(g_0, \gamma _0, \nu _0, z_0)$ is jointly continuous in its four variables, is $C^1$ in $(g_0, \nu _0, z_0)$, and (when $\gamma _0 \ne 0$) is $C^1$ in $(g_0, \gamma _0, \nu _0, z_0)$, with partial derivatives with respect to $t = g_0$, $\nu _0$, and $z_0$ satisfying

$$\begin{aligned} \Vert \partial K_0 / \partial t\Vert _{\mathcal {W}_0} = O(|\gamma _0| \mathfrak {h}_0^8). \end{aligned}$$

(4.69)

Moreover, $K_0$ is left- and right-differentiable in $\gamma _0$ at $\gamma _0 = 0$.

Proof

Let t denote any one of the coupling constants $g_0, \gamma _0, \nu _0$ or $z_0$. We drop the subscript 0, and let K(t) denote $K_0$ viewed as a function of t, with the remaining coupling constants fixed. Then $K^\phi _X$ is smooth for any $\phi , X$. If t is $g_0, \nu _0$ or $z_0$, then

$$\begin{aligned} (K^\phi _x)'&= -M_x(\phi ) K^\phi _x, \quad (K^\phi _x)'' = M_x^2(\phi ) K^\phi _x, \end{aligned}$$

(4.70)

where $M_x$ is $\tau _x^2, \tau _x$ or $\tau _{\Delta ,x}$, respectively. The maximal degree of $M_x$ is 4, so (4.49) implies that

$$\begin{aligned} \Vert K'_x\Vert _{\mathcal {G}_0} \le O(|\gamma _0| \mathfrak {h}_0^{8}), \quad \Vert K''_x\Vert _{\mathcal {G}_0} \le O(|\gamma _0| \mathfrak {h}_0^{12}). \end{aligned}$$

(4.71)

For t denoting $\gamma _0$, we restrict attention to $\gamma _0 > 0$, and write $U = U^+$ and $V_0 = V^+_0$ (the case $\gamma _0 < 0$ is similar). Then

$$\begin{aligned} (K^\phi _x)' = -U_x(\phi ) e^{-V_x(\phi ) - \gamma _0 U_x(\phi )}, \quad (K^\phi _x)'' = U_x^2(\phi ) e^{-V_x(\phi ) - \gamma _0 U_x(\phi )}, \end{aligned}$$

(4.72)

and (4.28) and (4.42) imply that

$$\begin{aligned} \Vert K'_x\Vert _{\mathcal {G}_0} \le O(\mathfrak {h}_0^4), \quad \Vert K''_x\Vert _{\mathcal {G}_0} \le O(\mathfrak {h}_0^8). \end{aligned}$$

(4.73)

By definition, $K_X = \prod _{x \in X} K_x$, so, for derivatives with respect to any one of the four variables (with $\gamma _0 \ne 0$ when differentiating with respect to $\gamma _0$),

$$\begin{aligned} (K^\phi _X)' = \sum _{x \in X} (K^\phi _x)' K^\phi _{X \setminus x}, \quad (K^\phi _X)'' = \sum _{x \in X} ((K^\phi _x)'' K^\phi _{X \setminus x} + (K^\phi _x)' (K^\phi _{X \setminus x})'). \end{aligned}$$

(4.74)

Thus, by the product property, (4.71), and Proposition 4.1,

$$\begin{aligned} \Vert K'_X\Vert _{\mathcal {G}_0} \le O(|X|) |\gamma _0| \mathfrak {h}_0^8 (|\gamma _0| \mathfrak {h}_0^4)^{|X|-1}. \end{aligned}$$

(4.75)

when differentiating with respect to $g_0$, $\nu _0$, or $z_0$. The bound (4.69) then follows from the hypothesis on $\gamma _0$. Similarly, using (4.73),

$$\begin{aligned} \Vert K'_X\Vert _{\mathcal {G}_0} \le O(|X|) \mathfrak {h}_0^4 (|\gamma _0| \mathfrak {h}_0^4)^{|X|-1} \end{aligned}$$

(4.76)

when differentiating with respect to $\gamma _0$ away from $\gamma _0 = 0$. In both cases, we have

$$\begin{aligned} \Vert K''_X\Vert _{\mathcal {G}_0} \le O(|X|^2) \mathfrak {h}_0^8 (|\gamma _0| \mathfrak {h}_0^4)^{(|X|-2) \wedge 0}. \end{aligned}$$

(4.77)

Thus, by Lemma 4.5, K is $C^1$ in any of its variables. Therefore, K is $C^1$ in $(g_0, \nu _0, z_0)$ on the whole domain and in all the variables when $\gamma _0 \ne 0$.

To show right-continuity in $\gamma _0$ at $\gamma _0 = 0$, fix $(g_0, \nu _0, z_0)$ and define $F \in \mathcal {W}^*_0$ by

$$\begin{aligned} F(X) = {\left\{ \begin{array}{ll} -U_x e^{-V_{0,x}} &{} X = \{ x \} \\ 0 &{} |X| > 1, \end{array}\right. } \end{aligned}$$

(4.78)

where $U_x, V_{0,x}$ are defined above. Let $K'(\gamma _0)$ denote the $\gamma _0$ derivative of K evaluated at $\gamma _0 > 0$. Then (4.72) and (4.74) imply that

$$\begin{aligned} F(X) - K'_X(\gamma _0) = {\left\{ \begin{array}{ll} U_x K_x(\gamma _0) &{} X = \{ x \} \\ \sum _{x \in X} K'_x(\gamma _0) K_{X \setminus x}(\gamma _0) &{} |X| > 1. \end{array}\right. } \end{aligned}$$

(4.79)

Thus, by (4.49), (4.73), and Proposition 4.1,

$$\begin{aligned} \Vert F(X) - K'_X(\gamma _0)\Vert _{\mathcal {G}_0} \le {\left\{ \begin{array}{ll} O(\gamma _0 \mathfrak {h}_0^8) &{} X = \{ x \} \\ O(|X|) \mathfrak {h}_0^4 (\gamma _0 \mathfrak {h}_0^4)^{|X|-1} &{} |X| > 1. \end{array}\right. } \end{aligned}$$

(4.80)

It follows that

$$\begin{aligned} \lim _{\gamma _0\downarrow 0} \Vert F - K'(\gamma _0)\Vert _{\mathcal {W}_0} = 0, \end{aligned}$$

(4.81)

i.e., F is the right-derivative of K in $\gamma _0$ at $\gamma _0 = 0$. Left-continuity is handled similarly. $\square$

5 Existence of Critical Parameters

In Sects. 5.1–5.2, we recall some facts about the renormalisation group map defined in [14]. In Sect. 5.3, we discuss the existence and properties of the finite-volume renormalisation group flow (a consequence of the main result of [5]), which is crucial to proving Theorem 1.2. Using the results of Sect. 5.3, we identify critical initial conditions for iteration of the renormalisation group in Sect. 5.4. In Sect. 5.5, we identify the critical point and discuss an important change of parameters. Then in Sect. 5.6 we obtain the asymptotic behaviour of the two-point function, susceptibility, and correlation length of order p, and thereby prove Theorem 1.2. Finally, Sect. 5.7 contains a version of the implicit function theorem that we apply in Sects. 5.4–5.5.

5.1 Renormalisation Group Coordinates

As discussed in Sect. 3.4, the evolution of $Z_j$ defined in (3.24) is tracked via coordinates $(I_j, K_j)$. In order to discuss these, we make the following definitions. We partition $\Lambda$ into $L^{N-j}$ disjoint scale- j blocks of side $L^j$. We let $\mathcal {P}_j$ denote the set of scale- j polymers, which are unions of elements of $\mathcal {B}_j$. Given $X \in \mathcal {P}_j$, we denote the collection of scale-j blocks in X by $\mathcal {B}_j(X)$. Scale-0 blocks are simply elements of $\Lambda$, and scale-0 polymers are subsets of $\Lambda$, as in Section 4.4. Also, as in the scale-0 case, there is a version of blocks and polymers also on ${{\mathbb {Z}}}^{d}$ rather than $\Lambda$.

Given a polynomial $V_j$ of the form

$$\begin{aligned} V_{j;x} = g_j \tau _x^2 + \nu _j \tau _x + z_j \tau _{\Delta ,x}, \end{aligned}$$

(5.1)

the interaction $I_j(X)$ is defined for $X \in \mathcal {P}_j(\Lambda )$ by

$$\begin{aligned} I_j(X) = e^{-\sum _{x\in X} V_{j;x}} \prod _{B \in \mathcal {B}_j(X)} (1 + W_j(B)), \end{aligned}$$

(5.2)

where $W_j(B)$ is an explicit polynomial that is quadratic in $V_j$ and is defined in [4, (3.21)]. In [14, Definition 1.7], a space $\mathcal {K}_j = \mathcal {K}_j(\Lambda )$ of maps $\mathcal {P}_j \rightarrow \mathcal {N}$ required to satisfy several properties is defined. The coordinate $K_j$ is constructed in [14] as an element of $\mathcal {K}_j$. The renormalisation group is used to construct a sequence $(V_j, K_j)$ from which $Z_j$ can be recovered via the circle product

$$\begin{aligned} Z_j = (I_j \circ K_j)(\Lambda ) = \sum _{X\in \mathcal {P}_j(\Lambda )} I_j(\Lambda \setminus X) K_j(X). \end{aligned}$$

(5.3)

5.2 Renormalisation Group Map

We restrict the discussion in this section to a finite volume $\Lambda = \Lambda _N$ with $N > 1$.

For fixed $(\tilde{m}^2, \tilde{g}_0) \in [0, \delta ) \times (0, \delta )$, we define a sequence $\tilde{g}_j = \tilde{g}_j(\tilde{m}^2, \tilde{g}_0)$ as in [3, (6.15)]; in particular, $\tilde{g}_0(\tilde{m}^2, \tilde{g}_0) = \tilde{g}_0$. In [14, Section 1.7.3], a sequence of norms $\Vert \cdot \Vert _{\mathcal {W}_j} = \Vert \cdot \Vert _{\mathcal {W}_j(\tilde{m}^2, \tilde{g}_j, \Lambda )}$ parametrised by $(\tilde{m}^2, \tilde{g}_j)$ is defined on maps $\mathcal {P}_j \rightarrow \mathcal {N}$. We let $\mathcal {W}_j$ denote the subspace of $\mathcal {K}_j$ consisting of all elements having finite $\mathcal {W}_j$ norm. Note that $\mathcal {W}_0 = \mathcal {K}_0 \cap \mathcal {W}_0^*$, where $\mathcal {W}_0^*$ is defined in Sect. 4.5.

In [3, (6.6)–(6.7)], a function ${\vartheta }_j = {\vartheta }_j(m^2)$ (denoted $\chi _j$ in [3]) is defined in such a way that ${\vartheta }_j$ decays exponentially when j is sufficiently large depending on m. We write ${{\tilde{\vartheta }}}_j = {\vartheta }_j(\tilde{m}^2)$. Given a constant $\alpha > 0$, we define the (finite-volume) renormalisation group domains $\mathbb {D}_j \subset \mathbb {R}^3 \oplus \mathcal {W}_j$ by

$$\begin{aligned} \mathbb {D}_j(\tilde{m}^2, \tilde{g}_j, \Lambda ) = \mathcal {D}_j \times B_{\mathcal {W}_j(\tilde{m}^2, \tilde{g}_j, \Lambda )}(\alpha {{\tilde{\vartheta }}}_j \tilde{g}_j^3), \end{aligned}$$

(5.4)

$$\begin{aligned} \mathcal {D}_j = \mathcal {D}_j(\tilde{g}_j) = \{ (g, \nu , z) : C^{-1}_\mathcal {D}\tilde{g}_j< g< C_\mathcal {D}\tilde{g}_j; |z|, L^{2j} |\nu | < C_\mathcal {D}\tilde{g}_j \}. \end{aligned}$$

(5.5)

This definition of $\mathcal {D}_j$ is consistent with (4.23) when $j = 0$. We let $\tilde{\mathbb {I}}_j(\tilde{m}^2)$ be the neighbourhood of $\tilde{m}^2$ defined by

$$\begin{aligned} \tilde{\mathbb {I}}_j = \tilde{\mathbb {I}}_j(\tilde{m}^2) = {\left\{ \begin{array}{ll} \, [\frac{1}{2} \tilde{m}^2, 2 \tilde{m}^2] \cap \mathbb {I}_j &{} (\tilde{m}^2 \ne 0) \\ \, [0,L^{-2(j-1)}] \cap \mathbb {I}_j &{} (\tilde{m}^2 =0), \end{array}\right. } \end{aligned}$$

(5.6)

where $\mathbb {I}_j = [0, \delta ]$ if $j < N$ and $\mathbb {I}_N = [\delta L^{-2 (N - 1)}, \delta ]$. The main result of [14] is the construction of the renormalisation group map on the domains $\mathbb {D}_j$. Although [14] constructs finite- and infinite-volume versions of this map, we only discuss the finite-volume map here.

For $m^2 \in \tilde{\mathbb {I}}_j(\tilde{m}^2)$, the finite-volume renormalisation group map at scale $j = 1, \ldots , N - 1$ is a map $\mathbb {D}_j(\tilde{m}^2, \tilde{g}_j, \Lambda ) \rightarrow \mathbb {R}^3 \oplus \mathcal {W}_{j+1}(\tilde{m}^2, \tilde{g}_{j+1}, \Lambda )$, which we denote

$$\begin{aligned} (V_j, K_j) \mapsto (V_{j+1}, K_{j+1}). \end{aligned}$$

(5.7)

The first component of this map takes the form

$$\begin{aligned} V_{j+1} = V_{\mathrm{pt},j+1}(V_j) + R_{j+1}(V_j, K_j), \end{aligned}$$

(5.8)

where the map $V_{\mathrm{pt},j+1}$ defined in [4] captures the second-order evolution of $V_j$, and $R_{j+1}$ is a third-order contribution. The main properties of the map (5.7) are listed in [3, Section 6.4]. Importantly, the renormalisation group map preserves the circle product in the sense that

$$\begin{aligned} (I_{j+1} \circ K_{j+1})(\Lambda ) = \mathbb {E}_{C_{j+1}}\theta (I_j \circ K_j)(\Lambda ). \end{aligned}$$

(5.9)

Since $\mathcal {P}_N(\Lambda )=\{\varnothing ,\Lambda _N\}$, this means that, if $(V_0, K_0) = (V^\pm _0, K^\pm _0)$ and if the renormalisation group map can be iterated N times with this choice of initial condition, then

$$\begin{aligned} Z_N = I_N(\Lambda ) + K_N(\Lambda ) = e^{-\sum _{x\in \Lambda } V_{N;x}} (1 + W_N(\Lambda )) + K_N(\Lambda ). \end{aligned}$$

(5.10)

5.3 Renormalisation Group Flow

The following theorem is an extension of [3, Proposition 7.1] to non-trivial $K_0$. Such an extension is possible, with only minor modifications to the proof of the $K_0 = \mathbbm {1}_\varnothing$ case, due to the generality allowed by the main result of [5].

The theorem provides, for any $N \ge 1$ and for initial error coordinate $K_0$ in a specified domain, a choice of initial condition $(\nu _0^c,z_0^c)$ for which there exists a finite-volume renormalisation group flow $(V_j, K_j) \in \mathbb {D}_j$ for $0 \le j \le N$. In order to ensure a degree of consistency amongst the sequences $(V_j, K_j)$, which depend on the volume $\Lambda _N$, a notion of consistency must be imposed upon the collection of initial error coordinates $K_{0,\Lambda } \in \mathcal {K}_0(\Lambda )$ for varying $\Lambda$. Specifically, the family $K_{0,\Lambda }$ is required to satisfy the property $({{\mathbb {Z}}}^{d})$ of [14, Definition 1.15]. We refer to any such family as a $\Lambda$-family. As discussed in [14, Definition 1.15], any $\Lambda$-family induces an infinite-volume error coordinate $K_{0,{{\mathbb {Z}}}^{d}} \in \mathcal {K}_0({{\mathbb {Z}}}^{d})$ in a natural way.

Theorem 5.1

Let $d = 4$. There exists a constant $a_* > 0$ and continuous functions $\nu _0^c, z_0^c$ of $(m^2, g_0, K_0)$, defined for $(m^2, g_0) \in [0, \delta ]^2$ (for some $\delta > 0$ sufficiently small) and for any $K_0 \in \mathcal {W}_0(m^2, g_0, {{\mathbb {Z}}}^{d})$ with $\Vert K_0\Vert _{\mathcal {W}_0(m^2, g_0, {{\mathbb {Z}}}^{d})} \le a_* g_0^3$, such that the following holds for $g_0 > 0$: if $K_{0,\Lambda } \in \mathcal {K}_0(\Lambda )$ is a $\Lambda$-family that induces the infinite-volume coordinate $K_0$, and if

$$\begin{aligned} V_0 = V_0^c(m^2, g_0, K_0) = (g_0, \nu _0^c(m^2,g_0,K_0), z_0^c(m^2,g_0,K_0)), \end{aligned}$$

(5.11)

then for any $N \in \mathbb {N}$ and $m^2 \in [\delta L^{-2 (N - 1)}, \delta ]$, there exists a sequence $(V_j, K_j) \in \mathbb {D}_j(m^2, g_0, \Lambda )$ such that

$$\begin{aligned} (V_{j+1},K_{j+1}) = (V_{j+1}(V_j, K_j), K_{j+1}(V_j, K_j)) \ \text { for all } j < N \end{aligned}$$

(5.12)

and (5.3) is satisfied. Moreover, $\nu _0^c,z_0^c$ are continuously differentiable in $g_0 \in (0, \delta )$ and $K_0 \in B_{\mathcal {W}_0(m^2, g_0, \Lambda )}(a_* g_0^3)$, and

$$\begin{aligned}&\nu _0^c(m^2,0,0) = z_0^c(m^2,0,0) = 0, \quad \frac{\partial \nu _0^c}{\partial g_0} = O(1), \quad \frac{\partial z_0^c}{\partial g_0} = O(1), \end{aligned}$$

(5.13)

where the estimates above hold uniformly in $m^2 \in [0, \delta ]$.

Proof

The proof results from small modifications to the proofs of [3, Proposition 7.1] and then to [3, Proposition 8.1], where (in both cases) we relax the requirement that $K_0 = \mathbbm {1}_\varnothing$, which was chosen in [3] due to the fact that $K_0 = \mathbbm {1}_\varnothing$ when $\gamma =0$. The more general condition that $K_0 \in B_{\mathcal {W}_0(m^2, g_0, \Lambda )}(a_* g_0^3)$ comes from the hypothesis of [5, Theorem 1.4] when $(m^2, g_0) = (\tilde{m}^2, \tilde{g}_0)$. By [5, Remark 1.5], no major changes to the proof result from this choice of $K_0$. The following paragraph outlines in more detail the modifications to the proof of [3, Proposition 7.1].

By [5, Theorem 1.4] and [5, Corollary 1.8], for any $(\tilde{m}^2, \tilde{g}_0) \in (0, \delta )^2$ and ${\tilde{K}}_0 \in B_{\mathcal {W}_0(\tilde{m}^2, \tilde{g}_0, {{\mathbb {Z}}}^{d})}(a_* \tilde{g}_0^3)$, there is a neighbourhood $\mathsf{N}(\tilde{g}_0, {\tilde{K}}_0)$ of $(\tilde{g}_0, {\tilde{K}}_0)$ such that for all $(m^2, g_0, K_0) \in \tilde{\mathbb {I}}(\tilde{m}^2) \times \mathsf{N}(\tilde{g}_0, {\tilde{K}}_0)$, there is an infinite-volume renormalisation group flow

$$\begin{aligned} (\check{V}_j, K_j) = \check{x}^d_j(\tilde{m}^2, \tilde{g}_0, {\tilde{K}}_0; m^2, g_0, K_0) \end{aligned}$$

(5.14)

in transformed variables $(\check{V}_j, K_j)$. The transformed variables are defined in [3, Section 6.6] and a flow in the original variables can be recovered from the transformed flow. The global solution is defined by $\check{x}^c_j(m^2, g_0, K_0) = \check{x}^d_j(m^2, g_0, K_0; m^2, g_0, K_0)$ (or $\check{x}^c \equiv 0$ if $g_0 = 0$). By [5, Remark 1.5], the proof of regularity of $\check{x}^c$ can proceed as in [3]. The functions $(\nu _0^c, z_0^c)$ are given by the $(\nu _0, z_0)$ components of $\check{x}^c_0 = (\check{V}_0, K_0) = (V_0, K_0)$. $\square$

Remark 5.2

The proof of [3, Proposition 7.1], hence of Theorem 5.1, makes important use of the parameter $\tilde{g}_0$ in order to prove regularity of the renormalisation group flow in $g_0$. However, once the flow has been constructed, we can and do set $\tilde{g}_0 = g_0$.

Suppose now that we are given some sufficiently small ${\hat{g}}_0 > 0$ and a $\Lambda$-family $K_{0,\Lambda } \in \mathcal {W}_0(m^2, {\hat{g}}_0, \Lambda )$ that satisfies the bounds $\Vert K_{0,\Lambda }\Vert _{\mathcal {W}_0(m^2, {\hat{g}}_0, \Lambda )} \le a_* {\hat{g}}_0^3$. Then in any fixed volume $\Lambda = \Lambda _N$, we can generalise (3.14) by defining $Z_0 = (I_0 \circ K_0)(\Lambda )$ [(3.14) is recovered when we set $K_0 = K^\pm _0$]. We also generalise (3.15) as $Z_N = \mathbb {E}_C\theta Z_0$, and let ${\hat{\chi }}_N(m^2, {\hat{g}}_0, K_0, \nu _0, z_0)$ be defined as in (3.19) from this $Z_N$ [generalising (3.19)]. Then an analogue of [3, Theorem 4.1] (which corresponds to the case $K_0 = \mathbbm {1}_\varnothing$) follows from Theorem 5.1. That is, if $(\nu _0^c, z_0^c) = (\nu _0^c(m^2, {\hat{g}}_0, K_0), z_0^c(m^2, {\hat{g}}_0, K_0))$, then the limit ${\hat{\chi }} = \lim _{N\rightarrow \infty } {\hat{\chi }}_N$ exists and

$$\begin{aligned} {\hat{\chi }} \left( m^2,{\hat{g}}_0,K_0, \nu _0^c, z_0^c \right)&= \frac{1}{m^2}, \end{aligned}$$

(5.15)

$$\begin{aligned} \frac{\partial {\hat{\chi }}}{\partial \nu _0} \left( m^2,{\hat{g}}_0, K_0,\nu _0^c, z_0^c \right)&\sim - \frac{1}{m^4} \frac{c({\hat{g}}_0^*, K_0)}{({\hat{g}}_0^*\mathsf{B}_{m^2})^{1/4}} \quad \text {as} (m^2,{\hat{g}}_0) \rightarrow (0,{\hat{g}}_0^*), \end{aligned}$$

(5.16)

where c is a continuous function and the bubble diagram $\mathsf{B}_{m^2}$ is is asymptotic to $(2\pi ^2)^{-1} \log m^{-2}$, as $m^2 \downarrow 0$, when $d = 4$. For instance, (5.15) follows from (3.19), (5.10), the bound on $K_N$ in Theorem 5.1, and the bound on $W_N$ in [13, (4.57)]. See [3, Section 8.4] for details and for the proof of (5.16).

We wish to obtain a version of (5.15)–(5.16) with the initial conditions of Section 4.1, i.e. with $({\hat{g}}_0, K_0) = (g_0, K^+_0)$ (if $\gamma _0 > 0$) or $({\hat{g}}_0, K_0) = (g_0 + 4d\gamma _0, K^-_0)$ (if $\gamma _0 < 0$). It is straightforward to verify that $K^\pm _0 \in \mathcal {K}_0$. For instance, the fact that $K^\pm _0$ is supersymmetric (which is required of all elements of $\mathcal {K}_0$) follows from the fact that $K^\pm _{0,x}$ is a function of $\tau _x$ (see [4, Section 5.2.1] for more on this topic). It also follows from the definition that the finite-volume coordinates $K^\pm _{0,\Lambda }$ form a $\Lambda$-family.

Moreover, by Proposition 4.4, if $|\gamma _0|$ is sufficiently small (depending on $g_0$; we now take $\tilde{g}_0=g_0$) then $K_0 = K^\pm _0$ satisfies the bound required by Theorem 5.1. However, we cannot apply the theorem immediately with this choice of $K_0$, due to the fact that $K^\pm _0$ depends on $(g_0, \nu _0, z_0)$. We resolve this issue in the next section.

5.4 Critical Parameters

For convenience, let

$$\begin{aligned} {\hat{g}}_0 = {\hat{g}}_0(g_0, \gamma _0) = g_0 + 4 d \gamma _0 \mathbbm {1}_{\gamma _0 < 0}. \end{aligned}$$

(5.17)

Thus, ${\hat{g}}_0$ is the coefficient of $\tau _x^2$ in $V^+_{0,x}$ when $\gamma _0 \ge 0$, and in $V^-_{0,x}$ when $\gamma _0 < 0$. Recall the function $K_0(g_0, \gamma _0, \nu _0, z_0)$ defined in (4.68). We wish to initialise the renormalisation group with $(\nu _0, z_0)$ a solution to the system of equations

$$\begin{aligned}&\nu _0 = \nu _0^c(m^2, {\hat{g}}_0(g_0, \gamma _0), K_0(g_0, \gamma _0, \nu _0, z_0)), \end{aligned}$$

(5.18)

$$\begin{aligned}&z_0 = z_0^c(m^2, {\hat{g}}_0(g_0, \gamma _0), K_0(g_0, \gamma _0, \nu _0, z_0)) . \end{aligned}$$

(5.19)

Such a choice of $(\nu _0, z_0)$ will be critical for $K_0$, where $K_0$ is itself evaluated at this same choice of $(\nu _0, z_0)$.

When $\gamma _0 = 0$, we get $K_0 = \mathbbm {1}_\varnothing$, so $K_0$ no longer depends on $(\nu _0, z_0)$ and this system is solved by $(\nu _0^c(m^2, g_0, 0), z_0^c(m^2, g_0, 0))$ for any (small) $m^2, g_0 \ge 0$. Local solutions for $\gamma _0 \ne 0$ can then be constructed using a version of the implicit function theorem from [25] that allows for the continuous but non-smooth behaviour of $K_0$ in $\gamma _0$. In order to obtain global solutions with certain desired regularity properties (needed in the next section), we make use of Proposition 5.10, which is based on a version of the implicit function theorem from [25].

Suppose $\delta > 0$ and suppose $r : [0, \delta ] \rightarrow [0, \infty )$ is a continuous positive-definite function; the latter means that $r(x) > 0$ if $x > 0$ and $r(0) = 0$. We define

$$\begin{aligned} D(\delta , r) = \{ (w, x, y) \in [0, \delta ]^2 \times (-\delta , \delta ) : |y| \le r(x) \} \end{aligned}$$

(5.20)

and we let $C^{0,1,\pm }(D(\delta , r))$ denote the space of continuous functions on $D(\delta , r)$ that are $C^1$ in (x, y) away from $y = 0$, $C^1$ in x everywhere, and have left- and right-derivatives in y at $y = 0$. In our applications, we take $w = m^2$, $x = g_0$ or $\beta$, and $y = \gamma _0$ or $\gamma$.

Proposition 5.3

There exists a continuous positive-definite function ${\hat{r}} : [0, \delta ] \rightarrow [0, \infty )$ and continuous functions ${\hat{\mu }}_0^c, {\hat{z}}_0^c \in C^{0,1,\pm }(D(\delta , {\hat{r}}))$ such that the system (5.18)–(5.19) is solved by $(\nu _0, z_0) = ({\hat{\mu }}_0^c, {\hat{z}}_0^c)$ whenever $(m^2, g_0, \gamma _0) \in D(\delta , {\hat{r}})$. Moreover, these functions satisfy the bounds

$$\begin{aligned} {\hat{\mu }}_0^c = O(g_0), \quad {\hat{z}}_0^c = O(g_0) \end{aligned}$$

(5.21)

uniformly in $(m^2, \gamma _0)$.

Proof

Recall the definition of ${\hat{g}}_0$ in (5.17), and let

$$\begin{aligned} F(m^2, g_0, \gamma _0, \nu _0, z_0) = (\nu _0, z_0) - (\nu _0^c(m^2, {\hat{g}}_0, K_0), z_0^c(m^2, {\hat{g}}_0, K_0) ), \end{aligned}$$

(5.22)

where $K_0 = K_0(g_0, \gamma _0, \nu _0, z_0)$. Then for $\delta > 0$ small and an appropriate constant $c > 0$ (depending on $a_*$), F is well-defined on

$$\begin{aligned} \{ (m^2, g_0, \gamma _0, \nu _0, z_0) : (m^2, {\hat{g}}_0, \gamma _0) \in D(\delta , c g_0^3), |\nu _0|, |z_0| \le C_\mathcal {D}g_0 \}. \end{aligned}$$

(5.23)

Indeed, for $(m^2, g_0, \gamma _0, \nu _0, z_0)$ in this domain, Proposition 4.4 (with $\tilde{g}_0 = g_0$) implies that $(m^2, {\hat{g}}_0, K_0)$ is in the domain of $(\nu _0^c, z_0^c)$. By Theorem 5.1 and Proposition 4.6, F is $C^1$ in $(g_0, \nu _0, z_0)$ and also in $\gamma _0$ away from $\gamma _0 = 0$, continuous in $m^2$, and has one-sided derivatives in $\gamma _0$ at $\gamma _0 = 0$.

For fixed $({\bar{m}}^2, {{\bar{g}}_0}) \in [0, \delta ]^2$, set $({\bar{\nu }_0}, {\bar{z}}_0) = (\nu _0^c({\bar{m}}^2, {\bar{g}}_0, 0), z_0^c({\bar{m}}^2, {\bar{g}}_0, 0))$ so that

$$\begin{aligned} F({\bar{m}}^2, {\bar{g}}_0, 0, \bar{\nu }_0, {\bar{z}}_0) = (0, 0). \end{aligned}$$

(5.24)

By (4.69), at $({\bar{g}}_0, 0, \bar{\nu }_0, {\bar{z}}_0)$,

$$\begin{aligned} \frac{\partial K_{0,x}}{\partial \nu _0} = \frac{\partial K_{0,x}}{\partial z_0} = 0. \end{aligned}$$

(5.25)

It follows that $D_{\nu _0,z_0} F({\bar{m}}^2, {\bar{g}}_0, 0, \bar{\nu }_0, {\bar{z}}_0)$ is the identity map on $\mathbb {R}^2$. The existence of $\delta , {\hat{r}}$ and ${\hat{\mu }}_0^c, {\hat{z}}_0^c$ follows from Proposition 5.10 with $w = m^2, x = g_0, y = \gamma _0, z = (\nu _0, z_0)$, and with $r_1(g_0) = c g_0^3$, $r_2(g_0) = C_\mathcal {D}g_0$.

By the fundamental theorem of calculus, for any $0< a < \gamma _0$,

$$\begin{aligned} {\hat{\mu }}_0^c(m^2, g_0, \gamma _0) = {\hat{\mu }}_0^c(m^2, g_0, a) + \int _a^{\gamma _0} \frac{\partial {\hat{\mu }}_0^c}{\partial \gamma _0} (m^2, g_0, t) \; dt. \end{aligned}$$

(5.26)

Taking the limit $a\downarrow 0$ and using (5.13), we obtain

$$\begin{aligned} |{\hat{\mu }}_0^c(m^2, g_0, \gamma _0)| \le O(g_0) + \gamma _0 \sup _{t \in (0, \gamma _0]} \left| \frac{\partial {\hat{\mu }}_0^c}{\partial \gamma _0}(m^2, g_0, t)\right| . \end{aligned}$$

(5.27)

The supremum above is bounded by a constant and so the first estimate of (5.21) for $\gamma _0 \ge 0$ follows from the fact that $|\gamma _0| \le {\hat{r}}(g_0)$ (since ${\hat{r}}(g_0)$ can be taken as small as desired). The case $\gamma _0 < 0$ and the second estimate follow similarly. $\square$

Corollary 5.4

Fix $(m^2, g_0, \gamma _0) \in D(\delta , {\hat{r}})$ with $g_0 > 0$ and $m^2 \in [\delta L^{-2 (N - 1)}, \delta )$ and set $(V_0, K_0) = (V^\pm _0, K^\pm _0)$ with $(\nu _0, z_0) = ({\hat{\mu }}_0^c, {\hat{z}}_0^c)$. Then for any $N \in \mathbb {N}$, there exists a sequence $(V_j, K_j) \in \mathbb {D}_j(m^2, g_0, \Lambda )$ such that

$$\begin{aligned} (V_{j+1},K_{j+1}) = (V_{j+1}(V_j, K_j), K_{j+1}(V_j, K_j)) \text { for all } j < N \end{aligned}$$

(5.28)

and (5.3) is satisfied. Moreover, the second-order evolution equation for $V_j$ is independent of $\gamma _0$.

Proof

By Proposition 4.4, and by taking ${\hat{r}}$ smaller if necessary, $K_0 = K^\pm _0$ satisfies the estimate required by Theorem 5.1 whenever $(m^2, g_0, \gamma _0) \in D(\delta , {\hat{r}})$. The existence of the sequence (5.28) then follows from Theorem 5.1 and Proposition 5.3. Although the presence of $\gamma _0$ causes a shift in initial conditions, the second-order evolution of $V_j$ is still given by the map $V_\mathrm{pt}$ [see (5.8)], which is independent of $\gamma _0$. $\square$

By (3.19), ${\hat{\chi }}(m^2, g_0, \gamma _0, \nu _0, z_0) = {\hat{\chi }}(m^2, g_0, K_0, \nu _0, z_0)$, where $K_0 = K_0(g_0, \gamma _0, \nu _0, z_0)$ is defined in (4.68). Then by (5.15)–(5.16), Corollary 5.4, and (4.69), with ${\hat{g}}_0 = {\hat{g}}_0(g_0, \gamma _0)$, we have

$$\begin{aligned} {\hat{\chi }} \left( m^2,{\hat{g}}_0,\gamma _0, {\hat{\mu }}_0^c, {\hat{z}}_0^c \right)&= \frac{1}{m^2}, \end{aligned}$$

(5.29)

$$\begin{aligned} \frac{\partial {\hat{\chi }}}{\partial \nu _0} \left( m^2,{\hat{g}}_0, \gamma _0,{\hat{\mu }}_0^c, {\hat{z}}_0^c \right)&\sim - \frac{1}{m^4} \frac{c({\hat{g}}_0^*, \gamma _0)}{({\hat{g}}_0^*\mathsf{B}_{m^2})^{1/4}} \quad \text {as} (m^2,g_0,\gamma _0) \rightarrow (0,g_0^*,\gamma _0^*), \end{aligned}$$

(5.30)

where ${\hat{g}}_0^* = {\hat{g}}_0(g_0^*, \gamma _0^*)$ and we write $c(g_0, \gamma _0) = c(g_0, K_0)$. Although (5.30) depends on $\gamma _0$, this dependence ultimately only affects the computation of the critical point $\nu _c(\beta , \gamma )$ and the constants $A_{\beta ,\gamma }, B_{\beta ,\gamma }$ in the proof of Theorem 1.2. The asymptotic behaviour of the susceptibility in (1.21) results from the logarithmic divergence of the bubble diagram $\mathsf{B}_{m^2}$ and the exponent $\frac{1}{4}$ that appears in the denominator in (5.30).

Remark 5.5

We have invoked (4.69) above in order to satisfy the condition

$$\begin{aligned} \Vert \partial K_0/\partial \nu _0\Vert _{\mathcal {W}_0} \le O(g_0^3) \end{aligned}$$

(5.31)

required in the proof of [3, Lemma 8.6] (see [3, (8.34)]). This condition holds trivially when $K_0$ does not depend on $\nu _0$, as in (5.15)–(5.16).

5.5 Change of Parameters

Recall from (3.18) that

$$\begin{aligned} \chi _N(\beta , \gamma , \nu ) = (1 + z_0) {\hat{\chi }}_N(m^2, g_0, \gamma _0, \nu _0, z_0), \end{aligned}$$

(5.32)

whenever the variables on the left- and right-hand sides satisfy

$$\begin{aligned} g_0 = (\beta - \gamma ) (1 + z_0)^2, \quad \nu _0 = \nu (1 + z_0) - m^2, \quad \gamma _0 = \frac{1}{4d} \gamma (1 + z_0)^2. \end{aligned}$$

(5.33)

Given $\beta ,\gamma ,\nu$, these relations leave free two of the variables $(m^2, g_0, \gamma _0, \nu _0, z_0)$. More generally, if any three of the variables $(\beta , \gamma , \nu , m^2, g_0, \gamma _0, \nu _0, z_0)$ are fixed, then two of the remaining variables are free. In the following two propositions, which together form an extension of [3, Proposition 4.2], we fix three variables and show that the addition of the constraints

$$\begin{aligned} \nu _0 = {\hat{\nu }}_0^c(m^2, g_0, \gamma _0), \quad z_0 = {\hat{z}}_0^c(m^2, g_0, \gamma _0) \end{aligned}$$

(5.34)

allows us to uniquely specify the two remaining variables. First, in Proposition 5.6, the three fixed variables are $(m^2, \beta , \gamma )$.

Proposition 5.6

There exist $\delta _* > 0$, a continuous positive-definite function $r_* : [0, \delta _*] \rightarrow [0, \infty )$, and continuous functions $(\nu ^*, g_0^*, \gamma _0^*, \nu _0^*, z_0^*)$ defined for $(m^2, \beta , \gamma ) \in D(\delta _*, r_*)$, such that (5.33) and (5.34) hold with $\nu = \nu ^*$ and $(g_0, \gamma _0, \nu _0, z_0) = (g^*_0, \gamma ^*_0, \nu ^*_0, z^*_0)$. Moreover,

$$\begin{aligned} g_0^* = \beta + O(\beta ^2), \quad \nu _0^* = O(\beta ), \quad z_0^* = O(\beta ). \end{aligned}$$

(5.35)

Proof

Suppose we have found the desired continuous functions $(g_0^*, \gamma _0^*)$ and that $g_0^*$ satisfies the first bound in (5.35). Then the functions defined by

$$\begin{aligned}&\nu _0^* = {\hat{\mu }}_0^c(m^2, g_0^*, \gamma _0^*), \quad z_0^* = {\hat{z}}_0^c(m^2, g_0^*, \gamma _0^*), \quad \nu ^* = \frac{\nu _0^* + m^2}{1 + z_0^*} \end{aligned}$$

(5.36)

are continuous, (5.33) is satisfied, and the remaining bounds in (5.35) follow using (5.21).

We first solve the third equation of (5.33), and then solve the first equation of (5.33). To this end, we begin by defining

$$\begin{aligned} f_1(m^2, g_0, \gamma , \gamma _0) = \gamma _0 - (4d)^{-1} \gamma (1 + {\hat{z}}_0^c(m^2, g_0, \gamma _0))^2 \end{aligned}$$

(5.37)

for $(m^2, g_0, \gamma _0) \in D(\delta , {\hat{r}})$ and $|\gamma | \le {\hat{r}}(g_0)$ (recall that ${\hat{r}}$ is defined in Proposition 5.3); although $f_1$ is well-defined for any $\gamma \in \mathbb {R}$, we restrict the domain in preparation for our application of Proposition 5.10. Note that $f_1$ is $C^1$ in $\gamma$ and $f_1(\cdot , \cdot , \gamma , \cdot ) \in C^{0,1,\pm }(D(\delta , {\hat{r}}))$ for any $\gamma$. The equation $f_1(m^2, g_0, \gamma , \gamma _0) = 0$ has the solution $\gamma _0 = 0$ when $\gamma = 0$ and, for any $\gamma _0 \ne 0$,

$$\begin{aligned} \frac{\partial f_1}{\partial \gamma _0} = 1 - (2d)^{-1} \gamma (1 + {\hat{z}}_0^c(m^2, g_0, \gamma _0)) \frac{\partial {\hat{z}}_0^c}{\partial \gamma _0}. \end{aligned}$$

(5.38)

Since the one-sided $\gamma _0$ derivatives of ${\hat{z}}_0^c$ exist at $\gamma _0 = 0$, we can see that the $\gamma _0$ derivative of $f_1$ is well-defined and equal to 1 when $\gamma = 0$ for any small $\gamma _0$ (including $\gamma _0 = 0$). Thus, by Proposition 5.10 (with $w = m^2$, $x = g_0$, $y = \gamma$, $z = \gamma _0$ and $r_1 = r_2 = {\hat{r}}$), there exists a continuous function $\gamma ^{(1)}_0(m^2, g_0, \gamma )$ on $D(\delta , r^{(1)})$ (for some continuous positive-definite function $r^{(1)}$ on $[0, \delta ]$) such that $f_1(m^2, g_0, \gamma , \gamma ^{(1)}_0) = 0$. Moreover, $\gamma ^{(1)}_0$ is $C^1$ in $(g_0, \gamma )$.

Next, we define

$$\begin{aligned} f_2(m^2, \beta , \gamma , g_0) = g_0 - (\beta - \gamma ) (1 + {\hat{z}}_0^c(m^2, g_0, \gamma ^{(1)}_0(m^2, g_0, \gamma )))^2 \end{aligned}$$

(5.39)

for $(m^2, g_0, \gamma ) \in D(\delta , r^{(1)})$ and $\beta \in [0, \delta _*]$, where $\delta _* > 0$ will be made sufficiently small below. Then $f_2(m^2, \beta , \gamma , g_0) = 0$ is solved by $(\gamma , g_0) = (0, g_0^*(m^2, \beta , 0))$, where $g_0^*(m^2, \beta , 0)$ was constructed in [3, (4.35)]. By [3, (4.37)], $g_0^* = \beta + O(\beta ^2)$, so we may restrict the domain of $f_2$ so that $|g_0| \le 2 \beta$. Moreover,

$$\begin{aligned} \frac{\partial f_2}{\partial g_0} = 1 - 2 (\beta - \gamma ) (1 + {\hat{z}}_0^c(m^2, g_0, \gamma ^{(1)}_0)) \left( \frac{\partial {\hat{z}}_0^c}{\partial g_0} + \frac{\partial {\hat{z}}_0^c}{\partial \gamma _0} \frac{\partial \gamma ^{(1)}_0}{\partial g_0} \right) . \end{aligned}$$

(5.40)

Differentiating both sides of

$$\begin{aligned} \gamma ^{(1)}_0 = \frac{1}{4d} \gamma (1 + {\hat{z}}_0^c(m^2, g_0, \gamma ^{(1)}_0))^2, \end{aligned}$$

(5.41)

and solving for $\frac{\partial \gamma ^{(1)}_0}{\partial g_0}$, gives

$$\begin{aligned} \frac{\partial \gamma ^{(1)}_0}{\partial g_0} = \frac{\gamma (1 + {\hat{z}}_0^c) \frac{\partial {\hat{z}}_0^c}{\partial g_0}}{2 d - \gamma (1 + {\hat{z}}_0^c) \frac{\partial {\hat{z}}_0^c}{\partial \gamma _0}}, \end{aligned}$$

(5.42)

where ${\hat{z}}_0^c$ and its derivatives are evaluated at $(m^2, g_0, \gamma ^{(1)}_0)$. Thus, $\frac{\partial \gamma ^{(1)}_0}{\partial g_0} = 0$ when $\gamma = 0$. It follows that $\partial f_2/\partial g_0$ is well-defined when $(\gamma , g_0) = (0, g_0^*(m^2, \beta , 0))$ and equals

$$\begin{aligned} 1 - 2 \beta (1 + {\hat{z}}_0^c(m^2, g_0^*, 0)) \frac{\partial {\hat{z}}_0^c}{\partial g_0}(m^2, \beta , 0, g_0^*), \end{aligned}$$

(5.43)

which is positive when $\delta _*$ is small, by (5.21). Thus, by Proposition 5.10 (with $w = m^2$, $x = \beta$, $y = \gamma$, $z = g_0$ and $r_1 = r^{(1)}$, $r_2(\beta ) = 2\beta$), there exists a function $g_0^*(m^2, \beta , \gamma ) \in C^{0,1,\pm }(D(\delta _*, r^{(2)}))$ (for some continuous positive-definite function $r^{(2)}$ on $[0, \delta _*]$) such that $f_2(m^2, \beta , \gamma , g_0^*) = 0$.

By the fact that $g_0^*$ solves $f_2 = 0$,

$$\begin{aligned} g_0^* = (\beta - \gamma ) + O((\beta - \gamma )^2). \end{aligned}$$

(5.44)

Since $|\gamma | \le r^{(2)}(g_0)$ and $r^{(2)}(g_0)$ can be taken as small as desired, this implies the first estimate in (5.35). Thus, by taking $r_*$ sufficiently small, if $|\gamma | \le r_*(\beta )$, then $|\gamma | \le r^{(2)}(g_0^*(m^2, \beta , \gamma ))$. Thus, for $\beta < \delta _*$ and $|\gamma | \le r_*(\beta )$, we can define

$$\begin{aligned} \gamma _0^*(m^2, \beta , \gamma ) = \gamma ^{(1)}_0(m^2, g_0^*(m^2, \beta , \gamma ), \gamma ), \end{aligned}$$

(5.45)

which completes the proof. $\square$

Using Proposition 5.6, it is possible to identify the critical point $\nu _c$, as follows. By (5.29), (5.32), Proposition 2.2, and Proposition 5.6,

$$\begin{aligned} \chi (\beta , \gamma , \nu ^*) = \frac{1 + z_0^*}{m^2} = \frac{1 + O(\beta )}{m^2}. \end{aligned}$$

(5.46)

Thus, with $\nu = \nu ^*$, we see that $\chi < \infty$ when $m^2 > 0$, and $\chi = \infty$ when $m^2 = 0$. By (1.10), this implies that

$$\begin{aligned} \nu _c(\beta , \gamma ) = \nu ^*(0, \beta , \gamma ) = O(\beta ), \quad \nu _c(\beta , \gamma ) < \nu ^*(m^2, \beta , \gamma ) \quad (m^2 > 0). \end{aligned}$$

(5.47)

It follows that

$$\begin{aligned} \chi (\beta , \gamma , \nu _c) = \infty , \end{aligned}$$

(5.48)

which is a fact that cannot be concluded immediately from the definition (1.10).

In (5.46), $\chi$ is evaluated at $\nu ^* = \nu ^*(m^2, \beta , \gamma )$. However, in the setting of Theorem 1.2, we need to evaluate $\chi$ at a given value of $\nu$ and then take $\nu \downarrow \nu _c$. To do so, we must determine a choice of $m^2$ in terms of $\nu$ such that (5.33) is satisfied and this choice must approach 0 (as it should by (5.47)) right-continuously as $\nu \downarrow \nu _c$. The following proposition carries out this construction. In the following, the functions ${\tilde{m}}^2, {\tilde{g}}_0$ should not be confused with the parameter $\tilde{m}^2, \tilde{g}_0$ that appeared previously in the $\mathcal {W}_j$ norms.

Proposition 5.7

Write $\nu = \nu _c + \varepsilon$. There exist functions ${\tilde{m}}^2, {\tilde{g}}_0, {\tilde{\gamma }}_0, {\tilde{\nu }}_0, {\tilde{z}}_0$ of $(\varepsilon , \beta , \gamma ) \in D(\delta _*, r_*)$ (all right-continuous as $\varepsilon \downarrow 0$) such that (5.33) and (5.34) hold with

$$\begin{aligned} (m^2, g_0, \gamma _0, \nu _0, z_0) = ({\tilde{m}}^2, {\tilde{g}}_0, {\tilde{\gamma }}_0, {\tilde{\nu }}_0, {\tilde{z}}_0). \end{aligned}$$

(5.49)

Moreover,

$$\begin{aligned} {\tilde{m}}^2(0, \beta , \gamma ) = 0, \qquad {\tilde{m}}^2(\varepsilon , \beta , \gamma )> 0 \quad (\varepsilon > 0). \end{aligned}$$

(5.50)

$$\begin{aligned} {\tilde{g}}_0 = \beta + O(\beta ^2), \quad {\tilde{\nu }}_0 = O(\beta ), \quad {\tilde{z}}_0 = O(\beta ). \end{aligned}$$

(5.51)

Proof

The proof is a minor modification of the proof in [3], using Proposition 5.6. Define

$$\begin{aligned} {\tilde{m}}^2 = {\tilde{m}}^2 (\varepsilon ,\beta ,\gamma ) = \inf \{m^2 > 0 : \nu ^*(m^2, \beta , \gamma ) = \nu _c(\beta , \gamma ) + \varepsilon \}, \end{aligned}$$

(5.52)

on $D(\delta _*, r_*)$. By continuity of $\nu ^*$, the infimum is attained and

$$\begin{aligned} \nu _c(\beta , \gamma ) + \varepsilon = \nu ^*({\tilde{m}}^2(\varepsilon , \beta , \gamma ), \beta , \gamma ). \end{aligned}$$

(5.53)

From the above expression, continuity of $\nu ^*$, and (5.47), it follows that ${\tilde{m}}^2$ is right-continuous as $\varepsilon \downarrow 0$. It is immediate that (5.50) holds. Also, the functions of $(\varepsilon ,\beta ,\gamma )$ defined by

$$\begin{aligned}&{\tilde{\nu }}_0 = \nu _0^*({\tilde{m}}^2, \beta , \gamma ), \quad {\tilde{z}}_0 = z_0^*({\tilde{m}}^2, \beta , \gamma ), \end{aligned}$$

(5.54)

$$\begin{aligned}&{\tilde{g}}_0 = (\beta - \gamma ) (1 + {\tilde{z}}_0)^2 ,\quad {\tilde{\gamma }}_0 = \frac{1}{4d} \gamma (1 + {\tilde{z}}_0)^2 \end{aligned}$$

(5.55)

are right-continuous as $\varepsilon \downarrow 0$ and satisfy (5.33). The bounds (5.51) follow from the definitions and (5.35), and the proof is complete. $\square$

5.6 Conclusion of the Argument

By (5.29), (5.32), Propositions 2.2 and 5.7

$$\begin{aligned} \chi (\beta , \gamma , \nu ) = \frac{1 + {\tilde{z}}_0}{{\tilde{m}}^2}. \end{aligned}$$

(5.56)

Using this, (5.29), and (5.30), by exactly the same argument as in [3, Section 4.3], there is a differential relation between $\frac{\partial \chi }{\partial \nu }$ and $\chi$, whose solution yields Theorem 1.2(ii).

The reason the susceptibility is handled first is that its leading-order critical behaviour can be computed from the second-order flow of the bulk coupling constants $(g_j, \nu _j, z_j)$. In contrast, in order to study the two-point function, we begin by writing

$$\begin{aligned} {\bar{\phi }}_a \phi _b = \frac{\partial ^2}{\partial \sigma _a\partial \sigma _b} e^{\sigma _a{\bar{\phi }}_a + \sigma _b\phi _b} \Big |_{\sigma _a=\sigma _b=0} \end{aligned}$$

(5.57)

in (3.17). The incorporation of the exponential function $e^{\sigma _a{\bar{\phi }}_a + \sigma _b\phi _b}$ into $Z_0$ is equivalent to subtracting

$$\begin{aligned} \sigma _a {\bar{\phi }}_a \mathbbm {1}_{x=a} + \sigma _b {\bar{\phi }}_b \mathbbm {1}_{x=b} \end{aligned}$$

(5.58)

from $V^\pm _0$. The renormalisation group map now acts on a polynomial of the form

$$\begin{aligned} g_j \tau ^2 + \nu _j \tau + z_j \tau _\Delta&- \lambda _{a,j} \sigma _a {\bar{\phi }}_a \mathbbm {1}_{x=a} - \lambda _{b,j} \sigma _b \phi _b \mathbbm {1}_{x=b}\nonumber \\&- \frac{1}{2} \sigma _a \sigma _b (q_{a,j} \mathbbm {1}_{x=a} + q_{b,j} \mathbbm {1}_{x=b}). \end{aligned}$$

(5.59)

We have only included terms up to second order in $(\sigma _a, \sigma _b)$ because, by (5.57), only these are needed to study the two-point function. The coefficients $(\lambda _{a,j}, \lambda _{b,j}, q_{a,j}, q_{b,j})$ are referred to as observable coupling constants and the behaviour of these coupling constants under the action of the renormalisation group is studied in detail in [2, 27].

It was shown in [2] that the observable flow does not affect the bulk flow. Moreover, the second-order evolution of the observable flow remains identical to that of the case $\gamma _0 = 0$. This occurs for the same reason that the bulk flow is unaffected to second order by $\gamma _0$ (as in the statement of Corollary 5.4): namely, the second-order contributions to the observable flow are produced by an extension of the map $V_\mathrm{pt}$ [recall (5.8)], whose definition does not depend on $\gamma _0$. Thus, the analysis of the observable flow when $\gamma _0$ is small can proceed in the same way as when $\gamma _0 = 0$. That is, the same analysis that was carried out in [2] to study the two-point function applies directly here to prove Theorem 1.2(i).

The analysis of the correlation length of order p in [6] also applies directly here, and for the same reason: the second-order flow of coupling constants is independent of $\gamma _0$. This gives Theorem 1.2(iii).

5.7 A Version of the Implicit Function Theorem

We make use of [25, Chapter 4, Theorem 9.3], which is a version of the implicit function theorem that allows for a continuous, rather than differentiable, parameter. While the precise statement of [25, Chapter 4, Theorem 9.3] takes this parameter from an open subset of a Banach space, by [25, Chapter 4, Theorem 9.2], the parameter can in fact be taken from an arbitrary metric space. With this minor change, we restate [25, Chapter 4, Theorem 9.3] as the following proposition.

Proposition 5.8

Let A be a metric space, let W, X be Banach spaces, and let $B \subset W$ be an open subset. Let $F : A \times B \rightarrow X$ be continuous, and suppose that F is $C^1$ in its second argument. Let $(\alpha , \beta ) \in A \times B$ be a point such that $F(\alpha , \beta ) = 0$ and $D_2 F(\alpha , \beta )^{-1}$ exists. Then there are open balls $M \ni \alpha$ and $N \ni \beta$ and a unique continuous mapping $f : M \rightarrow N$ such that $F(\xi , f(\xi )) = 0$ for all $\xi \in M$.

We also use the following lemma, which is a small modification of [25, Chapter 3, Theorem 11.1]. In particular, it considers functions that may only be left- or right-differentiable.

Lemma 5.9

Let F be a mapping as in the previous proposition with $A \subset \mathbb {R}^{m_1} \times \mathbb {R}^{m_2}$. In addition, suppose that F is left-differentiable (respectively, right-differentiable) in $\alpha _2$ at $(\alpha , \beta )$, with $\alpha = (\alpha _1, \alpha _2)$. If f is a continuous mapping defined in a neighbourhood of $\alpha$, such that $F(\xi , f(\xi )) = 0$, then f is left-differentiable (respectively, right-differentiable) in $\alpha _2$ at $\alpha$.

The above results lead to the following proposition, which we apply in the proofs of Propositions 5.3 and 5.6. Recall that $D(\delta , r)$ is defined in (5.20).

Proposition 5.10

Let $\delta > 0$, and let $r_1, r_2$ be continuous positive-definite functions on $[0, \delta ]$. Set

$$\begin{aligned} D(\delta , r_1, r_2) = \{ (w, x, y, z) \in D(\delta , r_1) \times \mathbb {R}^n : |z| \le r_2(x) \}, \end{aligned}$$

(5.60)

and let F be a continuous function on $D(\delta , r_1, r_2)$ that is $C^1$ in (x, z). Suppose that for all $({\bar{w}}, {\bar{x}}) \in [0, \delta ]^2$ there exists ${\bar{z}}$ such that both $F({\bar{w}}, {\bar{x}}, 0, {\bar{z}}) = 0$ and $D_Y F({\bar{w}}, {\bar{x}}, 0, {\bar{z}})$ is invertible. Then there is a continuous positive-definite function r on $[0, \delta ]$ and a continuous map $f : D(\delta , r) \rightarrow \mathbb {R}^n$ that is $C^1$ in x and such that $F(w, x, y, f(w, x, y)) = 0$ for all $(w, x, y) \in D(\delta , r)$. Moreover, if F is left-differentiable (respectively, right-differentiable) in y at some point (w, x, y, z), then f is left-differentiable (respectively, right-differentiable) at (w, x, y).

Proof

Take any $({\bar{w}}, {\bar{x}}) \in [0, \delta ] \times (0, \delta ]$ and let $R({\bar{w}}, {\bar{x}})$ be the maximal radius s such that for all $(w, x, y) \in B({\bar{w}}, {\bar{x}}, 0; s)$ there exists z such that both $F(w, x, y, z) = 0$ and $D_Z F(w, x, y, z)$ is invertible. By continuity of $(D_Z F(w, x, y, z))^{-1}$ near $({\bar{w}}, {\bar{x}}, 0, {\bar{z}})$, and by Proposition 5.8 (applied to the restriction of F to $A \times B$, for some $A \ni ({\bar{w}}, {\bar{x}}, 0)$ and an open set $B \ni {\bar{z}}$), we have $R({\bar{w}}, {\bar{x}}) > 0$ and there is a continuous function

$$\begin{aligned} f_{{\bar{w}},{\bar{x}}} : B({\bar{w}}, {\bar{x}}, 0; R({\bar{w}}, {\bar{x}})) \rightarrow \mathbb {R}^n \end{aligned}$$

(5.61)

such that $F(w, x, y, f_{{\bar{w}},{\bar{x}}}(w, x, y)) = 0$ for all $(w, x, y) \in B({\bar{w}}, {\bar{x}}, 0; R({\bar{w}}, {\bar{x}}))$. Moreover, the unique solution to $F(w, x, y, z) = 0$ is given by $z = f_{{\bar{w}},{\bar{x}}}(w, x, y)$ for all such (w, x, y). By an application of Lemma 5.9 (with $\alpha _1 = (w, x), \alpha _2 = y$), we see that $f_{{\bar{w}}, {\bar{x}}}$ is left- or right-differentiable in y wherever F is. By another application of Lemma 5.9 (with $\alpha _1 = (w, y), \alpha _2 = x$), we see that $f_{{\bar{w}}, {\bar{x}}}$ is $C^1$ in x.

Set $R({\bar{w}}, 0) = 0$ for all ${\bar{w}} \in [0, \delta ]$, and let

$$\begin{aligned} D_f = \bigcup _{({\bar{w}},{\bar{x}})\in [0, \delta ]^2} B({\bar{w}}, {\bar{x}}, 0; R({\bar{w}}, {\bar{x}})). \end{aligned}$$

(5.62)

We define $f(w, 0, 0) = 0$ and, for $x > 0$,

$$\begin{aligned} f(w, x, y) = f_{{\bar{w}},{\bar{x}}}(w, x, y) \quad \text {for}\quad (w, x, y) \in B({\bar{w}}, {\bar{x}}, 0; R({\bar{w}}, {\bar{x}})). \end{aligned}$$

(5.63)

By uniqueness, this function is well-defined. Continuity of f at (w, 0, 0) follows from the fact that $|f(w, x, y)| \le r_2(x)$. The remaining desired regularity properties of f follow from those of the $f_{{\bar{w}},{\bar{x}}}$. It remains to show that $D(\delta ,r) \subset D_f$ for some continuous positive-definite function r on $[0, \delta ]$.

First, let us show that R is continuous on $[0, \delta ]^2$. Let ${\bar{x}} > 0$ and fix $0< \epsilon < R({\bar{w}}, {\bar{x}})$. Then for any $({\bar{w}}', {\bar{x}}') \in [0,\delta ] \times (0, \delta ]$ such that $|({\bar{w}}, {\bar{x}}) - ({\bar{w}}', {\bar{x}}')| < \epsilon$, we have $B({\bar{w}}', {\bar{x}}', 0; R({\bar{w}}, {\bar{x}}) - \epsilon ) \subset B({\bar{w}}, {\bar{x}}, 0; R({\bar{w}}, {\bar{x}}))$ by maximality of R. It follows that $R({\bar{w}}', {\bar{x}}') \ge R({\bar{w}}, {\bar{x}}) - \epsilon$. By a similar argument, $R({\bar{w}}', {\bar{x}}') \le R({\bar{w}}, {\bar{x}}) + \epsilon$, so $|R({\bar{w}}, {\bar{x}}) - R({\bar{w}}', {\bar{x}}')| \le \epsilon$. Thus, R is continuous on $[0, \delta ] \times (0, \delta ]$. Continuity at ${\bar{x}} = 0$ follows from the fact that $R({\bar{w}}, {\bar{x}}) \le r_1({\bar{x}})$ uniformly in ${\bar{w}}$.

For ${\bar{x}} \in [0,\delta ]$, let

$$\begin{aligned} r({\bar{x}}) = \inf (R({\bar{w}}, {\bar{x}}) : {\bar{w}} \in [0, \delta ]). \end{aligned}$$

(5.64)

Since $R(\cdot , {\bar{x}})$ is continuous, $r({\bar{x}}) > 0$ for ${\bar{x}} > 0$. Moreover, $0 \le r(0) \le r_1(0) = 0$, so r is positive-definite. Continuity of r follows from joint continuity of R. For any $(w, x, y) \in D(\delta , r)$ (with this choice of r),

$$\begin{aligned} |(w, x, y) - (w, x, 0)| = |y| < r(x) \le R(w, x), \end{aligned}$$

(5.65)

so $(w, x, y) \in B(w, x, 0; R(w, x))$. We conclude that $D(\delta , r) \subset D_f$. $\square$