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Marginal triviality of the scaling limits of critical 4D Ising and $\phi_4^4$ models

Michael Aizenman* and Hugo Duminil-Copin†

25 January 2021

Abstract

We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the $\lambda\phi^4$ fields over $\mathbb{R}^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

1 Introduction

The results presented below address questions pertaining to two distinct research agendas: one aims at Constructive Field Theory and the other at the understanding of the critical behavior in Statistical Mechanics. While these two goals are somewhat different the questions and the answers are related. We start with their brief presentation.

1.1 Constructive Quantum Field Theory and Functional Integration

Quantum field theories with local interaction play an important role in the physics discourse, where they appear in subfields ranging from high energy to condensed matter physics. The mathematical challenge of proper formulation of this concept led to programs of Constructive Quantum Field Theory (CQFT). A path towards that goal was charted through the proposal to define quantum fields as operator valued distributions whose essential properties are formulated as the Wightman axioms [50]. Wightman's reconstruction theorem allows one to recover this structure from the collection of the corresponding correlation functions, defined over the Minkowski space-time. By the Osterwalder-Schrader theorem [39, 40], correlation functions with the required properties may potentially be obtained through analytic continuation from those of random distributions defined over the corresponding Euclidean space that meet a number of conditions: suitable analyticity, permutation symmetry, Euclidean covariance, and reflection-positivity.

• aizenman@princeton.edu Departments of Physics and Mathematics, Princeton University † duminil@ihes.fr Institut des Hautes Études Scientifiques and Université de Genève ---PAGE_BREAK---

Seeking natural candidates for such Euclidean fields, one ends up with the task of constructing probability averages over random distributions $\Phi(x)$, for which the expectation value of functionals $F(\Phi)$ would have properties fitting the formal expression

$$⟨F(\mathrm{\Phi })⟩\approx \frac{1}{\text{norm}}\int F(\mathrm{\Phi })\exp [-H(\mathrm{\Phi })]\prod _{x\in {\mathbb{R}}^d}d\mathrm{\Phi }(x),(1.1)\backslash langle\; F(\backslash Phi)\; \backslash rangle\; \backslash approx\; \backslash frac\{1\}\{\backslash text\{norm\}\}\; \backslash int\; F(\backslash Phi)\; \backslash exp[-H(\backslash Phi)]\; \backslash prod\_\{x\; \backslash in\; \backslash mathbb\{R\}^d\}\; d\backslash Phi(x),\; \backslash quad\; (1.1)$$

where $H(\Phi)$ is the Hamiltonian. In this context, it seems natural to consider expressions of the form

$$H(\mathrm{\Phi })(\mathrm{\Phi },A\mathrm{\Phi })+{\int }_{{\mathbb{R}}^d}P(\mathrm{\Phi }(x))dx(1.2)H(\backslash Phi)\; \backslash coloneqq\; (\backslash Phi,\; A\backslash Phi)\; +\; \backslash int\_\{\backslash mathbb\{R\}^d\}\; P(\backslash Phi(x))\; dx\; \backslash quad\; (1.2)$$

with $(\Phi, A\Phi)$ a positive definite and reflection-positive quadratic form, and $P(\Phi(x))$ a polynomial (or a more general function) whose terms of order $\Phi(x)^{2k}$ are interpreted heuristically as representing $k$-particle interactions. An example of a quadratic form with the above properties (at $K, b > 0$) and also rotation invariance is

$$(\mathrm{\Phi },A\mathrm{\Phi }):={\int }_{{\mathbb{R}}^d}(K\mathrm{\mid }\mathrm{\nabla }\mathrm{\Phi }{\mathrm{\mid }}^2(x)+b\mathrm{\mid }\mathrm{\Phi }(x){\mathrm{\mid }}^2)dx\mathrm{.}(1.3)(\backslash Phi,\; A\backslash Phi)\; :=\; \backslash int\_\{\backslash mathbb\{R\}^d\}\; (K\; |\backslash nabla\backslash Phi|^2(x)\; +\; b\; |\backslash Phi(x)|^2)\; dx.\; \backslash quad\; (1.3)$$

The functionals $F(\Phi)$ to which (1.1) is intended to apply include the smeared averages

$$T_f(\mathrm{\Phi }):={\int }_{{\mathbb{R}}^d}f(x)\mathrm{\Phi }(x)dx(1.4)T\_f(\backslash Phi)\; :=\; \backslash int\_\{\backslash mathbb\{R\}^d\}\; f(x)\backslash Phi(x)dx\; \backslash quad\; (1.4)$$

associated with continuous functions of compact support $f \in C_0(\mathbb{R}^d)$. By linearity, the expectation values of products of such variables take the form

$$⟨\prod _{j=1}^n{T}_{f_j}(\mathrm{\Phi })⟩:={\int }_{({\mathbb{R}}^d{)}^n}d{x}_1\dots d{x}_n{S}_n(x_1,\dots ,x_n)\prod _{j=1}^{n}f(x_j),(1.5)\backslash left\backslash langle\; \backslash prod\_\{j=1\}^\{n\}\; T\_\{f\_j\}(\backslash Phi)\; \backslash right\backslash rangle\; :=\; \backslash int\_\{(\backslash mathbb\{R\}^d)^n\}\; dx\_1\; \backslash dots\; dx\_n\; S\_n(x\_1,\; \backslash dots,\; x\_n)\; \backslash prod\_\{j=1\}^\{n\}\; f(x\_j),\; \backslash quad\; (1.5)$$

with $S_n(x_1, \dots, x_n)$ characterizing the probability measure on the space of distribution which corresponds to the expectation value $\langle - - \rangle$. This is summarized by saying that in a distributional sense

$$⟨\prod _{j=1}^n\mathrm{\Phi }(x_j)⟩=S_n(x_1,\dots ,x_n),(1.6)\backslash left\backslash langle\; \backslash prod\_\{j=1\}^\{n\}\; \backslash Phi(x\_j)\; \backslash right\backslash rangle\; =\; S\_n(x\_1,\; \backslash dots,\; x\_n),\; \backslash qquad\; (1.6)$$

with $S_n$ referred to as the Schwinger functions of the corresponding euclidean field theory.

A relatively simple class of Euclidean fields are the Gaussian fields, for which $H$ con- tains only quadratic terms. Gaussian fields (whether reflection-positive or not) are alter- natively characterized by having their structure determined by just the two-point function, with the $2n$-point Schwinger functions computable through Wick's law:

$$S_{2n}(x_1,\dots ,x_{2n})=\sum _{\pi }\prod _{j=1}^n{S}_2(x_{\pi (2j-1)},x_{\pi (2j)}):={\mathcal{G}}_n[S_2](x_1,\dots ,x_{2n}),(1.7)S\_\{2n\}(x\_1,\; \backslash ldots,\; x\_\{2n\})\; =\; \backslash sum\_\{\backslash pi\}\; \backslash prod\_\{j=1\}^\{n\}\; S\_2(x\_\{\backslash pi(2j-1)\},\; x\_\{\backslash pi(2j)\})\; :=\; \backslash mathcal\{G\}\_n[S\_2](x\_1,\; \backslash ldots,\; x\_\{2n\}),\; \backslash quad\; (1.7)$$

where $\pi$ ranges over pairing permutations of ${1, \ldots, 2n}$. The field theoretic interpretation of (1.7) is the absence of interaction. Due to that, and to their algebraically simple structure, such fields have been referred to as trivial.

When interpreting (1.1), one quickly encounters a number of problems. Even in the generally understood case of the Gaussian free field, with $H$ consisting of just the quadratic term (1.3), Equation (1.1) is not to be taken literally as the measure is supported by non- differentiable functions for which the integral in the exponential is almost surely divergent.

A natural step to tackle next seems to be the addition of the lowest order even term, i.e. $\lambda\Phi^4$. However, in dimensions $d > 1$, the free field is no longer a random function but a ---PAGE_BREAK---

random distribution which even locally is unbounded. Thus such simple looking proposals lead to additional divergences, whose severity increases with the dimension.

The heuristic “renormalization group” approach to the problem by K. Wilson [51] indi- cates that in low enough dimensions, specifically $d < 4$ for $\lambda\Phi^4$, the problem could be tack- led through cutoff-dependent counter-terms. Partially successful attempts to carry such a project rigorously have been the focus of a substantial body of works. The means em- ployed have included: counter-terms, which are allowed to depend on regularizing cutoffs, scale decomposition, renormalization group flows, the theory or regularity structures [27], etc.

A natural starting point towards such a construction of a $\Phi_d^4$ functional integral (1.1) is to regularize it with a pair of cutoffs: at the short distance (ultraviolet) scale and the large distance (infrared) scale. A lattice version of that is the restriction of $\Phi(\cdot)$ to the vertices of a finite graph with the vertex set

$$\begin{array}{cccc}& {\mathcal{V}}_{a,R}=(a\mathbb{Z}{)}^d\cap {\mathrm{\Lambda }}_R,{\mathrm{\Lambda }}_R:=[-R,R{]}^d\mathrm{.}& & \text{(1.8)}\end{array}\backslash mathcal\{V\}\_\{a,R\}\; =\; (a\backslash mathbb\{Z\})^d\; \backslash cap\; \backslash Lambda\_R,\; \backslash quad\; \backslash Lambda\_R\; :=\; [-R,\; R]^d.\; \backslash tag\{1.8\}$$

For the corresponding finite collection of variables ${\Phi(x)}{x \in \mathcal{V}{a,R}}$ the Hamiltonian (1.2) is initially interpreted in terms of the Riemann-sum style discrete analog of the integral expressions. Moments of $\Phi(x)$ are to be accompanied by lower order counter-terms. In particular, the fourth power addition takes the form

$$\begin{array}{cccc}& P(\mathrm{\Phi }(x))=\lambda {\mathrm{\Phi }}^4-c(\lambda ,a,R){\mathrm{\Phi }}^2,& & \text{(1.9)}\end{array}P(\backslash Phi(x))\; =\; \backslash lambda\backslash Phi^4\; -\; c(\backslash lambda,\; a,\; R)\backslash Phi^2,\; \backslash tag\{1.9\}$$

The cutoffs are removed, through the limit $R \nearrow \infty$ followed by $a \searrow 0$. Parameter such as $c(\lambda, a, R)$ are allowed to be adjusted in the process, so as to stabilize the Schwinger functions $S_n(x_1, \dots, x_n)$ on the continuum limit scale.

The constructive field theory program has yielded non-trivial scalar field theories over $\mathbb{R}^2$ and $\mathbb{R}^3$ [11, 21, 26, 40]. (Here we do not discuss here gauge field theories, cf. [31]). However, the progression of constructive results was halted when it was proved that for dimensions $d > 4$ the attempt to construct $\Phi_d^4$ with

$$\begin{array}{cccc}& \underset{\mathrm{\mid }x-y\mathrm{\mid }\to \mathrm{\infty }}{\lim }{S}_2(x,y)=0& & \text{(1.10)}\end{array}\backslash lim\_\{|x-y|\; \backslash to\; \backslash infty\}\; S\_2(x,y)\; =\; 0\; \backslash tag\{1.10\}$$

by the method outlined above (in essence: taking the scaling limit of the lattice models at $\beta \le \beta_c$) yields only Gaussian fields [1, 17].

Various partial results have indicated that the same may hold true for the critical dimension $d = 4$ (cf. [7, 8, 9, 20, 29]), however a sweeping statement such as proved for $d > 4$ has remained open. In this work we address this case.

For clarity let us note that, like the no-go statements of [1, 17], the results presented here do not involve explicit computations of the counterterms along the above scheme. Instead, they are based on dimension-dependent relations among the Schwinger functions which may emerge in any such limit.

1.2 Statement of the main result

The probability measures which correspond to (1.1) with the lattice and finite volume cutoffs (1.8) take the form of a statistical-mechanics Gibbs equilibrium state average

$$\begin{array}{cccc}& ⟨F(\varphi )⟩=\frac{1}{\text{norm}}\int F(\varphi )\exp [-H(\varphi )]\prod _{x\in {\mathrm{\Lambda }}_R}\rho (d{\varphi }_x),& & \text{(1.11)}\end{array}\backslash langle\; F(\backslash phi)\; \backslash rangle\; =\; \backslash frac\{1\}\{\backslash text\{norm\}\}\; \backslash int\; F(\backslash phi)\; \backslash exp[-H(\backslash phi)]\; \backslash prod\_\{x\; \backslash in\; \backslash Lambda\_R\}\; \backslash rho(d\backslash phi\_x),\; \backslash tag\{1.11\}$$ ---PAGE_BREAK---

with a Hamiltonian $H(\phi)$ and an a-priori measure $\rho(d\phi)$ of the form

$$H(\varphi )=-\sum _{\{x,y\}\in {\mathrm{\Lambda }}_R}{J}_{x,y}{\varphi }_x{\varphi }_y,\rho (d{\varphi }_x)=e^{-\lambda {\varphi }_x^4+b{\varphi }_x^2}d{\varphi }_x,(1.12)H(\backslash phi)\; =\; -\; \backslash sum\_\{\backslash \{x,y\backslash \}\; \backslash in\; \backslash Lambda\_R\}\; J\_\{x,y\}\; \backslash phi\_x\; \backslash phi\_y,\; \backslash quad\; \backslash rho(d\backslash phi\_x)\; =\; e^\{-\backslash lambda\; \backslash phi\_x^4\; +\; b\; \backslash phi\_x^2\}\; d\backslash phi\_x,\; \backslash qquad\; (1.12)$$

where $d\phi_x$ is the Lebesgue measure on $\mathbb{R}$ and $J_{x,y}$ is zero for non-nearest neighbour vertices, and $J \ge 0$ otherwise. To keep the notation simple, the basic variables are written here as they appear from the perspective of the lattice but our attention is focused on the correlations at distances of the order of $L$, with

$$1\ll L\ll R\mathrm{.}(1.13)1\; \backslash ll\; L\; \backslash ll\; R.\; \backslash qquad\; (1.13)$$

In terms of the scaling limit discussed above, $a$ is equal to $1/L$.

A point of fundamental importance is that since the interaction through which the field variables are correlated is local (nearest neighbor on the lattice scale), for the field correlations functions to exhibit non-singular variation on the scales $L \gg 1$, the system's parameters $(J, \lambda, b)$ need to be very close to the critical manifold, along which the correlation length of the lattice system diverges¹.

Quantities whose joint distribution we track in the scaling limit are based on the collections of random variables of the form

$$T_{f,L}(\varphi ):=\frac{1}{\sqrt{\sum _L}}\sum _{x\in {\mathbb{Z}}^d}f(x\mathrm{/}L){\varphi }_x,(1.14)T\_\{f,L\}(\backslash phi)\; :=\; \backslash frac\{1\}\{\backslash sqrt\{\backslash sum\_L\}\}\; \backslash sum\_\{x\; \backslash in\; \backslash mathbb\{Z\}^d\}\; f(x/L)\; \backslash phi\_x,\; \backslash qquad\; (1.14)$$

where $f$ ranges over compactly supported continuous functions, whose collection is denoted $C_0(\mathbb{R}^d)$, and $\sum_L$ denotes the variance of the sum of spins over the box of size $L$, i.e.

$${\mathrm{\Sigma }}_L:=⟨{\left(\sum _{x\in {\mathrm{\Lambda }}_L}{\varphi }_x\right)}^2⟩\mathrm{.}(1.15)\backslash Sigma\_L\; :=\; \backslash left\backslash langle\; \backslash left(\; \backslash sum\_\{x\; \backslash in\; \backslash Lambda\_L\}\; \backslash phi\_x\; \backslash right)^2\; \backslash right\backslash rangle.\; \backslash qquad\; (1.15)$$

Definition 1.1 A discrete system as described above, parametrized by $(J, \lambda, b, R, L)$, converges in distribution, in the double limit $\lim_{L \to \infty} \lim_{R/L \to \infty}$ (with a possible restriction to a subsequence along which also the other parameters are allowed to vary) if for any finite collection of test functions $f \in C_0(\mathbb{R}^d)$ the joint distributions of the random variables ${T_{f,L}(\phi)}$ converge.

Through a standard probabilistic construction, the limit can be presented as a random field $\Phi$, to whose weighted averages $T_f(\Phi)$ the above variables converge in distribution. We omit here the detailed discussion of this point², but remark that for the models considered here the construction is simplified by i) the exclusion of delta functions $\delta(x)$ and their derivatives from the family of considered test functions, and ii) the uniform local integrability of the rescaled correlation functions (before and at the limit). This important condition is implied in the present case by the infrared bound, which is presented below in Section 5.3.

Our main result concerning the euclidean field theory is the following.

¹The scaling limit of a correlation function with exponential decay which on the lattice scale is of a fixed correlation length results in a white noise distribution in the limit.

²By the Kolmogorov extension theorem, one may start by selecting sequences of the parameter values so as to establish convergence in distribution for a countable collection of test functions $f$, which is dense in $C_0(\mathbb{R}^d)$, and then use the uniform local integrability of the rescaled correlation function and of the limiting Schwinger functions, to extend the statement by continuity arguments to all $f \in C_0(\mathbb{R}^d)$. One may then recast the limiting variables as associated with a single random $\Phi$, as in (1.4). ---PAGE_BREAK---

Theorem 1.2 (Gaussianity of $\Phi_4^d$) For dimension $d=4$, any random field reachable by the above constructions, and satisfying (1.10), is a generalized Gaussian process.

Let us mention that the precise asymptotic behaviour of scaling limits of lattice models which start from sufficiently small perturbations of the Gaussian free field, i.e. small enough $\lambda$, have been obtained through rigorous renormalization techniques [9, 16, 20, 29]. In comparison, our result also covers arbitrarily “hard” $\phi^4$ fields. However, we do not currently provide comparable analysis of the convergence in terms of the exact scale of the logarithmic corrections, and the exact expression for the covariance of the limiting Gaussian field.

Let us also note that what from the perspective of constructive field theory may be regarded as disappointment is a positive and constructive result from the perspective of statistical mechanics. The theoreticians' goal there is to understand the critical behavior in models which lie beyond the reach of exact solutions. The proven gaussianity of the limit is therefore also a constructive result.

1.3 The statistical mechanics perspective

Statistical mechanics provides a general approach for studying the behaviour of extensive systems of a divergent number of degrees of freedom. Among the theoretically gratifying observations in this field has been the discovery of “universality”. The term means that some of the key features of phase diagrams, and critical behavior (including the critical exponents), appear to be the same across broad classes of systems of rather different microscopic structure. This has accorded relevance to studies of the phase transitions in drastically streamlined mathematical models. The ferromagnetic Ising spin model to which we turn next are among the earliest, and most studied such systems.

An intuitive explanation of universality is that the large scale behavior of models of rich short scale structure is described by statistical field theories for which there are far fewer options. A heuristic perspective on this phenomenon is provided by the renormalization group theory, c.g. [51]. In particular, the mechanism underlying the simplicity of the scaling limit is related to simplicity of the critical exponents, which means that for $d \ge 4$ they assume their mean field values. Rigorous results for the latter (though still partial, in terms of logarithmic corrections) were presented in [46, 6].

The Ising spin model on $\Lambda \subset \mathbb{Z}^d$ has as its basic variables a collection of $\pm 1$ valued variables ${\sigma_x}_{x \in \Lambda}$, and a Hamiltonian (the energy function) of the form

$$H_{\mathrm{\Lambda },J,h}(\sigma ):=-\sum _{\{x,y\}\subset \mathrm{\Lambda }}{J}_{x,y}{\sigma }_x{\sigma }_y-\sum _{x\in \mathrm{\Lambda }}h{\sigma }_x\mathrm{.}(1.16)H\_\{\backslash Lambda,J,h\}(\backslash sigma)\; :=\; -\; \backslash sum\_\{\backslash \{x,y\backslash \}\; \backslash subset\; \backslash Lambda\}\; J\_\{x,y\}\; \backslash sigma\_x\; \backslash sigma\_y\; -\; \backslash sum\_\{x\; \backslash in\; \backslash Lambda\}\; h\; \backslash sigma\_x.\; \backslash quad\; (1.16)$$

The model's finite volume Gibbs equilibrium state $\langle \cdot \rangle_{\Lambda,J,h,\beta}$ at inverse temperature $\beta \ge 0$ is the probability measure under which the expectation value of any function $F: {\pm 1}^\Lambda \to \mathbb{R}$ is given by

$$⟨F{⟩}_{\mathrm{\Lambda },J,h,\beta }:=\frac{1}{Z(\mathrm{\Lambda },J,h,\beta )}\sum _{\sigma \in \{\pm 1{\}}^{\mathrm{\Lambda }}}F(\sigma )\exp [-\beta H_{\mathrm{\Lambda },J,h}(\sigma )],(1.17)\backslash langle\; F\; \backslash rangle\_\{\backslash Lambda,\; J,\; h,\; \backslash beta\}\; :=\; \backslash frac\{1\}\{Z(\backslash Lambda,\; J,\; h,\; \backslash beta)\}\; \backslash sum\_\{\backslash sigma\; \backslash in\; \backslash \{\backslash pm\; 1\backslash \}^\backslash Lambda\}\; F(\backslash sigma)\; \backslash exp[-\backslash beta\; H\_\{\backslash Lambda,\; J,\; h\}(\backslash sigma)],\; \backslash quad\; (1.17)$$

where the normalizing factor $Z(\Lambda, J, h, \beta)$ is the model's partition function. Infinite volume Gibbs states on $\mathbb{Z}^d$, which we shall denote by $\langle \cdot \rangle_{J,h,\beta}$, are defined through suitable limits (over sequences $\Lambda_n \nearrow \mathbb{Z}^d$) of the above.

We focus here on the nearest neighbor ferromagnetic interaction (n.n.f.)

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with $J > 0$. In dimensions $d > 1$, this model exhibits a line of first-order phase transitions (in the plane of the model's thermodynamics parameters $(\beta, h)$) along the line $h = 0$, $\beta \in (\beta_c(d), \infty)$. The line terminates at the critical point $(\beta_c, 0)$ at which the model's correlation length diverges. Our discussion concerns the scaling limits at, or near, this point. Since the phase transition occurs at zero magnetic field, we restrict the discussion to $h = 0$ and will omit $h$ from the notation.

Away from the critical point the model's truncated correlation functions decay exponentially fast [3, 14]. This leads to the definition of the correlation length $\xi(\beta)$ as:

$$\xi (\beta ):=\underset{n\to \mathrm{\infty }}{\lim }-n\mathrm{/}\log ⟨{\sigma }_0;{\sigma }_{n{e}_1}{⟩}_{\beta }(\text{with }{\mathbf{e}}_1=(1,0,\dots ,0))\mathrm{.}(1.19)\backslash xi(\backslash beta)\; :=\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; -n\; /\; \backslash log\; \backslash langle\; \backslash sigma\_0;\; \backslash sigma\_\{ne\_1\}\; \backslash rangle\_\backslash beta\; \backslash quad\; (\backslash text\{with\; \}\; \backslash mathbf\{e\}\_1\; =\; (1,\; 0,\; \backslash dots,\; 0)).\; \backslash qquad\; (1.19)$$

The correlation length is proven to be finite for any $\beta < \beta_c$ [3] and divergent in the limit $\beta \to \beta_c$ [44]. At the critical point $\xi(\beta_c) = +\infty$ as the decay of the 2-point function slows to a power-law (see [44] and the discussion around Corollary 5.8).

At this point, one may notice the similarity between the Ising model’s Gibbs equilibrium distribution (1.17) and the discretized functional integral (1.11). Furthermore, in view of the probability measures’ relation

$$\frac{1}{2}[\delta (\sigma -1)+\delta (\sigma +1)]d\sigma =2\underset{\lambda \to \mathrm{\infty }}{\lim }{e}^{-\lambda ({\varphi }^2-1{)}^2}d\varphi \mathrm{/}\text{Norm}(\lambda )(1.20)\backslash frac\{1\}\{2\}\; [\backslash delta(\backslash sigma\; -\; 1)\; +\; \backslash delta(\backslash sigma\; +\; 1)]\; d\backslash sigma\; =\; 2\; \backslash lim\_\{\backslash lambda\; \backslash to\; \backslash infty\}\; e^\{-\backslash lambda(\backslash phi^2-1)^2\}\; d\backslash phi\; /\; \backslash text\{Norm\}(\backslash lambda)\; \backslash quad\; (1.20)$$

the Ising spin’s a-priori (binary) distribution can be viewed as the “hard” limit of the $\phi^4$ measure. Hence included in Theorem 1.2 is the statement that for $d=4$ any scaling limit of the critical Ising model is Gaussian.

However, our analysis flows in the opposite direction. In essence, the argument is structured as follows:

1. deploying methods which take advantage of the Ising systems' structure, the stated results are first proven for the n.n.f. Ising model (in four dimensions);

2. the analysis is adapted to the model's extension, in which each spin is replaced by a block average of 'elemental' Ising spins with an intrablock ferromagnetic coupling;

3. through weak limits the statement is extended to systems of variables whose a-priori single spin distribution belongs to the Griffiths-Simon (G-S) class.

Included in the G-S class (defined below) are the $\Phi^4$ measure $\rho(d\varphi)$ of (1.12).

To reduce the repetition, some of the relevant relations are presented below in a form which may not be the simplest for n.n.f. but is suitable for the model's generalized version. However in the rest of this section we focus on the n.n.f. case.

As it is known, and made explicit in Section 6.3, for Ising models a bellwether for Gaussian behaviour at large distances is the asymptotic validity of Wick's law at the level of the four-point function [1, 38]. The deviation is expressed in the Ursell function

$$U_4^{\beta }(x,y,z,t):=⟨{\sigma }_x{\sigma }_y{\sigma }_z{\sigma }_t{⟩}_{\beta }-[⟨{\sigma }_x{\sigma }_y{⟩}_{\beta }⟨{\sigma }_z{\sigma }_t{⟩}_{\beta }+⟨{\sigma }_x{\sigma }_z{⟩}_{\beta }⟨{\sigma }_y{\sigma }_t{⟩}_{\beta }+⟨{\sigma }_x{\sigma }_t{⟩}_{\beta }⟨{\sigma }_y{\sigma }_z{⟩}_{\beta }](1.21)U\_4^\backslash beta(x,\; y,\; z,\; t)\; :=\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash sigma\_z\; \backslash sigma\_t\; \backslash rangle\_\backslash beta\; -\; \backslash left[\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_z\; \backslash sigma\_t\; \backslash rangle\_\backslash beta\; +\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_z\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_y\; \backslash sigma\_t\; \backslash rangle\_\backslash beta\; +\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_t\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_y\; \backslash sigma\_z\; \backslash rangle\_\backslash beta\; \backslash right]\; (1.21)$$

the relevant question being whether $U_4(x,y,z,t)/\langle\sigma_x\sigma_y\sigma_z\sigma_t\rangle_\beta$ vanishes asymptotically for quadruples of sites at large distances, of comparable order between the pairs.

Gaussianity of the scaling limits for $d > 4$ was previously established through the combination of the tree diagram bound of [1]:

$$\mathrm{\mid }{U}_4^{\beta }(x,y,z,t)\mathrm{\mid }\le 2\sum _{u\in {\mathbb{Z}}^d}⟨{\sigma }_u{\sigma }_x{⟩}_{\beta }⟨{\sigma }_u{\sigma }_y{⟩}_{\beta }⟨{\sigma }_u{\sigma }_z{⟩}_{\beta }⟨{\sigma }_u{\sigma }_t{⟩}_{\beta }(1.22)|U\_4^\backslash beta(x,\; y,\; z,\; t)|\; \backslash le\; 2\; \backslash sum\_\{u\; \backslash in\; \backslash mathbb\{Z\}^d\}\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_x\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_y\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_z\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_t\; \backslash rangle\_\backslash beta\; \backslash qquad\; (1.22)$$ ---PAGE_BREAK---

and the Infrared Bound of [19, 21]

$$\begin{array}{cccc}& ⟨{\sigma }_x{\sigma }_y{⟩}_{{\beta }_c}\le \frac{C}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d-2}}\mathrm{.}& & \text{(1.23)}\end{array}\backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash rangle\_\{\backslash beta\_c\}\; \backslash leq\; \backslash frac\{C\}\{|x-y|^\{d-2\}\}.\; \backslash tag\{1.23\}$$

At the heuristic level, the triviality of the scaling limit for $d > 4$ is indicated by the following dimension counting. Assume that at $\beta_c$ the two-point function is of comparable values for pairs of sites at similar distances (which is false for $\beta \neq \beta_c$ at distances much larger than $\xi(\beta)$). Then, for quadruples of points at mutual distances of order $L$, the sum in the tree diagram bound (1.22) contributes a factor $L^d$ while the summand has two extra correlation function factors, in comparison to $\langle \sigma_x \sigma_y \sigma_z \sigma_t \rangle_\beta$, each factor dominated by $1/L^{d-2}$. This suggests that $U_4(x,y,z,t)$ in comparison to the full correlation functions may be of the order $O(L^{4-d})$, which for $d > 4$ vanishes in the limit $L \to \infty$. Up to numerous technical details this is the essence of the argument presented in [1, 17]. However, the above estimate is clearly inconclusive for $d = 4$.

The key advance presented here is the following improvement of the tree diagram bound. The multiplicative factor by which it improves (1.22) is derived through a multi scale analysis which is of relevance at the marginal dimension $d = 4$.

Theorem 1.3 (Improved tree diagram bound inequality) For the n.n.f. Ising model in dimension $d=4$, there exist $c, C > 0$ such that for every $\beta \le \beta_c$, every $L \le \xi(\beta)$ and every $x, y, z, t \in \mathbb{Z}^d$ at a distance larger than $L$ of each other,

$$\mathrm{\mid }{U}_4^{\beta }(x,y,z,t)\mathrm{\mid }\le \frac{C}{B_L(\beta {)}^c}\sum _{u\in {\mathbb{Z}}^4}⟨{\sigma }_u{\sigma }_x{⟩}_{\beta }⟨{\sigma }_u{\sigma }_y{⟩}_{\beta }⟨{\sigma }_u{\sigma }_z{⟩}_{\beta }⟨{\sigma }_u{\sigma }_t{⟩}_{\beta },(1.24)|U\_4^\backslash beta(x,\; y,\; z,\; t)|\; \backslash leq\; \backslash frac\{C\}\{B\_L(\backslash beta)^c\}\; \backslash sum\_\{u\; \backslash in\; \backslash mathbb\{Z\}^4\}\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_x\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_y\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_z\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_t\; \backslash rangle\_\backslash beta,\; \backslash quad\; (1.24)$$

where $B_L(\beta)$ is the bubble diagram truncated at a distance $L$ defined by the formula

$$\begin{array}{cccc}& B_L(\beta ):=\sum _{x\in {\mathrm{\Lambda }}_L}({\sigma }_0{\sigma }_x{)}_{\beta }^2\mathrm{.}& & \text{(1.25)}\end{array}B\_L(\backslash beta)\; :=\; \backslash sum\_\{x\; \backslash in\; \backslash Lambda\_L\}\; (\backslash sigma\_0\; \backslash sigma\_x)\_\backslash beta^2\; .\; \backslash tag\{1.25\}$$

For a heuristic insight on the implications of this improvement for $d = 4$, one may consider separately the two following scenarios: the two-point function $\langle \sigma_0 \sigma_x \rangle_\beta$ may be roughly of the order $L^{2-d}$ (meaning that the Infrared Bound is saturated up to constant), or it may be much smaller. In the first case (which is conjectured to hold when $d = 4$), $B_L(\beta)$ is of order $\log L$, so that the improved tree diagram bound indicates that $|U_4|/S_4 = O(\log L)^{-c}$, and thus is asymptotically negligible. In the second case (which is not the one expected to hold), already the unadulterated tree diagram bound (1.22) suffices.

We derive (1.24) making extensive use of the Ising model's random current representation that is presented in Section 3. It enables combinatorial identities through which the deviations from Wick's law can be expressed in terms of intersection probabilities of the random clusters which link pairwise the specified source points.

Beyond the four point function, the full statement of the scaling limit’s gaussianity is established here through the following estimate of the characteristic function of smeared averages of spins.

Proposition 1.4 There exist $c, C > 0$ such that for the n.n.f. Ising model on $\mathbb{Z}^4$, every $\beta \le \beta_c$, every $L \le \xi(\beta)$, and test function $f \in C_0(\mathbb{R}^4)$,

$$\mid {⟨\exp [z{T}_{f,L}(\sigma )-\frac{z^2}{2}⟨T_{f,L}(\sigma {)}^2{⟩}_{\beta }]⟩}_{\beta }-1\mid \le \frac{C\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}^4{r}_f^{12}}{(\log L{)}^c}{z}^4,(1.26)\backslash left|\; \backslash left\backslash langle\; \backslash exp[z\; T\_\{f,L\}(\backslash sigma)\; -\; \backslash frac\{z^2\}\{2\}\; \backslash langle\; T\_\{f,L\}(\backslash sigma)^2\; \backslash rangle\_\{\backslash beta\}]\; \backslash right\backslash rangle\_\{\backslash beta\}\; -\; 1\; \backslash right|\; \backslash leq\; \backslash frac\{C\; \backslash |f\backslash |\_\{\backslash infty\}^4\; r\_f^\{12\}\}\{(\backslash log\; L)^c\}\; z^4,\; \backslash quad\; (1.26)$$

with $|f|_{\infty} := \max{|f(x)| : x \in \mathbb{R}^4}$ and $r_f$ the diameter of the function's support. ---PAGE_BREAK---

The claimed gaussianity follows since (by the Infrared Bound, applied on the left-hand side) for any non-negative continuous function $f \neq 0$ with bounded support,

$$C{r}_f^2\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}^2\ge ⟨T_{f,L}(\sigma {)}^2{⟩}_{\beta }\ge c_f>0,(1.27)Cr\_f^2\; \backslash |f\backslash |\_\backslash infty^2\; \backslash geq\; \backslash langle\; T\_\{f,L\}(\backslash sigma)^2\; \backslash rangle\_\backslash beta\; \backslash geq\; c\_f\; >\; 0,\; \backslash quad\; (1.27)$$

uniformly in $\beta \leq \beta_c$ and $L$, we get that for $L \gg 1$ the distribution of $T_{f,L}(\sigma)$ is approximately Gaussian of variance $\langle T_{f,L}(\sigma)^2 \rangle_\beta$.

Organization of the proof: The result proven here is unconditional. However, to better convey the argument's structure, we first establish the claimed result for the scaling limits of critical models ($\beta = \beta_c$) under the auxiliary assumption that the two-point function behaves regularly on all scales, in a sense defined below. We then present an unconditional proof for $\beta \leq \beta_c$ in which we add to the above analysis the proof that the two-point function is regular on a sufficiently large collection of distance scales, up to the correlation length $\xi(\beta)$.

Organization of the article: In the next section, we present the Griffiths-Simon construction of random variables which can be obtained as local aggregates of ferromagnetically coupled Ising spins. It yields a useful link between the $\phi^4$ and Ising variables. Following that, in Section 3 we present the basics of Ising models' random current representation, and the intuition based on random walk intersection probabilities. Section 4 contains a conditional proof of the improved tree diagram bound at criticality, derived under a power-law decay assumption on the two-point function. Next, as a preparation for the unconditional proof, in Section 5 we present some relevant properties of Ising model's two-point function. These estimates are stated and proved in the context of systems of real valued variables with the single-spin distribution in the aforementioned Griffiths-Simon class. Included there are mostly known but also some new results. Section 6 contains the unconditional proof of our main results for the Ising model. Section 7 is devoted to its extension to the Griffiths-Simon class. The appendix contains some auxiliary technical statements that are of independent interest.

2 The Griffiths-Simon class of measures

The discrete approximations of the $\varphi^4$ functional integral and the Gibbs states of an Ising model are not only analogous, as explained above, but are actually related.

In one direction one has (1.20) and the implications mentioned next to it. However, in this work we shall make use of another relation, which permits us to apply tools which are initially developed for general Ising models to the study of the $\varphi^4$ functional integral. This relation is based on a construction which was initiated by Griffiths [23], and advanced further by Simon-Griffiths [45].

Definition 2.1 A probability measure on $\rho(d\varphi)$ on $\mathbb{R}$ is said to belong to the Griffiths-Simon (GS) class if either of the following conditions is satisfied

1. the expectation values with respect to $\rho$ can be presented as

$$\int F(\phi )\rho (d\phi )=\sum _{\sigma \in \{-1,1{\}}^N}F\left(\alpha \sum _{n=1}^N{b}_n{\sigma }_n\right)e^{\sum _{n,m=1}^N{K}_{n,m}{\sigma }_n{\sigma }_m}\mathrm{/}\text{Norm}\mathrm{.}(2.1)\backslash int\; F(\backslash varphi)\; \backslash rho(d\backslash varphi)\; =\; \backslash sum\_\{\backslash sigma\; \backslash in\; \backslash \{-1,1\backslash \}^N\}\; F\backslash left(\backslash alpha\; \backslash sum\_\{n=1\}^\{N\}\; b\_n\; \backslash sigma\_n\backslash right)\; e^\{\backslash sum\_\{n,m=1\}^\{N\}\; K\_\{n,m\}\; \backslash sigma\_n\; \backslash sigma\_m\}\; /\; \backslash text\{Norm\}.\; \backslash quad\; (2.1)$$

with some ${b_n} \subset \mathbb{R}$, and $K_{n,m} \geq 0$.

2. $\rho$ can be presented as a (weak) limit of probability measures of the above type, and ---PAGE_BREAK---

Figure 1: The decorated graph, in which the sites $x \in \Lambda$ of a graph of interest are re- placed by “blocks” $\mathcal{B}x$ of sites indexed as $(x, n)$. The Ising “constituent spins” $\sigma{x,n}$ are coupled pairwise through intra-block couplings $\delta_{x,y} K_{n,m}$ and inter-block couplings $J_{x,y}$. The depicted lines indicate a possible realization of the corresponding random current.

is of sub-gaussian growth:

A random variable is said to be of Griffiths-Simon type if its probability distribution is in the GS class.

The construction (1) was employed by Griffiths [23] for a proof that the Ising model’s Lee-Yang property as well as the Griffiths correlation inequalities hold also for a broader class of similar models with other notable spin variables. Subsequently, Simon and Grif- fiths [45] pointed out that upon taking weak limits this can be extended to cover alsothe $\phi^4$ a-priori measures, spelled in (1.12).

More specifically, a finite collection of the variables {$\varphi_x$}${x\in\Lambda}$ with the a-priori measure $\rho(d\varphi) = e^{-\lambda\varphi^4+b\varphi^2}d\varphi/\text{norm can be produced as the } N\to\infty \text{ limit (in distribution) of the} $ collection of the block averages of elemental Ising spins {$\sigma{x,n}$} (the dots in Fig. 1 )

$${\phi }_x^{(N)}={\alpha }_N(\lambda ,b)\sum _{n=1}^N{\sigma }_{x,n}(2.3)\backslash varphi\_x^\{(N)\}\; =\; \backslash alpha\_N(\backslash lambda,\; b)\; \backslash sum\_\{n=1\}^\{N\}\; \backslash sigma\_\{x,n\}\; \backslash qquad\; (2.3)$$

under the “ultra-local” coupling (which is to be added to the intersite interaction $H$ of (1.12))

$$H_{\text{inner}}=-\frac{g_N(\lambda ,b)}{N}\sum _{x\in \mathrm{\Lambda }}\sum _{n,m}{\sigma }_{x,n}{\sigma }_{x,m}(2.4)H\_\{\backslash text\{inner\}\}\; =\; -\; \backslash frac\{g\_N(\backslash lambda,\; b)\}\{N\}\; \backslash sum\_\{x\; \backslash in\; \backslash Lambda\}\; \backslash sum\_\{n,m\}\; \backslash sigma\_\{x,n\}\; \backslash sigma\_\{x,m\}\; \backslash quad\; (2.4)$$

with suitably adjusted $(\alpha_N, g_N)$. Their exact values are not important for our discussion, but let us note that $H_{\text{inner}}$ is a mean field interaction and thus it is easy to see that for each $(\lambda, b)$ with $\lambda \neq 0$: $g_N(\lambda, b)$ tends to 1 as $N$ tends to infinity, at a $(\lambda, b)$ dependent rate.

In this representation, any system of $\phi^4$ variables associated with the sites of a graph $\mathcal{V}$, and coupled through the graph's edges, is presentable as the limit ($N \to \infty$) of a system of constituent Ising spins associated with the Cartesian graph product $\mathbb{Z}^d \times \mathcal{K}_N$, with $\mathcal{K}_N$ denoting the complete graph of $N$ vertices. ---PAGE_BREAK---

3 Random current intersection probabilities

3.1 Definition and switching lemma

Starting with the Ising model, in this section we briefly introduce its random current representation, which allows to express the model’s subtle correlation effects in more tangible stochastic geometric terms. The utility of the random current representation is enhanced by the combinatorial symmetry expressed in its switching lemma, which enables to structure some of the essential truncated correlations in terms guided by the analysis of the intersection properties of the traces of random walks.

Definition 3.1 A current configuration n on Λ is an integer-valued function defined over unordered pairs (x, y) ∈ Λ. The current's set of sources is defined as the set

$$\mathrm{\partial }\mathbf{n}:=\{x\in \mathrm{\Lambda }:(-1{)}^{\sum _{y\in \mathrm{\Lambda }}\mathbf{n}(x,y)}=-1\}\mathrm{.}(3.1)\backslash partial\; \backslash mathbf\{n\}\; :=\; \backslash \{\; x\; \backslash in\; \backslash Lambda\; :\; (-1)^\{\backslash sum\_\{y\; \backslash in\; \backslash Lambda\}\; \backslash mathbf\{n\}(x,y)\}\; =\; -1\; \backslash \}.\; \backslash quad\; (3.1)$$

For a given Ising model on $\Lambda$, we associate to a current configuration the weight

$$w(\mathbf{n})=w_{\mathrm{\Lambda },J,\beta }(\mathbf{n}):=\prod _{\{x,y\}\subset V}\frac{(\beta J_{x,y}{)}^{\mathbf{n}(x,y)}}{\mathbf{n}(x,y)!}\mathrm{.}(3.2)w(\backslash mathbf\{n\})\; =\; w\_\{\backslash Lambda,\; J,\; \backslash beta\}(\backslash mathbf\{n\})\; :=\; \backslash prod\_\{\backslash \{x,y\backslash \}\; \backslash subset\; V\}\; \backslash frac\{(\backslash beta\; J\_\{x,y\})^\{\backslash mathbf\{n\}(x,y)\}\}\{\backslash mathbf\{n\}(x,y)!\}.\; \backslash quad\; (3.2)$$

Starting from Taylor's expansion

$$\exp (\beta J_{x,y}{\sigma }_x{\sigma }_y)=\sum _{\mathbf{n}(x,y)\ge 0}\frac{(\beta J_{x,y}{\sigma }_x{\sigma }_y{)}^{\mathbf{n}(x,y)}}{\mathbf{n}(x,y)!},(3.3)\backslash exp(\backslash beta\; J\_\{x,y\}\; \backslash sigma\_x\; \backslash sigma\_y)\; =\; \backslash sum\_\{\backslash mathbf\{n\}(x,y)\; \backslash ge\; 0\}\; \backslash frac\{(\backslash beta\; J\_\{x,y\}\; \backslash sigma\_x\; \backslash sigma\_y)^\{\backslash mathbf\{n\}(x,y)\}\}\{\backslash mathbf\{n\}(x,y)!\},\; \backslash quad\; (3.3)$$

one can see that the Ising model’s partition function (defined below (1.17)) can be expressed in terms of the corresponding random current:

$$Z(\mathrm{\Lambda },\beta )=2^{\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}\sum _{\mathbf{n}:\mathrm{\partial }\mathbf{n}=\mathrm{\varnothing }}w(\mathbf{n})\mathrm{.}(3.4)Z(\backslash Lambda,\; \backslash beta)\; =\; 2^\{|\backslash Lambda|\}\; \backslash sum\_\{\backslash mathbf\{n\}:\backslash partial\backslash mathbf\{n\}=\backslash emptyset\}\; w(\backslash mathbf\{n\}).\; \backslash quad\; (3.4)$$

Furthermore, the spin-spin correlation functions can be represented as

$$⟨\prod _{x\in A}{\sigma }_x{⟩}_{\mathrm{\Lambda },\beta }=\frac{\sum _{\mathbf{n}:\mathrm{\partial }\mathbf{n}=A}w(\mathbf{n})}{\sum _{\mathbf{n}:\mathrm{\partial }\mathbf{n}=\mathrm{\varnothing }}w(\mathbf{n})}\mathrm{.}(3.5)\backslash langle\; \backslash prod\_\{x\; \backslash in\; A\}\; \backslash sigma\_x\; \backslash rangle\_\{\backslash Lambda,\; \backslash beta\}\; =\; \backslash frac\{\backslash sum\_\{\backslash mathbf\{n\}:\; \backslash partial\backslash mathbf\{n\}=A\}\; w(\backslash mathbf\{n\})\}\{\backslash sum\_\{\backslash mathbf\{n\}:\; \backslash partial\backslash mathbf\{n\}=\backslash emptyset\}\; w(\backslash mathbf\{n\})\}.\; \backslash quad\; (3.5)$$

At this point, it helps to note that any configuration with $\partial\mathbf{n} = \emptyset$, i.e. without sources, can be viewed as the edge count of a multigraph which is decomposable into a union of loops. In contrast, any configuration with $\partial\mathbf{n} = A$, such as the one appearing in the numerator of (3.5), can be viewed as describing the edge count of a multigraph which is decomposable into a collection of loops and of paths connecting pairwise the sources, i.e. sites of $A$. In particular, a configuration with $\partial\mathbf{n} = {u, v}$ can be viewed as giving the “flux numbers” of a family of loops together with a path from $u$ to $v$. Thus, the random current representation allows to present the spin-spin correlation as the effect on the partition function of a loop system with the addition of a path linking the two sources. In these terms, the spin-spin correlation $\langle \sigma_{x_1} \cdots \sigma_{x_{2n}} \rangle_\beta$ represents the sum of the multiplicative effect of the introduction of $n$ paths pairing the sources.

Connectivity properties of currents play a significant role in our analysis. To express those we shall employ the following terminology and notation. ---PAGE_BREAK---

Definition 3.2 i) We say that $x$ is connected to $y$ (in $\mathbf{n}$), and denote the event by $x \xrightarrow{\mathbf{n}} y$, if there exists a path of vertices $x = u_0, u_1, \dots, u_k = y$ with $\mathbf{n}(u_i, u_{i+1}) > 0$ for every $0 \le i < k$. We say that $x$ is connected to a set $S$ if it is connected to a vertex in $S$.

ii) The cluster of $x$, denoted by $C_n(x)$, is the set of vertices connected to $x$ in $\mathbf{n}$.

iii) For a set of vertices $B$, we denote by $\mathcal{F}_B$ the set of $\mathbf{n}$ satisfying that there exists a sub-current $\mathbf{m} \le \mathbf{n}$ such that $\partial\mathbf{m} = B$.

Some of the most powerful properties of the random current representation are best seen when considering pairs of random currents and using the following lemma.

Lemma 3.3 (Switching lemma) For any $A, B \subset \Lambda$ and any function $F$ from the set of currents into $\mathbb{R}$,

$$\sum _{\begin{array}{c}{\mathbf{n}}_1:\mathrm{\partial }{\mathbf{n}}_1=A\\ {\mathbf{n}}_2:\mathrm{\partial }{\mathbf{n}}_2=B\end{array}}F({\mathbf{n}}_1+{\mathbf{n}}_2)w({\mathbf{n}}_1)w({\mathbf{n}}_2)=\sum _{\begin{array}{c}{\mathbf{n}}_1:\mathrm{\partial }{\mathbf{n}}_1=A\mathrm{\Delta }B\\ {\mathbf{n}}_2:\mathrm{\partial }{\mathbf{n}}_2=\mathrm{\varnothing }\end{array}}F({\mathbf{n}}_1+{\mathbf{n}}_2)w({\mathbf{n}}_1)w({\mathbf{n}}_2){\mathbf{1}}_{{\mathbf{n}}_1+{\mathbf{n}}_2\in {\mathcal{F}}_B}\mathrm{.}(3.6)\backslash sum\_\{\backslash substack\{\backslash mathbf\{n\}\_1:\backslash partial\backslash mathbf\{n\}\_1=A\; \backslash \backslash \; \backslash mathbf\{n\}\_2:\backslash partial\backslash mathbf\{n\}\_2=B\}\}\; F(\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2)w(\backslash mathbf\{n\}\_1)w(\backslash mathbf\{n\}\_2)\; =\; \backslash sum\_\{\backslash substack\{\backslash mathbf\{n\}\_1:\backslash partial\backslash mathbf\{n\}\_1=A\backslash Delta\; B\; \backslash \backslash \; \backslash mathbf\{n\}\_2:\backslash partial\backslash mathbf\{n\}\_2=\backslash emptyset\}\}\; F(\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2)w(\backslash mathbf\{n\}\_1)w(\backslash mathbf\{n\}\_2)\backslash mathbf\{1\}\_\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\; \backslash in\; \backslash mathcal\{F\}\_B\}.\; \backslash quad\; (3.6)$$

where $A\Delta B$ denotes the symmetric difference of sets, $A\Delta B := (A \setminus B) \cup (B \setminus A)$.

The switching lemma appeared as a combinatorial identity in Griffiths-Hurst-Sherman's derivation of the GHS inequality [24]. Its greater potential for the geometrization of the correlation functions was developed in [1], and works which followed. In this paper, we employ two generalizations of this useful identity. In the first, the two currents n₁ and n₂ need not be defined on the same graph (see [4, Lemma 2.2] for details). The second will involve a slightly more general switching statement, which was used in several occasions in the past (cf. [5, Lemma 2.1] and reference therein).

It should be recognized that other stochastic geometric representations of spin correlations and/or interactions can be found (e.g. the Symanzik representation of the $\phi^4$ action [48], and the BFS random walk representation of the correlation functions [11]). It is conceivable that the overall strategy could be applied also through other means. However we find the random current representation particularly useful for our purpose.

3.2 Representation of Ursell's four-point function

The switching lemma enables one to rewrite spin-spin correlation ratios in terms of probabilities of events expressed in terms of the random currents. The first of these is the relation

$$\frac{⟨{\sigma }_A{⟩}_{\mathrm{\Lambda },\beta }⟨{\sigma }_B{⟩}_{\mathrm{\Lambda },\beta }}{⟨{\sigma }_A{\sigma }_B{⟩}_{\mathrm{\Lambda },\beta }}:={\mathbf{P}}_{\mathrm{\Lambda },\beta }^{A\mathrm{\Delta }B,\mathrm{\varnothing }}[{\mathbf{n}}_1+{\mathbf{n}}_2\in {\mathcal{F}}_B],(3.7)\backslash frac\{\backslash langle\; \backslash sigma\_A\; \backslash rangle\_\{\backslash Lambda,\; \backslash beta\}\; \backslash langle\; \backslash sigma\_B\; \backslash rangle\_\{\backslash Lambda,\; \backslash beta\}\}\{\backslash langle\; \backslash sigma\_A\; \backslash sigma\_B\; \backslash rangle\_\{\backslash Lambda,\; \backslash beta\}\}\; :=\; \backslash mathbf\{P\}\_\{\backslash Lambda,\; \backslash beta\}^\{A\backslash Delta\; B,\; \backslash emptyset\}\; [\backslash mathbf\{n\}\_1\; +\; \backslash mathbf\{n\}\_2\; \backslash in\; \backslash mathcal\{F\}\_B],\; \backslash qquad\; (3.7)$$

for which we denote by $\mathbf{P}_{\Lambda,\beta}^A (\mathbf{n})$ the probability distribution on random currents constrained by the source condition $\partial\mathbf{n} = A$, or more explicitly

$${\mathbf{P}}_{\mathrm{\Lambda },\beta }^A(\mathbf{n}):=\frac{2^{\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}w(\mathbf{n})}{⟨\prod _{x\in A}{\sigma }_x{⟩}_{\mathrm{\Lambda },\beta }Z(\mathrm{\Lambda },\beta )}\mathbb{I}[\mathrm{\partial }\mathbf{n}=A],(3.8)\backslash mathbf\{P\}\_\{\backslash Lambda,\; \backslash beta\}^\{A\}(\backslash mathbf\{n\})\; :=\; \backslash frac\{2^\{|\backslash Lambda|\}\; w(\backslash mathbf\{n\})\}\{\backslash langle\; \backslash prod\_\{x\; \backslash in\; A\}\; \backslash sigma\_x\; \backslash rangle\_\{\backslash Lambda,\; \backslash beta\}\; Z(\backslash Lambda,\; \backslash beta)\}\; \backslash mathbb\{I\}[\backslash partial\backslash mathbf\{n\}\; =\; A],\; \backslash quad\; (3.8)$$

and by $\mathbf{P}_{\Lambda,\beta}^{A_1, \dots, A_i}$ we denote the law of an independent family of currents $(\mathbf{n}_1, \dots, \mathbf{n}_i)$

$${\mathbf{P}}_{\mathrm{\Lambda },\beta }^{A_1,\dots ,A_i}:={\mathbf{P}}_{\mathrm{\Lambda },\beta }^{A_1}\otimes \cdots \otimes {\mathbf{P}}_{\mathrm{\Lambda },\beta }^{A_i}\mathrm{.}(3.9)\backslash mathbf\{P\}\_\{\backslash Lambda,\; \backslash beta\}^\{A\_1,\; \backslash dots,\; A\_i\}\; :=\; \backslash mathbf\{P\}\_\{\backslash Lambda,\; \backslash beta\}^\{A\_1\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash mathbf\{P\}\_\{\backslash Lambda,\; \backslash beta\}^\{A\_i\}.\; \backslash quad\; (3.9)$$

For two-point sets we may write $A = xy$ instead of ${x,y}$. ---PAGE_BREAK---

As we will also work with the infinite volume Gibbs measures, let us note that random currents and the switching lemma admit a generalization to infinite volume³. Existing continuity results [4] permit to extend (3.7) to the infinite volume, expressed in terms of the weak limits of the random current measures $\mathbf{P}{\Lambda_n, \beta}^A$ and $\mathbf{P}{\Lambda_n, \beta}^{A_1, \dots, A_i}$, in the limit $\Lambda_n \nearrow \mathbb{Z}^d$. The limiting statement is similar to (3.7) but without the finite volume subscript $\Lambda$:

$$\frac{⟨{\sigma }_A{⟩}_{\beta }⟨{\sigma }_B{⟩}_{\beta }}{⟨{\sigma }_A{\sigma }_B{⟩}_{\beta }}={\mathbf{P}}_{\beta }^{A\mathrm{\Delta }B,\mathrm{\varnothing }}[{\mathbf{n}}_1+{\mathbf{n}}_2\in {\mathcal{F}}_B]\mathrm{.}(3.10)\backslash frac\{\backslash langle\; \backslash sigma\_A\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_B\; \backslash rangle\_\backslash beta\}\{\backslash langle\; \backslash sigma\_A\; \backslash sigma\_B\; \backslash rangle\_\backslash beta\}\; =\; \backslash mathbf\{P\}\_\backslash beta^\{A\backslash Delta\; B,\; \backslash emptyset\}[\backslash mathbf\{n\}\_1\; +\; \backslash mathbf\{n\}\_2\; \backslash in\; \backslash mathcal\{F\}\_B].\; \backslash quad\; (3.10)$$

Combining (3.10) for the different values of the product of spin-spin correlations leads to

$$U_4^{\beta }(x,y,z,t)=-2⟨{\sigma }_x{\sigma }_y{⟩}_{\beta }⟨{\sigma }_z{\sigma }_t{⟩}_{\beta }{\mathbf{P}}_{\beta }^{xy,zt}[{\mathbf{C}}_{n_1+n_2}(x)\cap {\mathbf{C}}_{n_1+n_2}(z)\mathrm{\ne }\mathrm{\varnothing }]\mathrm{.}(3.11)U\_4^\backslash beta(x,\; y,\; z,\; t)\; =\; -2\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_z\; \backslash sigma\_t\; \backslash rangle\_\backslash beta\; \backslash mathbf\{P\}\_\backslash beta^\{xy,zt\}\; [\backslash mathbf\{C\}\_\{n\_1+n\_2\}(x)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_1+n\_2\}(z)\; \backslash neq\; \backslash emptyset].\; \backslash quad\; (3.11)$$

This equality is of fundamental importance to the question discussed here. It was the basis of the analysis of [1], and is the starting point for our discussion.

By (3.11), the relative magnitude of the deviation of the four-point function $\langle \sigma_x \sigma_y \sigma_z \sigma_t \rangle_\beta$ from the Gaussian law (i.e. the discrepancy in Wick's formula) is bounded in terms of intersection properties of the two clusters that link the indicated sources pairwise:

$$\frac{\mathrm{\mid }{U}_4^{\beta }(x,y,z,t)\mathrm{\mid }}{⟨{\sigma }_x{\sigma }_y{\sigma }_z{\sigma }_t{⟩}_{\beta }}\le 2{\mathbf{P}}_{\beta }^{xy,zt}[{\mathbf{C}}_{n_1+n_2}(x)\cap {\mathbf{C}}_{n_1+n_2}(z)\mathrm{\ne }\mathrm{\varnothing }]\mathrm{.}(3.12)\backslash frac\{|U\_4^\backslash beta(x,\; y,\; z,\; t)|\}\{\backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash sigma\_z\; \backslash sigma\_t\; \backslash rangle\_\backslash beta\}\; \backslash le\; 2\; \backslash mathbf\{P\}\_\backslash beta^\{xy,zt\}\; [\backslash mathbf\{C\}\_\{n\_1+n\_2\}(x)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_1+n\_2\}(z)\; \backslash neq\; \backslash emptyset].\; \backslash quad\; (3.12)$$

The random sets $\mathbf{C}{n_1+n_2}(x)$ and $\mathbf{C}{n_1+n_2}(z)$ are not independently distributed. However (3.12) can be further simplified through a monotonicity property of random currents. As proved in [1], and recalled here in the Appendix, the probability of an intersection can only increase upon the two sets' replacement by a pair of independently distributed clusters defined through the addition of two sourceless currents:

$${\mathbf{P}}_{\beta }^{xy,zt}[{\mathbf{C}}_{n_1+n_2}(x)\cap {\mathbf{C}}_{n_1+n_2}(z)\mathrm{\ne }\mathrm{\varnothing }]\le {\mathbf{P}}_{\beta }^{xy,zt,\mathrm{\varnothing },\mathrm{\varnothing }}[{\mathbf{C}}_{n_1+n_3}(x)\cap {\mathbf{C}}_{n_2+n_4}(z)\mathrm{\ne }\mathrm{\varnothing }]\mathrm{.}(3.13)\backslash mathbf\{P\}\_\backslash beta^\{xy,zt\}\; [\backslash mathbf\{C\}\_\{n\_1+n\_2\}(x)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_1+n\_2\}(z)\; \backslash neq\; \backslash emptyset]\; \backslash leq\; \backslash mathbf\{P\}\_\backslash beta^\{xy,zt,\backslash emptyset,\backslash emptyset\}\; [\backslash mathbf\{C\}\_\{n\_1+n\_3\}(x)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_2+n\_4\}(z)\; \backslash neq\; \backslash emptyset].\; \backslash quad\; (3.13)$$

This leads to the simpler upper bound in which the two random sets are independent:

$$\mathrm{\mid }{U}_4^{\beta }(x,y,z,t)\mathrm{\mid }\le 2⟨{\sigma }_x{\sigma }_y{⟩}_{\beta }⟨{\sigma }_z{\sigma }_t{⟩}_{\beta }{\mathbf{P}}_{\beta }^{xy,zt,\mathrm{\varnothing },\mathrm{\varnothing }}[{\mathbf{C}}_{n_1+n_3}(x)\cap {\mathbf{C}}_{n_2+n_4}(z)\mathrm{\ne }\mathrm{\varnothing }]\mathrm{.}(3.14)|U\_4^\backslash beta(x,\; y,\; z,\; t)|\; \backslash le\; 2\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_z\; \backslash sigma\_t\; \backslash rangle\_\backslash beta\; \backslash mathbf\{P\}\_\backslash beta^\{xy,zt,\backslash emptyset,\backslash emptyset\}\; [\backslash mathbf\{C\}\_\{n\_1+n\_3\}(x)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_2+n\_4\}(z)\; \backslash neq\; \backslash emptyset].\; \backslash quad\; (3.14)$$

Bounding the intersection probability by the expected number of intersection sites and applying the switching lemma leads directly to the tree diagram bound (1.22). However, as was explained above, to tackle the marginal dimension $d=4$ one needs to improve on that.

While $\mathbf{C}{n_1+n_3}(x)$ and $\mathbf{C}{n_2+n_4}(z)$ are bulkier and exhibit less independence than simple random walks linking the sources ${x,y}$ and ${z,t}$, the analogy is of help in guiding the intuition towards useful estimate strategies. In particular, it is classical that in dimension $d=4$ the probability that the traces of two random walks starting at distance $L$ of each other intersect, tends to 0 (as $1/\log L$, see [2, (2.8)] and [34]), but nevertheless the expected number of points of intersection remains of order $\Omega(1)$. The discrepancy is explained by the fact that although the intersections occur rarely, the conditional expectation of the number of intersection sites, conditioned on there being at least one, diverges logarithmically in $L$. The thrust of our analysis will be to establish similar behaviour in the system considered here. More explicitly, we will prove that the conditional expectation of the clusters' intersection size, conditioned on it being non-empty, grows at least as $(\log L)^c$.

The analysis of clusters’ intersection properties is more difficult than that of the paths of simple random walks for at least two reasons:

³The extension of the switching lemma to $\mathbb{Z}^d$ is straightforward for $\beta \le \beta_c$ since then $\mathbf{n}_1 + \mathbf{n}_2$ does not contain infinite paths of positive currents, almost surely under $\mathbf{P}_\beta^{A,B}$. For $\beta < \beta_c$ this is implied by the discussion of [1] for $\beta < \beta_c$, and for $\beta = \beta_c$ it follows from the continuity result of [4] for $\beta = \beta_c$. ---PAGE_BREAK---

• Missing information on the two-point function: Most analyses of intersection properties of random walks involve estimates on the Green function. In our system its role is to some extent taken by the two-point spin-spin correlation function. However, unlike the former case we do not a priori know the two-point function's exact order of magnitude (though a good one-sided inequality is provided by the Infrared Bound). This raises a difficulty that we address by studying the regularity properties of the two-point function in Section 5.

• The lack of a simple Markov property: in one way or another, the analysis of intersections for random walks involves the random walk's Markov property. Among its other applications, the walk's renewal property facilitates de-correlating the walks' behaviour at different places. In comparison, the random current clusters exhibit only a multidimensional domain Markov property. One of the main contributions of this paper will be to show a mixing property of random currents which will enable us to bypass the difficulty raised by the lack of a renewal property.

We expect that both the regularity estimates and the mixing properties established here are of independent interest, and may be of help in studies of the model also in three dimensions.

4 A conditional improvement of the tree diagram bound for

$$\beta ={\beta }_c\backslash beta\; =\; \backslash beta\_c$$

To better convey the strategy by which the tree diagram bound is improved, we start with a conditional proof of (1.24) for the Ising model on $\mathbb{Z}^4$ at criticality (i.e. when $\beta = \beta_c$), under the following assumption on the model’s two-point function. The removal of this assumption will raise substantial problems which are presented in the sections that follow. Below, $|·|$ denotes the infinity-norm

$$\mathrm{\mid }x\mathrm{\mid }:=\max \{\mathrm{\mid }{x}_i\mathrm{\mid },1\le i\le d\}\mathrm{.}(4.1)|x|\; :=\; \backslash max\backslash \{|x\_i|,\; 1\; \backslash le\; i\; \backslash le\; d\backslash \}.\; \backslash quad\; (4.1)$$

Assumption 4.1 (Power-law decay) There exist $\eta$ and $c, C \in (0, \infty)$ such that for every $x \in \mathbb{Z}^d$,

$$\frac{c}{\mathrm{\mid }x{\mathrm{\mid }}^{d-2+\eta }}\le ⟨{\sigma }_0{\sigma }_x{⟩}_{{\beta }_c}\le \frac{C}{\mathrm{\mid }x{\mathrm{\mid }}^{d-2+\eta }}\mathrm{.}(4.2)\backslash frac\{c\}\{|x|^\{d-2+\backslash eta\}\}\; \backslash le\; \backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\_\{\backslash beta\_c\}\; \backslash le\; \backslash frac\{C\}\{|x|^\{d-2+\backslash eta\}\}.\; \backslash quad\; (4.2)$$

The Infrared Bound (5.37) guarantees that $\eta \ge 0$ in any dimension $d > 2$. Note that if $\eta > 0$ for $d = 4$, then $B_L(\beta_c)$ is bounded uniformly in $L$ in which case the tree diagram bound implies the improved one. Thus, under this assumption the case requiring attention is just $\eta = 0$ (which is the generally expected value).

4.1 Intersection clusters

Our starting point is (3.14) in which $U_4^{\beta_c}$ is bounded by the probability of intersection of two independently distributed clusters $C_{n_1+n_3}(x)$ and $C_{n_2+n_4}(z)$, of which $\mathbf{n}_1$ and $\mathbf{n}_2$ include paths linking pairwise widely separated sources, $\partial \mathbf{n}_1 = {x,y}$ and $\partial \mathbf{n}_2 = {z,t}$. Introduce the notation

$$\mathcal{T}:=C_{n_1+n_3}(x)\cap C_{n_2+n_4}(z),(4.3)\backslash mathcal\{T\}\; :=\; C\_\{n\_1+n\_3\}(x)\; \backslash cap\; C\_\{n\_2+n\_4\}(z),\; \backslash quad\; (4.3)$$ ---PAGE_BREAK---

and let $|\mathcal{T}|$ be the set's cardinality. The tree diagram bound corresponds to the first moment estimate:

$${\mathbf{P}}_{{\beta }_c}^{xy,zt,\mathrm{\varnothing },\mathrm{\varnothing }}[\mathrm{\mid }\mathcal{T}\mathrm{\mid }>0]\le {\mathbf{E}}_{{\beta }_c}^{xy,zt,\mathrm{\varnothing },\mathrm{\varnothing }}[\mathrm{\mid }\mathcal{T}\mathrm{\mid }],(4.4)\backslash mathbf\{P\}\_\{\backslash beta\_c\}^\{xy,zt,\backslash emptyset,\backslash emptyset\}\; [|\backslash mathcal\{T\}|\; >\; 0]\; \backslash leq\; \backslash mathbf\{E\}\_\{\backslash beta\_c\}^\{xy,zt,\backslash emptyset,\backslash emptyset\}\; [|\backslash mathcal\{T\}|],\; \backslash quad\; (4.4)$$

in which the intersection probability is bounded by the intersection set's expected size.

Although the set $\mathcal{T}$ is less tractable than the intersection of a pair of Markovian random walks, their intuitive example provides a useful guide. The intersection of the traces of two simple random walks in dimension $d = 4$ has a Cantor-set like structure. Guided by this analogy, and taking advantage of the switching lemma, we show that conditioned on the event that $u$ belongs to $\mathcal{T}$, the intersection $|\mathcal{T}|$ is typically very large. This is in line with our expectation that the vertices in the intersection set occur in large (disconnected) clusters, causing the expected size of $|\mathcal{T}|$ to be much larger than the probability of it being non-zero.

Below and in the rest of this article, we introduce the annulus of sizes $k \le n$ and the boundary of a box as follows:

$$\begin{array}{cccc}& \mathrm{A}\mathrm{n}\mathrm{n}(k,n):={\mathrm{\Lambda }}_n\times {\mathrm{\Lambda }}_{k-1}\text{and}\mathrm{\partial }{\mathrm{\Lambda }}_n:=\mathrm{A}\mathrm{n}\mathrm{n}(n,n)& & \text{(4.5)}\end{array}\backslash mathrm\{Ann\}(k,n)\; :=\; \backslash Lambda\_n\; \backslash times\; \backslash Lambda\_\{k-1\}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash partial\backslash Lambda\_n\; :=\; \backslash mathrm\{Ann\}(n,n)\; \backslash tag\{4.5\}$$

(cf. Fig. 2).

In the proof, we apply the following deterministic covering lemma, which links the number of points in a set $\mathcal{X} \subset \mathbb{Z}^d$ with the number of concentric annuli of the form $u + \mathrm{Ann}(\ell_k, \ell_{k+1})$, with $u \in \mathcal{X}$, which it takes to cover $\mathcal{X}$. To state it we denote, for any (possibly finite) increasing sequence of lengths $\mathcal{L} = (\ell_k)$, every $u \in \mathbb{Z}^d$, and every integer $K$,

$$\begin{array}{cccc}& {\mathbf{M}}_u(\mathcal{X};\mathcal{L},K)=\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}\{k\le K:\mathcal{X}\cap [u+\mathrm{A}\mathrm{n}\mathrm{n}({\mathrm{\ell }}_k,{\mathrm{\ell }}_{k+1})]\mathrm{\ne }\mathrm{\varnothing }\}& & \text{(4.6)}\end{array}\backslash mathbf\{M\}\_u(\backslash mathcal\{X\};\; \backslash mathcal\{L\},\; K)\; =\; \backslash mathrm\{card\}\backslash \{k\; \backslash le\; K\; :\; \backslash mathcal\{X\}\; \backslash cap\; [u\; +\; \backslash mathrm\{Ann\}(\backslash ell\_k,\; \backslash ell\_\{k+1\})]\; \backslash ne\; \backslash emptyset\backslash \}\; \backslash tag\{4.6\}$$

(cf. Fig. 2).

Lemma 4.2 (Annular covering) In the above notation, for any sequence $\mathcal{L} = (\ell_k)$ with $\ell_1 \ge 1$ and $\ell_{k+1} \ge 2\ell_k$

$$\begin{array}{cccc}& \mathrm{\mid }\mathcal{X}\mathrm{\mid }\ge 2^{\min \{{\mathbf{M}}_u(\mathcal{X};\mathcal{L},K)\mathrm{/}5:u\in \mathcal{X}\}}\mathrm{.}& & \text{(4.7)}\end{array}|\backslash mathcal\{X\}|\; \backslash ge\; 2^\{\backslash min\backslash \{\backslash mathbf\{M\}\_u(\backslash mathcal\{X\};\backslash mathcal\{L\},K)/5:\; u\; \backslash in\; \backslash mathcal\{X\}\backslash \}\}.\; \backslash tag\{4.7\}$$

Proof It suffices to show that if $|\mathcal{X}| < 2^r$ for some $r$, then there exists a site $u \in \mathcal{X}$ for which $\mathbf{M}_u(\mathcal{X};\mathcal{L},K) < 5r$.

We prove the following stronger statement: For every set $\mathcal{X}$ containing the origin and every $K$, if $|\mathcal{X} \cap \Lambda_{\ell_K}| < 2^r$, then there exists $u \in \mathcal{X} \cap \Lambda_{\ell_K}$ with $M_u(\mathcal{X};\mathcal{L},K) < 5r$.

The assertion is obviously true for $r=1$ as one can pick $u$ to be the origin. Next, consider the case of $r > 1$ assuming the statement holds for all smaller values. If the intersection of $\mathcal{X}$ and $\Lambda_{\ell_{K-1}}$ is reduced to the origin, then $M_0(\mathcal{X};\mathcal{L},K) \le 2$ (only the annuli $\mathrm{Ann}(\ell_l, \ell_{l+1})$ with $l$ equal to $K-1$ or $K$ can intersect $\mathcal{X}$) as required so we now assume that this is not the case. Consider $0 \le k \le K-2$ maximal such that there exists $u \in \mathcal{X}$ with $\ell_k < |u| \le \ell_{k+1}$.

Since $\mathcal{X} \cap \Lambda_{\ell_{k-1}}$ and $\mathcal{X} \cap (u + \Lambda_{\ell_{k-1}})$ are disjoint (we use that $\ell_k \ge 2\ell_{k-1}$), one of the two sets has cardinality strictly smaller than $2^{r-1}$. Assume first that it is $\mathcal{X} \cap \Lambda_{\ell_{k-1}}$. The induction hypothesis implies the existence of $v \in \mathcal{X} \cap \Lambda_{\ell_{k-1}}$ such that

By our choice of $k$, every site in $\mathcal{X}$ is either in $\Lambda_{\ell_{k+1}}$ or outside of $\Lambda_{\ell_{K-1}}$. This implies that only the annuli $\mathrm{Ann}(\ell_l, \ell_{l+1})$ with $l$ equal to $k, k+1, K-2, K-1$ or $K$ can intersect $\mathcal{X}$, so that

---PAGE_BREAK---

Figure 2: The two (duplicated)-currents $\mathbf{n}_1+\mathbf{n}_3$ and $\mathbf{n}_2+\mathbf{n}_4$ in blue and black respectively. The clusters of $x$ (or equivalently $y$) in $\mathbf{n}_1+\mathbf{n}_3$ and $z$ (or equivalently $t$) in $\mathbf{n}_2+\mathbf{n}_4$ are depicted in bold. The red vertices are the elements of the intersection $\mathcal{T}$. We illustrated the annuli around one element, denoted $u$, of $\mathcal{T}$ and draw them in gray when an intersection occurs. Here, we therefore have $M_u(\mathcal{T}; \mathcal{L}, 5) = 3$ since three annuli contain an intersection.

If it is $\mathcal{X} \cap (u + \Lambda_{\ell_{k-1}})$ which has small cardinality, simply translate the set by $u$ and apply the same reasoning. The distance between the vertex $v$ obtained by the procedure and 0 is at most $\ell_{k-1} + \ell_k \le \ell_K$, so that the claim follows in this case as well. □

In the following conditional statement, we denote by $\mathcal{L}\alpha$ a sequence of integers defined recursively so that $\ell_{k+1} = \ell_k^\alpha$ with a specified $\alpha > 1$ and $\ell_0$ a large enough integer.

Proposition 4.3 (Conditional intersection-clustering bound) Under the assumption that the Ising model on $\mathbb{Z}^4$ satisfies (4.2) with $\eta = 0$ and restricting to $\alpha > 3^8$: there exist $\ell_0 = \ell_0(\alpha)$ and $\delta = \delta(\alpha) > 0$ such that for every $K > 2$ and every $u, x, y, z, t \in \mathbb{Z}^4$ with mutual distance between $x, y, z, t$ larger than $2\ell_K$,

Before deriving this estimate, which is proven in the next section, let us show how it leads to the improved tree diagram bound.

Proof of Theorem 1.3 under the assumption (4.2). As the discussion is limited here to $\beta = \beta_c$, we omit it from the notation. If $\eta > 0$ the bubble diagram is finite and hence the desired statement is already contained in the tree diagram bound (1.22). Focus then on the case $\eta = 0$, for which the bubble diagram diverges logarithmically. Fix $\alpha > 3^8$ and let $\ell_0$ and $\delta$ be given by Proposition 4.3. Since $x, y, z, t$ are at mutual distances at least $L$, there exists $c = c(\alpha) > 0$ such that one may pick

$$\begin{array}{cccc}& K=K(L)\ge c\log \log L& & \text{(4.11)}\end{array}K\; =\; K(L)\; \backslash geq\; c\; \backslash log\; \backslash log\; L\; \backslash tag\{4.11\}$$

in such a way that $L \ge 2\ell_K$. ---PAGE_BREAK---

Using Lemma 4.2, then the switching lemma, and finally Proposition 4.3, we get

For the larger values of $|\mathcal{T}|$, the Markov inequality and the switching lemma give

Adding (4.12) and (4.13) gives an improved tree diagram bound which, in view of (4.11) and of the logarithmic divergence of $B_L(\beta_c)$ implied by $\eta = 0$, yields (1.24). $\square$

4.2 Derivation of the conditional intersection-clustering bound (Proposition 4.3)

The intuition underlying the conditional intersection-clustering bound and the choice of $\ell_k$ is guided by the aforementioned example of simple random walks. In dimension 4, the traces of two independent random walks starting at the origin intersect in an annulus of the form $\text{Ann}(n, n^\alpha)$ with probability at least $c(\alpha) > 0$ uniformly in $n$. Since the paths traced by these random walks within different annuli are roughly independent, one may expect the number of annuli among the $K$ first ones in which the paths intersect to be, with large probability, of the order of $\delta K$.

However, in the case considered here, the clusters of $u$ in $\mathbf{n}_1 + \mathbf{n}_3$ and $\mathbf{n}_2 + \mathbf{n}_4$ do not have the renewal structure of Markovian random walks. We shall compensate for that in two steps:

(i) reformulate the intersection property,

(ii) derive an asymptotic mixing statement.

For the first step, let $I_k$ be the event (with $I$ standing for intersection) that there exist unique clusters of $\text{Ann}(\ell_k, \ell_{k+1})$ in $\mathbf{n}_1 + \mathbf{n}_3$ and $\mathbf{n}_2 + \mathbf{n}_4$ crossing the annulus from the inner boundary to the outer boundary and that these two clusters are intersecting. Lemma 4.4 presents the statement that the probability that the event occurs and that these clusters intersect, is bounded away from 0 uniformly in $k$.

Note that the annuli $\text{Ann}(\ell_k, \ell_{k+1})$ are wide enough so that sourceless currents will typically have no radial crossing, and when such crossings are forced by the placement of sources (for instance when one source, is at the common center of a family of nested annuli and the other at a distant site outside), in each annulus there will most likely be only one crossing cluster. It then follows that all the crossing clusters of $\mathbf{n}_1 + \mathbf{n}_3$ belong to the $\mathbf{n}_1 + \mathbf{n}_3$ cluster of the sources, and a similar property holds for the crossing clusters of $\mathbf{n}_2 + \mathbf{n}_4$.

For the second step, we prove that events observed within sufficiently separated annuli are roughly independent. The exact assertion is presented below in Proposition 4.6 and will be the crux of the whole paper.

Following is the first of these two statements. ---PAGE_BREAK---

Lemma 4.4 (Conditional intersection-clustering property) Assume (4.2) holds for the Ising model on $\mathbb{Z}^4$ with $\eta = 0$. For $\alpha > 3/4$, there exist $\ell_0 = \ell_0(\alpha)$ and $c = c(\alpha, \ell_0) > 0$ such that for every $x, z \notin \Lambda_{2\ell_{k+1}}$,

$$\begin{array}{cccc}& {\mathbf{P}}_{{\beta }_c}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[I_k]\ge c\mathrm{.}& & \text{(4.14)}\end{array}\backslash mathbf\{P\}\_\{\backslash beta\_c\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[I\_k]\; \backslash geq\; c.\; \backslash tag\{4.14\}$$

The main ingredient in the proof is a second moment method on the number of inter- sections in $\text{Ann}(\ell_k, \ell_{k+1})$ of the clusters of the origin in $\mathbf{n}_1 + \mathbf{n}_3$ and $\mathbf{n}_2 + \mathbf{n}_4$. A second part of the proof is devoted to the uniqueness of the clusters crossing the annulus. This makes the event under consideration measurable in terms of the currents within just the specified annulus, allowing us to apply the mixing property for the proof of Proposition 4.3, which follows further below.

Proof Drop $\beta_c$ from the notation. Fix $\alpha > 3^4$ and set $\varepsilon > 0$ so that $\alpha > (1+\varepsilon)(3+\varepsilon)^4$. The constants $c_i$ below depend on $\varepsilon$ only. Introduce the intermediary integers $n \le m \le M \le N$ satisfying

$$\begin{array}{cccc}& n\ge {\mathrm{\ell }}_k^{3+ϵ},m\ge n^{3+ϵ},M\ge m^{1+ϵ},N\ge M^{3+ϵ},{\mathrm{\ell }}_{k+1}\ge N^{3+ϵ}\mathrm{.}& & \text{(4.15)}\end{array}n\; \backslash ge\; \backslash ell\_k^\{3+\backslash epsilon\},\; \backslash quad\; m\; \backslash ge\; n^\{3+\backslash epsilon\},\; \backslash quad\; M\; \backslash ge\; m^\{1+\backslash epsilon\},\; \backslash quad\; N\; \backslash ge\; M^\{3+\backslash epsilon\},\; \backslash quad\; \backslash ell\_\{k+1\}\; \backslash ge\; N^\{3+\backslash epsilon\}.\; \backslash tag\{4.15\}$$

We start by proving that $\mathcal{M} := \mathbf{C}{n_1+n_3}(0) \cap \mathbf{C}{n_2+n_4}(0) \cap \text{Ann}(m, M)$ is non-empty with positive probability by applying a second-moment method on $|\mathcal{M}|$. Namely, the switching lemma (more precisely (A.10)) and (4.2) imply that

On the other hand, we find that

$${\mathbf{E}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[\mathrm{\mid }\mathcal{M}{\mathrm{\mid }}^2]=\sum _{v,w\in \text{Ann}(m,M)}{\mathbf{P}}^{0x,\mathrm{\varnothing }}[v,w0]{\mathbf{P}}^{0z,\mathrm{\varnothing }}[v,w0]\mathrm{.}(4.17)\backslash mathbf\{E\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[|\backslash mathcal\{M\}|^2]\; =\; \backslash sum\_\{v,w\; \backslash in\; \backslash text\{Ann\}(m,M)\}\; \backslash mathbf\{P\}^\{0x,\backslash emptyset\}[v,w\; \backslash stackrel\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\}\{\backslash longleftrightarrow\}\; 0]\; \backslash mathbf\{P\}^\{0z,\backslash emptyset\}[v,w\; \backslash stackrel\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\}\{\backslash longleftrightarrow\}\; 0].\; \backslash quad\; (4.17)$$

Now, by a delicate application of the switching lemma and a monotonicity argument we have the following inequality (stated and proven as Proposition A.3 in the Appendix),

$${\mathbf{P}}^{0x,\mathrm{\varnothing }}[v,w\stackrel{}{\leftarrow }0]\le \frac{⟨{\sigma }_0{\sigma }_v⟩⟨{\sigma }_v{\sigma }_w⟩⟨{\sigma }_w{\sigma }_x⟩}{⟨{\sigma }_0{\sigma }_x⟩}+\frac{⟨{\sigma }_0{\sigma }_w⟩⟨{\sigma }_w{\sigma }_v⟩⟨{\sigma }_v{\sigma }_x⟩}{⟨{\sigma }_0{\sigma }_x⟩}\mathrm{.}(4.18)\backslash mathbf\{P\}^\{0x,\backslash emptyset\}[v,\; w\; \backslash xleftarrow\{\backslash hspace\{2em\}\}\; 0]\; \backslash leq\; \backslash frac\{\backslash langle\backslash sigma\_0\backslash sigma\_v\backslash rangle\backslash langle\backslash sigma\_v\backslash sigma\_w\backslash rangle\backslash langle\backslash sigma\_w\backslash sigma\_x\backslash rangle\}\{\backslash langle\backslash sigma\_0\backslash sigma\_x\backslash rangle\}\; +\; \backslash frac\{\backslash langle\backslash sigma\_0\backslash sigma\_w\backslash rangle\backslash langle\backslash sigma\_w\backslash sigma\_v\backslash rangle\backslash langle\backslash sigma\_v\backslash sigma\_x\backslash rangle\}\{\backslash langle\backslash sigma\_0\backslash sigma\_x\backslash rangle\}.\; \backslash quad\; (4.18)$$

Together with (4.2), this gives

$${\mathbf{E}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[\mathrm{\mid }\mathcal{M}{\mathrm{\mid }}^2]\le C_3(B_M-B_{m-1})B_{2M}\le C_4(\log M{)}^2\mathrm{.}(4.19)\backslash mathbf\{E\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[|\backslash mathcal\{M\}|^2]\; \backslash le\; C\_3(B\_M\; -\; B\_\{m-1\})\; B\_\{2M\}\; \backslash le\; C\_4(\backslash log\; M)^2.\; \backslash quad\; (4.19)$$

The second moment (or Cauchy-Schwarz) inequality, and the bound $M \ge m^{1+\epsilon}$ thus imply

$${\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[M\mathrm{\ne }1]\ge \frac{{\mathbf{E}}^{0x,0z,\mathrm{\varnothing },[\mathrm{\mid }M\mathrm{\mid }{]}^2}}{{\mathbf{E}}^{0x,0z,\mathrm{\varnothing },[\mathrm{\mid }M{\mathrm{\mid }}^2]}}\ge c_5>0.(4.20)\backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[M\; \backslash neq\; 1]\; \backslash geq\; \backslash frac\{\backslash mathbf\{E\}^\{0x,0z,\backslash emptyset,[|M|]^2\}\}\{\backslash mathbf\{E\}^\{0x,0z,\backslash emptyset,[|M|^2]\}\}\; \backslash geq\; c\_5\; >\; 0.\; \backslash quad\; (4.20)$$

At this stage, one may feel that the main point of the lemma was established: we showed that with uniformly positive probability the clusters of 0 in n₁ + n₃ and n₂ + n₄ intersect in Ann(m, M). However, to conclude the argument we need to establish the uniqueness, with large probability, of the crossing cluster in n₁ + n₃ (the same then holds true for ---PAGE_BREAK---

n₂ + n₄). This part of the proof is slightly more technical and may be omitted in a first reading. It is here that we shall need α to be large enough.

To prove the uniqueness of crossings, we employ the notion of the current's backbone⁴, on which more can be found in [1, 3, 12, 13, 15]. If the event {M ≠ Ø} occurs but not Iₖ, then one of the following four events must occur (see e.g. Fig. 3):

$F_1 := \text{the backbone } \Gamma(\mathbf{n}_1) \text{ of } \mathbf{n}_1 \text{ does two successive crossings of } \mathrm{Ann}(\ell_k, n);$

$F_2 := \mathbf{n}_1 + \mathbf{n}_2 \text{ contains a cluster crossing } \mathrm{Ann}(n, m) \setminus \Gamma(\mathbf{n}_1);$

$F_3 := \mathbf{n}_1 + \mathbf{n}_2 \text{ contains a cluster crossing } \mathrm{Ann}(M, N) \setminus \Gamma(\mathbf{n}_1);$

$F_4 := \text{the backbone } \Gamma(\mathbf{n}_1) \text{ of } \mathbf{n}1 \text{ does two successive crossings of } \mathrm{Ann}(N, \ell{k+1}).$

We bound the probabilities of these events separately. For $F_1$ to occur, the backbone $\Gamma(\mathbf{n}1)$ must do a zigzag: to go from 0 to a vertex $v \in \partial\Lambda_n$, then to a vertex $w \in \partial\Lambda{\ell_k}$, and finally to $x$. The chain rule for backbones (see e.g. [3]) combined with the assumed condition (4.2), jointly imply that

$${\mathbf{P}}^{0x,\mathrm{\varnothing }}[F_1]\le \sum _{\begin{array}{c}v\in \mathrm{\partial }{\mathrm{\Lambda }}_n\\ w\in \mathrm{\partial }{\mathrm{\Lambda }}_{{\mathrm{\ell }}_k}\end{array}}\frac{⟨{\sigma }_0{\sigma }_v⟩⟨{\sigma }_v{\sigma }_w⟩⟨{\sigma }_w{\sigma }_x⟩}{⟨{\sigma }_0{\sigma }_x⟩}\le C_6{n}^3{\mathrm{\ell }}_k^3{n}^{-4}\le C_7{\mathrm{\ell }}_k^{-ϵ}\mathrm{.}(4.21)\backslash mathbf\{P\}^\{0x,\backslash emptyset\}[F\_1]\; \backslash leq\; \backslash sum\_\{\backslash substack\{v\; \backslash in\; \backslash partial\backslash Lambda\_n\; \backslash \backslash \; w\; \backslash in\; \backslash partial\backslash Lambda\_\{\backslash ell\_k\}\}\}\; \backslash frac\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_v\; \backslash rangle\; \backslash langle\; \backslash sigma\_v\; \backslash sigma\_w\; \backslash rangle\; \backslash langle\; \backslash sigma\_w\; \backslash sigma\_x\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\}\; \backslash leq\; C\_6\; n^3\; \backslash ell\_k^3\; n^\{-4\}\; \backslash leq\; C\_7\; \backslash ell\_k^\{-\backslash epsilon\}.\; \backslash quad\; (4.21)$$

To bound the probability of $F_2$, condition on $\Gamma(\mathbf{n}_1)$. The remaining current in $\mathbf{n}_1$ is a sourceless current with depleted coupling constants (see [3, 12, 13] for details on this type of reasoning). The probability that some $v \in \partial\Lambda_n$ and $w \in \partial\Lambda_m$ are connected in $Z^4 \setminus \Gamma(\mathbf{n}_1)$ to each other can then be bounded by $\langle\sigma_v\sigma_w\rangle\langle\sigma_v\sigma_w\rangle'$ where the $\langle\cdot\rangle'$ denotes an Ising measure with depleted coupling constants (the depletion depends on $\Gamma(\mathbf{n}_1)$ and the switching lemma concerns one current with depletion and one without; we refer to [4] for the statement and proof of the switching lemma in this context, and some applications). At the risk of repeating ourselves, we refer to [3] for an illustration of this line of reasoning. The Griffiths inequality [22] implies that this probability is bounded by $\langle\sigma_v\sigma_w\rangle^2$, which together with (4.2), immediately leads to the following sequence of inequalities:

$$\begin{array}{cccc}& {\mathbf{P}}^{0x,\mathrm{\varnothing }}[F_2]\le \sum _{\begin{array}{c}v\in \mathrm{\partial }{\mathrm{\Lambda }}_n\\ w\in \mathrm{\partial }{\mathrm{\Lambda }}_m\end{array}}({\sigma }_v{\sigma }_w{)}^2\le C_8{n}^{-ϵ}\mathrm{.}& & \text{(4.22)}\end{array}\backslash mathbf\{P\}^\{0x,\backslash emptyset\}[F\_2]\; \backslash leq\; \backslash sum\_\{\backslash substack\{v\; \backslash in\; \backslash partial\backslash Lambda\_n\; \backslash \backslash \; w\; \backslash in\; \backslash partial\backslash Lambda\_m\}\}\; (\backslash sigma\_v\; \backslash sigma\_w)^2\; \backslash leq\; C\_8\; n^\{-\backslash epsilon\}.\; \backslash tag\{4.22\}$$

The event $F_4$ is bounded similarly to $F_1$, and $F_3$ similarly to $F_2$. For $\ell_0 = \ell_0(\epsilon)$ large enough the sum of the four probabilities does not exceed half of the constant $c_5$ in (4.20), and the main statement follows. $\square$

Remark 4.5 The condition $\alpha > 3^4$ is used in the second part of the proof, where we need the exponent connecting the inner and outer radii of annuli to be strictly larger than 3. We did not try to improve on this exponent.

The second of the above described statements is one of the main innovations of this paper. It concerns a mixing property, which in Section 6.1 will be stated under a more general form and derived unconditionally for every $d \ge 4$.

⁴We mentioned that a current n with sources x and y can be seen as the superposition of one path from x to y and loops. The backbone Γ(n) is an appropriate choice of such a path induced by an ordering of the edges. Again, we refrain ourselves from providing more details here and refer to the relevant literature for details on this notion. ---PAGE_BREAK---

Figure 3: In this picture, $\Gamma(\mathbf{n}_1)$ does only one crossing of $\text{Ann}(\ell_k, n)$, and $\mathbf{n}_1 + \mathbf{n}_2 - \Gamma(\mathbf{n}_1)$ does not cross $\text{Ann}(n, m) \setminus \Gamma(\mathbf{n}_1)$. This prevents the fact that the cluster in red, made of loops in $\mathbf{n}_1+\mathbf{n}_2-\Gamma(\mathbf{n}_1)$ would connect an excursion of $\Gamma(\mathbf{n}1)$ outside of $\Lambda{\ell_k}$ but not reaching $\partial\Lambda_n$ to $\partial\Lambda_m$ (which would potentially create an additional cluster crossing $\text{Ann}(\ell_k, m)$).

Proposition 4.6 (Conditional mixing property) Assume that the complementary pair of power law bounds (4.2) holds for the Ising model on $Z^4$ with $\eta = 0$, and fix $\alpha > 3^8$. Then there exists $C > 0$ such that for every $n^\alpha \le N$, every $x \notin \Lambda_N$, and every pair of events E and F depending on the restriction of n to edges within $\Lambda_n$ and outside of $\Lambda_N$ respectively,

$$\mathrm{\mid }{\mathbf{P}}_{{\beta }_c}^{0x}[E\cap F]-{\mathbf{P}}_{{\beta }_c}^{0x}[E]{\mathbf{P}}_{{\beta }_c}^{0x}[F]\mathrm{\mid }\le \frac{C}{\sqrt{\log (N\mathrm{/}n)}}\mathrm{.}(4.23)|\; \backslash mathbf\{P\}\_\{\backslash beta\_c\}^\{0x\}[E\; \backslash cap\; F]\; -\; \backslash mathbf\{P\}\_\{\backslash beta\_c\}^\{0x\}[E]\; \backslash mathbf\{P\}\_\{\backslash beta\_c\}^\{0x\}[F]\; |\; \backslash le\; \backslash frac\{C\}\{\backslash sqrt\{\backslash log(N/n)\}\}.\; \backslash quad\; (4.23)$$

The heart of the proof will be the use of a (random) resolution of identity $\mathbb{N}$, meaning a random variable which is concentrated around 1, given by a weighted sum of indicator functions $\Pi[y \xleftarrow{n_1+n_2} 0]$ with $y \in \mathbb{Z}^d$, where $\partial \mathbf{n}_1 = {0, x}$ and $\partial \mathbf{n}_2 = \emptyset$, which will enable us to write

$${\mathbf{P}}^{0x}[E\cap F]\approx {\mathbf{E}}^{0x,\mathrm{\varnothing }}[\mathbb{N}\mathrm{\Pi }({\mathbf{n}}_1\in E\cap F)]\mathrm{.}(4.24)\backslash mathbf\{P\}^\{0x\}[E\; \backslash cap\; F]\; \backslash approx\; \backslash mathbf\{E\}^\{0x,\backslash emptyset\}[\backslash mathbb\{N\}\backslash Pi(\backslash mathbf\{n\}\_1\; \backslash in\; E\; \backslash cap\; F)].\; \backslash quad\; (4.24)$$

Since $\mathbb{N}$ will be a certain convex combination of the random variables $\Pi[y \underset{\mathbf{n}_1+\mathbf{n}_2}{\stackrel{\hspace{2em}}{\rightleftharpoons}} 0] / (\sigma_0 \sigma_y)$, the term on the right will be a convex sum of $\mathbf{P}^{0x,\emptyset}$-probabilities of the events ${y \underset{\mathbf{n}_1+\mathbf{n}_2}{\stackrel{\hspace{2em}}{\rightleftharpoons}} 0, \mathbf{n}_1 \in E \cap F}$. For each fixed $y$, we will use the switching principle to transform the sources ${0, x}$ and $\emptyset$ of $\mathbf{n}_1$ and $\mathbf{n}_2$ into ${0, y}$ and ${y, x}$, exchanging at the same time the roles of $\mathbf{n}_1$ and $\mathbf{n}_2$ inside $\Lambda_n$ without changing anything outside $\Lambda_N$. This useful operation has a nice byproduct: the event $\mathbf{n}_1 \in F$ becomes $\mathbf{n}_2 \in F$ which is independent of $\mathbf{n}_1 \in E$. Deducing the mixing from there will be a matter of elementary algebraic manipulations.

The error term will be (almost entirely) due to how concentrated around 1 N is. In order to prove this fact, we will implement a refined second moment method in which we estimate the expectation and the second moment of N sharply. The proof will require some regularity assumptions on the gradient of the two-point function: for every $x \in \mathbb{Z}^d$,

$$\begin{array}{cccc}& \mathrm{\mid }{\mathrm{\nabla }}_x⟨{\sigma }_0{\sigma }_x⟩\mathrm{\mid }\le \frac{C}{\mathrm{\mid }x\mathrm{\mid }}⟨{\sigma }_0{\sigma }_x⟩,& & \text{(4.25)}\end{array}|\backslash nabla\_x\; \backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle|\; \backslash leq\; \backslash frac\{C\}\{|x|\}\; \backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle,\; \backslash tag\{4.25\}$$

which follows from (4.2) by an argument that we choose to postpone to Section 5.5 (after the required technology has been introduced). ---PAGE_BREAK---

Proof Let us recall that we are discussing here $\beta = \beta_c$, omitting the symbol from the notation. Fix $\alpha > 3^4$ (the power 4 instead of 8 suffices at this stage) and choose $\varepsilon > 0$ so that $\alpha > (1+\varepsilon)(9+\varepsilon)^2$. Below, the constants $C_i$ are independent of $\beta$ and $n^\alpha = N \le \xi(\beta)$ (we may assume equality between $N$ and $n^\alpha$ without loss of generality). Introduce two intermediary integers $m \le M$ satisfying that

$$\begin{array}{cccc}& m\ge n^{9+\epsilon },M\ge m^{1+\epsilon },N\ge M^{9+\epsilon }& & \text{(4.26)}\end{array}m\; \backslash ge\; n^\{9+\backslash varepsilon\},\; \backslash quad\; M\; \backslash ge\; m^\{1+\backslash varepsilon\},\; \backslash quad\; N\; \backslash ge\; M^\{9+\backslash varepsilon\}\; \backslash tag\{4.26\}$$

as well as the notation $n_k = 2^k m$ for $k \ge 1$. Set $K$ such that $n_{K+1} \le M < n_{K+2}$. The key to our proof will be the random variable

$$\begin{array}{cccc}& \mathbf{N}:=\frac{1}{K}\sum _{k=1}^K\frac{1}{{\alpha }_k}\sum _{y\in \text{Ann}(n_k,n_{k+1})}\mathbb{I}[y^{{\mathbf{n}}_1+{\mathbf{n}}_2}\stackrel{}{\leftarrow }0]\text{where }{\alpha }_k:=\sum _{y\in \text{Ann}(n_k,n_{k+1})}⟨{\sigma }_0{\sigma }_y⟩\mathrm{.}& & \text{(4.27)}\end{array}\backslash mathbf\{N\}\; :=\; \backslash frac\{1\}\{K\}\; \backslash sum\_\{k=1\}^\{K\}\; \backslash frac\{1\}\{\backslash alpha\_k\}\; \backslash sum\_\{y\; \backslash in\; \backslash text\{Ann\}(n\_k,\; n\_\{k+1\})\}\; \backslash mathbb\{I\}[y^\{\backslash mathbf\{n\}\_1\; +\; \backslash mathbf\{n\}\_2\}\; \backslash xleftarrow\{\backslash hspace\{2em\}\}\; 0]\; \backslash quad\; \backslash text\{where\; \}\; \backslash alpha\_k\; :=\; \backslash sum\_\{y\; \backslash in\; \backslash text\{Ann\}(n\_k,\; n\_\{k+1\})\}\; \backslash langle\; \backslash sigma\_0\; \backslash sigma\_y\; \backslash rangle.\; \backslash tag\{4.27\}$$

Combining the regularity assumptions (4.25) and (4.2) with Proposition A.3 (the precise computation is presented in Section 6.2), we find

$$\begin{array}{cccc}& {\mathbf{E}}^{0x,\mathrm{\varnothing }}[\mathbf{N}]=\frac{1}{K}\sum _{k=1}^K\frac{1}{{\alpha }_k}\sum _{y\in \text{Ann}(n_k,n_{k+1})}\frac{⟨{\sigma }_0{\sigma }_y⟩⟨{\sigma }_y{\sigma }_x⟩}{⟨{\sigma }_0{\sigma }_x⟩}\ge 1-\frac{C_1}{K},& & \text{(4.28)}\end{array}\backslash mathbf\{E\}^\{0x,\backslash emptyset\}[\backslash mathbf\{N\}]\; =\; \backslash frac\{1\}\{K\}\; \backslash sum\_\{k=1\}^\{K\}\; \backslash frac\{1\}\{\backslash alpha\_k\}\; \backslash sum\_\{y\; \backslash in\; \backslash text\{Ann\}(n\_k,\; n\_\{k+1\})\}\; \backslash frac\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_y\; \backslash sigma\_x\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\}\; \backslash geq\; 1\; -\; \backslash frac\{C\_1\}\{K\},\; \backslash tag\{4.28\}$$

$$\begin{array}{cccc}& {\mathbf{E}}^{0x,\mathrm{\varnothing }}[{\mathbf{N}}^2]\le \frac{1}{K^2}\sum _{k,l=1}^K\frac{1}{{\alpha }_k{\alpha }_l}\sum _{\begin{array}{c}y\in \text{Ann}(n_k,n_{k+1})\\ z\in \text{Ann}(n_l,n_{l+1})\end{array}}\frac{⟨{\sigma }_0{\sigma }_y⟩⟨{\sigma }_y{\sigma }_z⟩⟨{\sigma }_z{\sigma }_x⟩+⟨{\sigma }_0{\sigma }_z⟩⟨{\sigma }_z{\sigma }_y⟩⟨{\sigma }_y{\sigma }_x⟩}{⟨{\sigma }_0{\sigma }_x⟩}\le 1+\frac{C_2}{K}\mathrm{.}& & \text{(4.29)}\end{array}\backslash mathbf\{E\}^\{0x,\backslash emptyset\}[\backslash mathbf\{N\}^2]\; \backslash leq\; \backslash frac\{1\}\{K^2\}\; \backslash sum\_\{k,l=1\}^\{K\}\; \backslash frac\{1\}\{\backslash alpha\_k\; \backslash alpha\_l\}\; \backslash sum\_\{\backslash substack\{y\; \backslash in\; \backslash text\{Ann\}(n\_k,\; n\_\{k+1\})\; \backslash \backslash \; z\; \backslash in\; \backslash text\{Ann\}(n\_l,\; n\_\{l+1\})\}\}\; \backslash frac\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_y\; \backslash sigma\_z\; \backslash rangle\; \backslash langle\; \backslash sigma\_z\; \backslash sigma\_x\; \backslash rangle\; +\; \backslash langle\; \backslash sigma\_0\; \backslash sigma\_z\; \backslash rangle\; \backslash langle\; \backslash sigma\_z\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_y\; \backslash sigma\_x\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\}\; \backslash leq\; 1\; +\; \backslash frac\{C\_2\}\{K\}.\; \backslash tag\{4.29\}$$

The Cauchy-Schwarz inequality and the fact that $\mathbf{P}^{0x}[E \cap F] = \mathbf{P}^{0x,\emptyset}[\mathbf{n}_1 \in E \cap F]$ thus imply that

$$\begin{array}{cccc}& \mathrm{\mid }{\mathbf{P}}^{0x,\mathrm{\varnothing }}[{\mathbf{n}}_1\in E\cap F]-{\mathbf{E}}^{0x,\mathrm{\varnothing }}[\mathbf{N}{\mathbb{I}}_{{\mathbf{n}}_1\in E\cap F}]\mathrm{\mid }\le \sqrt{{\mathbf{E}}^{0x,\mathrm{\varnothing }}[(\mathbf{N}-1{)}^2]}\le \frac{C_3}{\sqrt{K}}\mathrm{.}& & \text{(4.30)}\end{array}|\backslash mathbf\{P\}^\{0x,\backslash emptyset\}[\backslash mathbf\{n\}\_1\; \backslash in\; E\; \backslash cap\; F]\; -\; \backslash mathbf\{E\}^\{0x,\backslash emptyset\}[\backslash mathbf\{N\}\backslash mathbb\{I\}\_\{\backslash mathbf\{n\}\_1\; \backslash in\; E\; \backslash cap\; F\}]|\; \backslash leq\; \backslash sqrt\{\backslash mathbf\{E\}^\{0x,\backslash emptyset\}[(\backslash mathbf\{N\}-1)^2]\}\; \backslash leq\; \backslash frac\{C\_3\}\{\backslash sqrt\{K\}\}.\; \backslash tag\{4.30\}$$

Now, fix $y \in \text{Ann}(m, M)$ and let $G(y)$ be the event (depending on $\mathbf{n}_1 + \mathbf{n}_2$ only) that there exists $\mathbf{k} \le \mathbf{n}_1 + \mathbf{n}_2$ such that $\mathbf{k}=0$ on $\Lambda_n$, $\mathbf{k}=\mathbf{n}_1+\mathbf{n}_2$ outside $\Lambda_N$, and $\partial\mathbf{k} = {x,y}$. We find that

$$\begin{array}{cccc}& {\mathbf{P}}^{0x,\mathrm{\varnothing }}[{\mathbf{n}}_1\in E\cap F,y^{{\mathbf{n}}_1+{\mathbf{n}}_2}\stackrel{}{\leftarrow }0,G(y)]=\frac{⟨{\sigma }_0{\sigma }_y⟩⟨{\sigma }_y{\sigma }_x⟩}{⟨{\sigma }_0{\sigma }_x⟩}{\mathbf{P}}^{0y,yx}[{\mathbf{n}}_1\in E,{\mathbf{n}}_2\in F,G(y)],& & \text{(4.31)}\end{array}\backslash mathbf\{P\}^\{0x,\backslash emptyset\}[\backslash mathbf\{n\}\_1\; \backslash in\; E\; \backslash cap\; F,\; y^\{\backslash mathbf\{n\}\_1\; +\; \backslash mathbf\{n\}\_2\}\; \backslash xleftarrow\{\backslash hspace\{2em\}\}\; 0,\; G(y)]\; =\; \backslash frac\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_y\; \backslash sigma\_x\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\}\; \backslash mathbf\{P\}^\{0y,yx\}[\backslash mathbf\{n\}\_1\; \backslash in\; E,\; \backslash mathbf\{n\}\_2\; \backslash in\; F,\; G(y)],\; \backslash tag\{4.31\}$$

where we use the following reasoning: for $\mathbf{m} \in G(y)$, consider the multi-graph $\mathcal{M}$ obtained by duplicating every edge of the graph into $\mathbf{m}(x, y)$ edges. If $G(y)$ occurs, the existence of $\mathbf{k}$ guarantees the existence of a subgraph $\mathcal{K} \subset \mathcal{M}$ with $\partial\mathcal{K} = {x, y}$ containing no edge with endpoints in $\Lambda_n$ and all those of $\mathcal{M}$ with endpoints outside $\Lambda_N$, so that the generalized switching principle formulated in [5, Lemma 2.1] implies that

where we allow ourselves the latitude of calling $E$ and $F$ the events defined for multi-graphs corresponding to the events $E$ and $F$ for currents. One gets (4.31) when rephrasing this equality in terms of weighted currents (exactly like in standard proofs of the switching principle, see e.g. [1] or [5] for a closely related reasoning). ---PAGE_BREAK---

Observe now that forgetting about $G(y)$ on the right-hand side of (4.31) gives

$${\mathbf{P}}^{0y,yx}[{\mathbf{n}}_1\in E,{\mathbf{n}}_2\in F]={\mathbf{P}}^{0y}[E]{\mathbf{P}}^{yx}[F]\mathrm{.}(4.33)\backslash mathbf\{P\}^\{0y,yx\}[\backslash mathbf\{n\}\_1\; \backslash in\; E,\; \backslash mathbf\{n\}\_2\; \backslash in\; F]\; =\; \backslash mathbf\{P\}^\{0y\}[E]\backslash mathbf\{P\}^\{yx\}[F].\; \backslash quad\; (4.33)$$

Furthermore, since $x \notin \Lambda_N$ and $y \in \Lambda_m$, (4.25) implies that

$$\mid \frac{⟨{\sigma }_0{\sigma }_y⟩⟨{\sigma }_y{\sigma }_x⟩}{⟨{\sigma }_0{\sigma }_x⟩}-⟨{\sigma }_0{\sigma }_y⟩\mid \le \frac{C_{4}m}{N}\mathrm{.}(4.34)\backslash left|\; \backslash frac\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_y\; \backslash sigma\_x\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\}\; -\; \backslash langle\; \backslash sigma\_0\; \backslash sigma\_y\; \backslash rangle\; \backslash right|\; \backslash leq\; \backslash frac\{C\_4\; m\}\{N\}.\; \backslash quad\; (4.34)$$

Last but not least, we can bound (from below) $\mathbf{P}^{0x,\emptyset}[G(y)]$ and $\mathbf{P}^{0y,yx}[G(y)]$ as follows. We only briefly describe the argument since we will present it in full details in Section 6.2. The event $G(y)$ clearly contains the event that $\text{Ann}(M,N)$ is not crossed by a cluster in $\mathbf{n}_1$, and $\text{Ann}(n,m)$ is not crossed by a cluster in $\mathbf{n}_2$, since in such case $\mathbf{k}$ can be defined as the sum of $\mathbf{n}_1$ restricted to the clusters intersecting $\Lambda_N^c$ (this current has no sources) and $\mathbf{n}_2$ restricted to the clusters intersecting $\Lambda_m^c$ (this current has sources $x$ and $y$). Now, we can bound the probability of $\mathbf{n}_1$ crossing $\text{Ann}(M,N)$ in the same spirit as we bounded the probabilities for $F_1$ and $F_3$ in the previous proof by splitting $\text{Ann}(M,N)$ in two annuli $\text{Ann}(\sqrt{MN}, N)$ and $\text{Ann}(M, \sqrt{MN})$, then estimating the probability that the backbone of $\mathbf{n}_1$ crosses the inner annulus more than once, and then the probability that the remaining current (which is sourceless) crosses the outer annulus. Doing the same for the probability that a cluster of $\mathbf{n}_2$ crosses $\text{Ann}(n,m)$, we find that

$$\frac{⟨{\sigma }_0{\sigma }_x⟩}{⟨{\sigma }_0{\sigma }_y⟩⟨{\sigma }_y{\sigma }_x⟩}{\mathbf{P}}^{0x,\mathrm{\varnothing }}[G(y),y\underset{n_1+n_2=2}{\overset{\begin{array}{c}{n}_1+n_2\\ \leftrightarrow \end{array}}{\to }}0]={\mathbf{P}}^{0y,yx}[G(y)]\ge 1-\frac{C_5}{n^{ϵ}}\ge 1-C_6{\left(\frac{n}{N}\right)}^{ϵ\mathrm{/}(\alpha -1)}\mathrm{.}(4.35)\backslash frac\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_y\; \backslash sigma\_x\; \backslash rangle\}\; \backslash mathbf\{P\}^\{0x,\backslash emptyset\}[G(y),\; y\; \backslash xrightarrow[n\_1+n\_2=2]\{\backslash substack\{n\_1+\; n\_2\; \backslash \backslash \; \backslash leftrightarrow\}\}\; 0]\; =\; \backslash mathbf\{P\}^\{0y,yx\}[G(y)]\; \backslash geq\; 1\; -\; \backslash frac\{C\_5\}\{n^\backslash epsilon\}\; \backslash geq\; 1\; -\; C\_6\; \backslash left(\backslash frac\{n\}\{N\}\backslash right)^\{\backslash epsilon/(\backslash alpha-1)\}.\; \backslash quad\; (4.35)$$

Note that we use that $M \ge N^{9+\epsilon}$ in this part of the proof.

Overall, the value of K and (4.30)–(4.35) put together imply

$$\mathrm{\mid }{\mathbf{P}}^{0x}[E\cap F]-\sum _{y\in \mathrm{A}\mathrm{n}\mathrm{n}(m,M)}\delta (y){\mathbf{P}}^{0y,\mathrm{\varnothing }}[E]{\mathbf{P}}^{yx,\mathrm{\varnothing }}[F]\mathrm{\mid }\le \frac{C_7}{\sqrt{\log (N\mathrm{/}n)}},(4.36)|\backslash mathbf\{P\}^\{0x\}[E\; \backslash cap\; F]\; -\; \backslash sum\_\{y\; \backslash in\; \backslash mathrm\{Ann\}(m,M)\}\; \backslash delta(y)\; \backslash mathbf\{P\}^\{0y,\backslash emptyset\}[E]\; \backslash mathbf\{P\}^\{yx,\backslash emptyset\}[F]|\; \backslash leq\; \backslash frac\{C\_7\}\{\backslash sqrt\{\backslash log(N/n)\}\},\; \backslash quad\; (4.36)$$

with $\delta(y) = (\sigma_0\sigma_y)/(K\alpha_{k(y)})$ where $k(y)$ is such that $y \in \mathrm{Ann}(n_{k(y)}, n_{k(y)+1})$.

The end of the proof is now a matter of elementary algebraic manipulations. Applying this inequality twice (once with $x$ and once with $x'$) for $F$ being the full set, we obtain that for every $x, x' \notin \Lambda_N$ and every event $E$ which is depending on $\Lambda_n$ only,

$$\begin{array}{cccc}& \mathrm{\mid }{\mathbf{P}}^{0x}[E]-{\mathbf{P}}^{0x^{\mathrm{\prime }}}[E]\mathrm{\mid }\le \frac{2C_7}{\sqrt{\log (N\mathrm{/}n)}}\mathrm{.}& & \text{(4.37)}\end{array}|\backslash mathbf\{P\}^\{0x\}[E]\; -\; \backslash mathbf\{P\}^\{0x\text{'}\}[E]|\; \backslash leq\; \backslash frac\{2C\_7\}\{\backslash sqrt\{\backslash log(N/n)\}\}.\; \backslash tag\{4.37\}$$

Now, assume the stronger assumption that $\alpha > 3^8$ and fix $m = [\sqrt{Nn}]$. Applying

• (4.36) for m and N, the full event and F,

• then (4.37) for n, m and E (note that m ≥ n³),

• and (4.36) for m and N, E and F,

gives that for every $x \notin \Lambda_N$,

Using Lemma 4.4 and Proposition 4.6, we may now establish the clustering of inter-sections, under (4.2).

$\square$

---PAGE_BREAK---

Proof of Proposition 4.3 In view of the translation invariance of the claimed statement, we take $u$ to be the origin. Since $x$ and $y$ are at a distance larger than $2\ell_K$ of each other, one of them is at a distance (larger than or equal to) $\ell_K$ of $u$. Without loss of generality we take that to be $x$, and make a similar assumption about $z$.

Let $\mathcal{S}_K$ denote the set of subsets of ${1, \dots, K-2}$ containing even integers only and fix $S \in \mathcal{S}K$. Let $A_S$ be the event that no $I_k$ occurs for $k \in S$. If $s$ denotes the maximal element of $S$, the mixing property Proposition 4.6 used with $n = \ell{s-1}$ and $N = \ell_s$ gives

$${\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[A_S]\le {\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[I_s^c]{\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[A_{S\setminus \{s\}}]+\frac{C}{\sqrt{\log {\mathrm{\ell }}_{s-1}}}\mathrm{.}(4.39)\backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[A\_S]\; \backslash leq\; \backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[I\_s^c]\backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[A\_\{S\backslash setminus\backslash \{s\backslash \}\}]\; +\; \backslash frac\{C\}\{\backslash sqrt\{\backslash log\; \backslash ell\_\{s-1\}\}\}.\; \backslash quad\; (4.39)$$

To be precise and honest, we use a multi-current version, with four currents, of the mixing property. We will state and prove this property in Sections 6.1 and 6.2 and ignore this additional difficulty for now. Also, it is here that the stronger restriction $\alpha > 3^\delta$ is used, along with the choice of $\ell_0 = \ell_0(\alpha)$, to enable the mixing. Note that we used that the event $I_s$ is expressed in terms of just the restriction of the currents $\mathbf{n}_1, \dots, \mathbf{n}4$ to $\text{Ann}(\ell_s, \ell{s+1})$.

Now, the intersection property Lemma 4.4 and an elementary bound on $\ell_{s-1}$ gives the existence of $c_0 > 0$ small enough that

$${\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[A_S]\le (1-2c_0){\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[A_{S\setminus \{s\}}]+c_0(1-c_0{)}^{\mathrm{\mid }S\mathrm{\mid }-1}\mathrm{.}(4.40)\backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[A\_S]\; \backslash le\; (1-2c\_0)\backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[A\_\{S\backslash setminus\backslash \{s\backslash \}\}]\; +\; c\_0(1-c\_0)^\{|S|-1\}.\; \backslash quad\; (4.40)$$

An induction gives immediately that for every $S \in \mathcal{S}_K$,

$${\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[A_S]\le (1-c_0{)}^{\mathrm{\mid }S\mathrm{\mid }}\mathrm{.}(4.41)\backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[A\_S]\; \backslash le\; (1-c\_0)^\{|S|\}.\; \backslash quad\; (4.41)$$

Let $B_S \subset A_S$ be the event that the clusters of 0 in $\mathbf{n}_1 + \mathbf{n}_3$ and $\mathbf{n}_2 + \mathbf{n}4$ do not intersect in any of the annuli $\text{Ann}(\ell_s, \ell{s+1})$ for $s \in S$. Thanks to Corollary A.2, the probability of $B_S$ increases when removing sources, so that

$${\mathbf{P}}^{0x,0z,0y,0t}[B_S]\le {\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[B_S]\le {\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[A_S]\le (1-c_0{)}^{\mathrm{\mid }S\mathrm{\mid }}\mathrm{.}(4.42)\backslash mathbf\{P\}^\{0x,0z,0y,0t\}[B\_S]\; \backslash le\; \backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[B\_S]\; \backslash le\; \backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[A\_S]\; \backslash le\; (1-c\_0)^\{|S|\}.\; \backslash quad\; (4.42)$$

To conclude, observe that if $\mathbf{M}_0(\mathcal{T}; \mathcal{L}_\alpha, K) \le \delta K$, then there must exist a set $S \in \mathcal{S}_K$ of cardinality at least $(\frac{1}{2} - \delta)K$ such that $B_S$ occurs. We deduce that

which implies the claim by appropriately choosing the value of $\delta$. $\square$

5 Weak regularity of the two-point function

Progressing towards the unconditional proof of Theorem 1.3 we establish in this section the abundance, below the correlation length, of regular scales at which the two-point function has properties similar to those it would have under the power-law decay assumption (4.2). This auxiliary result is stated here as Theorem 5.12.

Towards this goal we focus here on the two-point function, and present some old and new observations. In particular, we discuss the following three properties of the two-point function:

(i) monotonicity (Section 5.1) ---PAGE_BREAK---

(ii) sliding-scale spatial Infrared Bound (Section 5.3),

(iii) gradient estimate (Section 5.5),

(iv) a lower bound for the two point function at $\beta_c$.

The first three are based on the reflection-positivity of the n.n.f. interaction, and apply to systems of real valued variables of arbitrary (but common) distribution, of sub-gaussian growth, i.e. satisfying (2.2). That includes the Ising and $\varphi^4$ variables which are of partic- ular interest for us. The last item is proven for systems with spins in the GS class.

Some unifying notation: In statements which apply to both the Ising and $\varphi^4$ systems, we shall refer to the spin/field variables by the “neutral” symbol $\tau$. Its a-priori distribution is denoted $\rho(d\tau)$. It may be displayed as a subscript, but also will often be omitted.

The expectation value functional with respect to the Gibbs measure, or functional integral, for a system in the domain $\Lambda$ is denoted $\langle \cdot \rangle_{\Lambda,\rho,\beta}$, with $\langle \cdot \rangle_{\rho,\beta}$ denoting the states' natural infinite volume limit.

We also denote by $\beta_c(\rho)$ (or just $\beta_c$ where the spins' a-priori distribution $\rho$ is clear from the context) is the critical inverse temperature and $\xi(\rho,\beta)$ the correlation length.

Throughout this section $|J| := \sum_y J_{0,y}$ and

$$\begin{array}{cccc}& S_{\rho ,\beta }(x):=⟨{\tau }_0{\tau }_x{⟩}_{\rho ,\beta }\mathrm{.}& & \text{(5.1)}\end{array}S\_\{\backslash rho,\backslash beta\}(x)\; :=\; \backslash langle\; \backslash tau\_0\; \backslash tau\_x\; \backslash rangle\_\{\backslash rho,\backslash beta\}.\; \backslash tag\{5.1\}$$

We refer to points in $\mathbb{R}^d$ as $x = (x_1, \dots, x_d)$ and denote by $e_j$ the unit vector with $x_j = 1$.

5.1 Messager-Miracle-Sole monotonicity for the two-point function

The Messenger-Miracle-Solé (MMS) inequality [30, 37, 41] states that for models with n.n.f. interactions (and more generally reflection-positive interactions) in a region $\Lambda$ endowed with reflection symmetry, the correlation function $\langle \prod_{x \in A} \tau_x \prod_{x \in B} \tau_x \rangle_{\Lambda, \rho, \beta}$ at sets of sites A and B which are on the same side of a reflection plane, can only decrease when B is replaced by its reflected image, $\mathcal{R}(B)$, i.e.

$${⟨\prod _{x\in A}{\tau }_x\prod _{x\in B}{\tau }_x⟩}_{\mathrm{\Lambda },\rho ,\beta }\ge {⟨\prod _{x\in A}{\tau }_x\prod _{x\in \mathcal{R}(B)}{\tau }_x⟩}_{\mathrm{\Lambda },\rho ,\beta }\mathrm{.}(5.2)\backslash left\backslash langle\; \backslash prod\_\{x\; \backslash in\; A\}\; \backslash tau\_x\; \backslash prod\_\{x\; \backslash in\; B\}\; \backslash tau\_x\; \backslash right\backslash rangle\_\{\backslash Lambda,\; \backslash rho,\; \backslash beta\}\; \backslash geq\; \backslash left\backslash langle\; \backslash prod\_\{x\; \backslash in\; A\}\; \backslash tau\_x\; \backslash prod\_\{x\; \backslash in\; \backslash mathcal\{R\}(B)\}\; \backslash tau\_x\; \backslash right\backslash rangle\_\{\backslash Lambda,\; \backslash rho,\; \backslash beta\}.\; \backslash quad\; (5.2)$$

In the infinite volume limit on $\mathbb{Z}^d$, this principle can be invoked for reflections with respect to

• hyperplanes passing through vertices or mid-edges, i.e. reflections changing only one coordinate $x_i$, which is sent to $L - x_i$ for some fixed $L \in \frac{1}{2}\mathbb{Z}$,

• “diagonal” hyperplanes, i.e. reflections changing only two coordinates $x_i$ and $x_j$, which are sent to $x_j \pm L$ and $x_i \mp L$ respectively, for some $L \in \mathbb{Z}$.

In particular, this implies the following useful comparison principle.

Proposition 5.1 (MMS monotonicity) For the n.n.f. model on $\mathbb{Z}^d$ ($d \ge 1$) with real valued spin variables satisfying (2.2):

i) along the principal axes the two-point function is monotone decreasing in $|x|_\infty$

$$\mathit{\text{ii)}}\mathit{\text{for}}\mathit{\text{any}}x=(x_1,\dots ,x_d)\in {\mathbb{Z}}^d,\backslash textit\{ii)\; for\; any\; \}\; x\; =\; (x\_1,\; \backslash dots,\; x\_d)\; \backslash in\; \backslash mathbb\{Z\}^d,$$

$$S_{\rho ,\beta }((\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }},0_{\perp }))\ge S_{\rho ,\beta }(x)\ge S_{\rho ,\beta }((\mathrm{\parallel }x{\mathrm{\parallel }}_1,0_{\perp })),(5.3)S\_\{\backslash rho,\backslash beta\}((\backslash |x\backslash |\_\{\backslash infty\},\; 0\_\{\backslash perp\}))\; \backslash geq\; S\_\{\backslash rho,\backslash beta\}(x)\; \backslash geq\; S\_\{\backslash rho,\backslash beta\}((\backslash |x\backslash |\_\{1\},\; 0\_\{\backslash perp\})),\; \backslash quad\; (5.3)$$

where $|x|1 := \sum{j=1}^{d} |x_j|$, $|x|{\infty} := \max_j |x_j|$, and $0{\perp}$ is the null vector in $\mathbb{Z}^{d-1}$. ---PAGE_BREAK---

The above carries the useful implication that for any $x, y \in \mathbb{Z}^d$ with $|y|{\infty} \ge d |x|{\infty}$,

$$S_{\rho ,\beta }(x)\ge S_{\rho ,\beta }(y)(5.4)S\_\{\backslash rho,\backslash beta\}(x)\; \backslash ge\; S\_\{\backslash rho,\backslash beta\}(y)\; \backslash quad\; (5.4)$$

since $|x|_1 \le d |x|_\infty \le |y|\infty$, by (5.3) the two quantities are on the correspondingly opposite sides of $S{\rho,\beta}((|x|_1, 0_\perp))$.

We shall encounter below also monotonicity statements of the Fourier transform. Both are useful in extracting point-wise implications from bounds on the corresponding two point function's bulk averages (as in Corollary 5.8, below).

5.2 The two-point function’s Fourier transform

In view of the model’s translation invariance it is natural to consider the system’s behavior also through its Fourier spin-wave modes. These are defined as

$${\tilde{\tau }}_{\beta }(p):=\frac{1}{\sqrt{(2L{)}^d}}\sum _{x\in (-L,L{]}^d}{e}^{ip\cdot x}{\tau }_x(5.5)\backslash tilde\{\backslash tau\}\_\{\backslash beta\}(p)\; :=\; \backslash frac\{1\}\{\backslash sqrt\{(2L)^d\}\}\; \backslash sum\_\{x\; \backslash in\; (-L,L]^d\}\; e^\{ip\; \backslash cdot\; x\}\; \backslash tau\_x\; \backslash qquad\; (5.5)$$

with $p$ ranging over $\Lambda_L^* := [-\pi, \pi)^d \cap \frac{\pi}{L}\mathbb{Z}^d$.

These variables are especially relevant in case the Hamiltonian is taken with the periodic boundary conditions, under which sites $x, y \in \Lambda_L$ are neighbors if either $|x - y|_1 = 1$ or $|y_i - x_i| = 2L - 1$ for some $i \in {1, \dots, d}$. With these boundary conditions, the model is invariant under cyclic shifts, and its Hamiltonian decomposes into a sum of single-mode contributions:

$$H_{{\mathrm{\Lambda }}_R}(\tau )=\sum _{p\in {\mathrm{\Lambda }}_L^{\ast }}\mathcal{E}(p)\mathrm{\mid }{\widehat{\tau }}_{\beta }(p){\mathrm{\mid }}^2,(5.6)H\_\{\backslash Lambda\_R\}(\backslash tau)\; =\; \backslash sum\_\{p\; \backslash in\; \backslash Lambda\_L^*\}\; \backslash mathcal\{E\}(p)\; |\backslash hat\{\backslash tau\}\_\backslash beta(p)|^2,\; \backslash quad\; (5.6)$$

with

$$\mathcal{E}(p):=2\sum _{j=1}^d[1-\cos (p_j)]=4\sum _{j=1}^d{\sin }^2(p_j\mathrm{/}2)\mathrm{.}(5.7)\backslash mathcal\{E\}(p)\; :=\; 2\; \backslash sum\_\{j=1\}^\{d\}\; [1\; -\; \backslash cos(p\_j)]\; =\; 4\; \backslash sum\_\{j=1\}^\{d\}\; \backslash sin^2(p\_j/2).\; \backslash quad\; (5.7)$$

Among the various relations in which the Fourier transform plays a useful role, the following statements will be of relevance for our discussion.

i) The spin-wave modes' second moment coincides with the finite volume Fourier transform of the two-point correlation function ($S^{(L)}(p)$):

$${\tilde{S}}_{\rho ,\beta }^{(L)}(p):=\sum _{x\in {\mathrm{\Lambda }}_L}{e}^{ip\cdot x}⟨{\tau }_0{\tau }_x{⟩}_{{\mathrm{\Lambda }}_L,\rho ,\beta }^{(b\mathrm{.}c\mathrm{.})}=(⟨{\widehat{\tau }}_{\beta }(p){⟩}^2{)}_{{\mathrm{\Lambda }}_L,\rho ,\beta }^{(b\mathrm{.}c\mathrm{.})}\ge 0.(5.8)\backslash widetilde\{S\}\_\{\backslash rho,\backslash beta\}^\{(L)\}(p)\; :=\; \backslash sum\_\{x\; \backslash in\; \backslash Lambda\_L\}\; e^\{ip\; \backslash cdot\; x\}\; \backslash langle\; \backslash tau\_0\; \backslash tau\_x\; \backslash rangle\_\{\backslash Lambda\_L,\; \backslash rho,\; \backslash beta\}^\{(b.c.)\}\; =\; (\backslash langle\; \backslash hat\{\backslash tau\}\_\backslash beta(p)\; \backslash rangle^2)\_\{\backslash Lambda\_L,\; \backslash rho,\; \backslash beta\}^\{(b.c.)\}\; \backslash ge\; 0.\; \backslash quad\; (5.8)$$

ii) For the n.n.f. interaction, and more generally reflection-positive interactions, the following gaussian-domination (aka infrared) bound holds [18, 19]:

$$\mathcal{E}(p){\tilde{S}}_{\rho ,\beta }^{(L)}(p)\le \frac{1}{2\mathrm{\mid }J\mathrm{\mid }\beta }\mathrm{.}(5.9)\backslash mathcal\{E\}(p)\; \backslash widetilde\{S\}\_\{\backslash rho,\backslash beta\}^\{(L)\}(p)\; \backslash leq\; \backslash frac\{1\}\{2|J|\backslash beta\}.\; \backslash quad\; (5.9)$$

The bound appeals to the physicists’ intuition, reminding one of the equipartition law. Alas, it has so far been proven only for reflection-positive interactions.

iii) The Parseval-Plancherel identity yields the sum rule:

$$⟨{\tau }_0^2{⟩}_{{\mathrm{\Lambda }}_L,\rho ,\beta }^{(b\mathrm{.}c\mathrm{.})}=\frac{1}{\mathrm{\mid }{\mathrm{\Lambda }}_L\mathrm{\mid }}\sum _{p\in {\mathrm{\Lambda }}_L^{\ast }}{\tilde{S}}_{\rho ,\beta }^{(L)}(p)\mathrm{.}(5.10)\backslash langle\; \backslash tau\_0^2\; \backslash rangle\_\{\backslash Lambda\_L,\; \backslash rho,\; \backslash beta\}^\{(b.c.)\}\; =\; \backslash frac\{1\}\{|\backslash Lambda\_L|\}\; \backslash sum\_\{p\; \backslash in\; \backslash Lambda\_L^*\}\; \backslash widetilde\{S\}\_\{\backslash rho,\; \backslash beta\}^\{(L)\}(p).\; \backslash quad\; (5.10)$$ ---PAGE_BREAK---

As was pointed out in [19], the combination of (5.10) with (5.9) yields a (then novel) way to prove the occurrence of spontaneous magnetization in dimensions $d > 2$, at high enough $\beta$.

More explicitly, in (5.10) one may note that the Infrared Bound (5.9) does not provide any direct control on the $p=0$ term, since $\mathcal{E}(0) = 0$. And, in fact, the hallmark of the low temperature phase ($\beta > \beta_c(\rho)$) is that this single value of the summand attains macroscopic size:

$${\widehat{S}}_{\rho ,\beta }^{(L)}(0)\approx \mathrm{\mid }{\mathrm{\Lambda }}_L\mathrm{\mid }M(\rho ,\beta {)}^2(5.11)\backslash widehat\{S\}\_\{\backslash rho,\backslash beta\}^\{(L)\}(0)\; \backslash approx\; |\backslash Lambda\_L|\; M(\backslash rho,\; \backslash beta)^2\; \backslash quad\; (5.11)$$

with $M(\rho, \beta)$ the model's spontaneous magnetization.

We shall also use the following statement on the relation between the finite volume and the infinite volume states.

Proposition 5.2 For a model in the GS class on $\mathbb{Z}^d$, $d > 2$, with translation invariant finite range interactions, for any $\beta < \beta_c(\rho)$:

1. the system has only one infinite volume Gibbs equilibrium state.

2. the correlation functions of that state satisfy, for any finite $A \subset \mathbb{Z}^d$, and any sequence of finite volumes $V_n \subset \mathbb{Z}^d$ which asymptotically cover any finite region,

$$⟨\prod _{x\in A}{\tau }_x{⟩}_{\rho ,\beta }=\underset{V_n\to {\mathbb{Z}}^d}{\lim }⟨\prod _{x\in A}{\tau }_x{⟩}_{V_n,\rho ,\beta }^{(b\mathrm{.}c\mathrm{.})}(5.12)\backslash langle\; \backslash prod\_\{x\; \backslash in\; A\}\; \backslash tau\_x\; \backslash rangle\_\{\backslash rho,\; \backslash beta\}\; =\; \backslash lim\_\{V\_n\; \backslash to\; \backslash mathbb\{Z\}^d\}\; \backslash langle\; \backslash prod\_\{x\; \backslash in\; A\}\; \backslash tau\_x\; \backslash rangle\_\{V\_n,\; \backslash rho,\; \backslash beta\}^\{(b.c.)\}\; \backslash quad\; (5.12)$$

with $\langle - \rangle^{(b.c.)}{V_n, \rho, \beta}$ denoting the correlation function under boundary conditions which may include either cross-boundary spin couplings (e.g. periodic), or arbitrary specified values of $\tau{|\partial V_n}$.

3. with the finite volumes taken as the rectangular domains $\Lambda_L$, also the Fourier Transform functions converge, i.e. for any $p \in [-\pi, \pi]^d$, and sequence as in (5.12)

$$\underset{n\to \mathrm{\infty }}{\lim }\sum _{x\in V_n}{e}^{ip\cdot x}⟨{\tau }_0{\tau }_x{⟩}_{V_n,\rho ,\beta }^{(b\mathrm{.}c\mathrm{.})}=\sum _{x\in {\mathbb{Z}}^d}{e}^{ip\cdot x}{S}_{\rho ,\beta }(x)=:{\widehat{S}}_{\rho ,\beta }(p)(5.13)\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash sum\_\{x\; \backslash in\; V\_n\}\; e^\{ip\; \backslash cdot\; x\}\; \backslash langle\; \backslash tau\_0\; \backslash tau\_x\; \backslash rangle\_\{V\_n,\; \backslash rho,\; \backslash beta\}^\{(b.c.)\}\; =\; \backslash sum\_\{x\; \backslash in\; \backslash mathbb\{Z\}^d\}\; e^\{ip\; \backslash cdot\; x\}\; S\_\{\backslash rho,\; \backslash beta\}(x)\; =:\; \backslash hat\{S\}\_\{\backslash rho,\; \backslash beta\}(p)\; \backslash quad\; (5.13)$$

The statement follows by standard arguments that we omit here. The main ingredients are the exponential decay of correlations, which at any $\beta < \beta_c(\rho)$ are exponentially bounded, uniformly in the volume, and the FKG inequality. The first two points hold also for $\beta = \beta_c(\rho)$ [4]. However not the last, (5.13), since at the critical temperature the correlation function is not summable.

We shall employ the freedom which Proposition 5.2 provides in establishing the different monotonicity properties of $S(p)$ in $p$.

Furthermore, for the disordered regime, where $M(\rho, \beta) = 0$, the sum rule combined with the Infrared Bound implies that for every $\beta < \beta_c(\rho)$,

$$⟨{\tau }_0^2{⟩}_{\beta }={\int }_{[-\pi ,\pi {]}^d}{\widehat{S}}_{\rho ,\beta }(p)dp\le {\int }_{[-\pi ,\pi {]}^d}\frac{dp}{2\mathrm{\mid }J\mathrm{\mid }\beta \mathcal{E}(p)}\mathrm{.}(5.14)\backslash langle\; \backslash tau\_0^2\; \backslash rangle\_\backslash beta\; =\; \backslash int\_\{[-\backslash pi,\; \backslash pi]^d\}\; \backslash hat\{S\}\_\{\backslash rho,\; \backslash beta\}(p)\; dp\; \backslash leq\; \backslash int\_\{[-\backslash pi,\; \backslash pi]^d\}\; \backslash frac\{dp\}\{2|J|\backslash beta\; \backslash mathcal\{E\}(p)\}.\; \backslash quad\; (5.14)$$

Since $\mathcal{E}(p)$ vanishes only at $p=0$ and there at the rate $\mathcal{E}(p) \sim |p|^2$, the integral is convergent for $d > 2$ and one gets

$$⟨{\tau }_0^2{⟩}_{\rho ,{\beta }_c(\rho )}\le \frac{C_d}{2\mathrm{\mid }J\mathrm{\mid }{\beta }_c(\rho )}(5.15)\backslash langle\; \backslash tau\_0^2\; \backslash rangle\_\{\backslash rho,\; \backslash beta\_c(\backslash rho)\}\; \backslash leq\; \backslash frac\{C\_d\}\{2|J|\backslash beta\_c(\backslash rho)\}\; \backslash quad\; (5.15)$$

with $C_d < \infty$ for $d > 2$. This bound will be used in Section 7. ---PAGE_BREAK---

5.3 The spectral representation and a sliding-scale Infrared Bound

We next present a Fourier transform counterpart (though one derived by different means) of the Messager-Miracle-Sole monotonicity stated in Section 5.1, and use it for a sliding-scale extension of the Infrared Bound (5.9). The results, which include both old [21, 47] and new observations, are based on the relation of the two-point function with the transfer matrix, and the positivity of the latter.

The transfer matrix has been the source of many insights on the structure of statistical mechanical systems with finite range interactions. Its appearance can be seen in Ising's study of one dimensional systems, for which it permits a simple proof of the absence of phase transition. Also, in higher dimensions it has played an essential role in many important developments [19, 21, 42], some of which rely on positivity properties. Here we shall use the following consequences of its spectral representation for the two-point function.

Proposition 5.3 (Spectral Representation) In the n.n.f. model on $\mathbb{Z}^d$ ($d \ge 1$), at $\beta < \beta_c(\rho)$, for every square summable function $v: \mathbb{Z}^{d-1} \to \mathbb{C}$, there exists a positive measure $\mu_{v,\beta}$ of finite mass

$${\int }_{1\mathrm{/}\xi (\rho ,\beta )}^{\mathrm{\infty }}d{\mu }_{v,\beta }(a)=\sum _{x_{\perp },y_{\perp }\in {\mathbb{Z}}^{d-1}}{v}_{x_{\perp }}\overline{v_{y_{\perp }}}{S}_{\rho ,\beta }((0,y_{\perp }-x_{\perp }))(5.16)\backslash int\_\{1/\backslash xi(\backslash rho,\; \backslash beta)\}^\{\backslash infty\}\; d\backslash mu\_\{v,\backslash beta\}(a)\; =\; \backslash sum\_\{x\_\{\backslash perp\},\; y\_\{\backslash perp\}\; \backslash in\; \backslash mathbb\{Z\}^\{d-1\}\}\; v\_\{x\_\{\backslash perp\}\}\; \backslash overline\{v\_\{y\_\{\backslash perp\}\}\}\; S\_\{\backslash rho,\; \backslash beta\}((0,\; y\_\{\backslash perp\}\; -\; x\_\{\backslash perp\}))\; \backslash quad\; (5.16)$$

such that for every $n \in \mathbb{Z}$

$$\sum _{x_{\perp },y_{\perp }\in {\mathbb{Z}}^{d-1}}{v}_{x_{\perp }}\overline{v_{y_{\perp }}}{S}_{\rho ,\beta }((n,x_{\perp }-y_{\perp }))={\int }_{1\mathrm{/}\xi (\rho ,\beta )}^{\mathrm{\infty }}{e}^{-a\mathrm{\mid }n\mathrm{\mid }}d{\mu }_{v,\beta }(a)\mathrm{.}(5.17)\backslash sum\_\{x\_\{\backslash perp\},\; y\_\{\backslash perp\}\; \backslash in\; \backslash mathbb\{Z\}^\{d-1\}\}\; v\_\{x\_\{\backslash perp\}\}\; \backslash overline\{v\_\{y\_\{\backslash perp\}\}\}\; S\_\{\backslash rho,\; \backslash beta\}((n,\; x\_\{\backslash perp\}\; -\; y\_\{\backslash perp\}))\; =\; \backslash int\_\{1/\backslash xi(\backslash rho,\; \backslash beta)\}^\{\backslash infty\}\; e^\{-a|n|\}\; d\backslash mu\_\{v,\backslash beta\}(a).\; \backslash quad\; (5.17)$$

And for every $p_{\perp} \in [-\pi, \pi]^{d-1}$ there exists a positive measure $\mu_{p_{\perp},\beta}$ of finite mass such that for every $p_1 \in [-\pi, \pi]$

$${\widehat{S}}_{\rho ,\beta }(p)={\int }_0^{\mathrm{\infty }}\frac{e^a-e^{-a}}{{\mathcal{E}}_1(p_1)+(e^{a\mathrm{/}2}-e^{-a\mathrm{/}2}{)}^2}d{\mu }_{p_1,\beta }(a),(5.18)\backslash hat\{S\}\_\{\backslash rho,\; \backslash beta\}(p)\; =\; \backslash int\_0^\backslash infty\; \backslash frac\{e^a\; -\; e^\{-a\}\}\{\backslash mathcal\{E\}\_1(p\_1)\; +\; (e^\{a/2\}\; -\; e^\{-a/2\})^2\}\; d\backslash mu\_\{p\_1,\; \backslash beta\}(a),\; \backslash quad\; (5.18)$$

with $\mathcal{E}_1(k) := 2[1 - \cos(k)] = 4\sin^2(k/2)$.

Although the spectral representation is quite well-known (cf. [21] and references therein) for completeness of the presentation we include the derivation of (5.17) in the Appendix. Equation (5.18) then follows by applying (5.17) to the function

$$v_{p_1}(x_{\perp }):=\frac{1}{\sqrt{\mathrm{\mid }{\mathrm{\Lambda }}_{\mathrm{\ell }}^{(d-1)}\mathrm{\mid }}}{e}^{i{p}_1\cdot x_{\perp }}I[x\in {\mathrm{\Lambda }}_{\mathrm{\ell }}^{(d-1)}]v\_\{p\_1\}(x\_\backslash perp)\; :=\; \backslash frac\{1\}\{\backslash sqrt\{|\backslash Lambda\_\backslash ell^\{(d-1)\}|\}\}\; e^\{ip\_1\; \backslash cdot\; x\_\backslash perp\}\; I[x\; \backslash in\; \backslash Lambda\_\backslash ell^\{(d-1)\}]$$

and taking the limit $\ell \to \infty$. Here $\Lambda_\ell^{(d-1)}$ is the $d-1$ dimensional version of the box $\Lambda_L$ and $I[\cdot]$ is the indicator function. The convergence is facilitated by the exponential decay of correlations at $\beta < \beta_c$.

Of particular interest for us are the following implications of (5.18) (the first was noted and applied in [21]).

Proposition 5.4 For a n.n.f. model on $\mathbb{Z}^d$ ($d \ge 1$), at any $\beta < \beta_c(\rho)$:

1. $\hat{S}_{\rho,\beta}(p_1, p_2, \dots, p_d)$ is monotone decreasing in each $|p_j|$, over $[-\pi, \pi]$,

2. $\mathcal{E}1(p_1)\hat{S}{\rho,\beta}(p)$ and $|p_1|^2\hat{S}_{\rho,\beta}(p_1)$ are monotone increasing in $|p_1|,$ ---PAGE_BREAK---

3. the function

$${\widehat{S}}_{\rho ,\beta }^{(\text{mod})}(p):={\widehat{S}}_{\rho ,\beta }(p)+{\widehat{S}}_{\rho ,\beta }(p+\pi (1,1,0,\dots ,0))(5.19)\backslash hat\{S\}\_\{\backslash rho,\backslash beta\}^\{(\backslash text\{mod\})\}(p)\; :=\; \backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(p)\; +\; \backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(p\; +\; \backslash pi(1,1,0,\backslash dots,0))\; \backslash quad\; (5.19)$$

is monotone decreasing in $\delta$ along the line of constant ${p_3, \dots, p_d}$ and

$$\begin{array}{cccc}& (p_1,p_2)=(\mathrm{\mid }{p}_1-p_2\mathrm{\mid }+\delta ,\mathrm{\mid }{p}_1-p_2\mathrm{\mid }-\delta ),\delta \in [0,\mathrm{\mid }{p}_1-p_2\mathrm{\mid }]\mathrm{.}& & \text{(5.20)}\end{array}(p\_1,\; p\_2)\; =\; (|p\_1\; -\; p\_2|\; +\; \backslash delta,\; |p\_1\; -\; p\_2|\; -\; \backslash delta),\; \backslash quad\; \backslash delta\; \backslash in\; [0,\; |p\_1\; -\; p\_2|].\; \backslash tag\{5.20\}$$

and the above remains true under any permutation of the indices.

The correction in (5.19) is insignificant in the regime where $\hat{S}{\rho,\beta}(p)$ is large. That is so since $|\hat{S}{\rho,\beta}(p+\pi(1,1,0,\dots,0))| \le C/\beta$ uniformly for $|p| \le \pi/2$. (The main term diverges in the limit $\beta \nearrow \beta_c(\rho)$ and $p \to 0$.)

Proof The first two statements are implied by the combination of (5.18) with the observation that each of the following functions is monotone in $k \in [0, \pi]: k \mapsto \mathcal{E}_1(k)$, $k \mapsto k^2/\mathcal{E}_1(k)$, and for each $a \ge 0$, $k \mapsto \mathcal{E}_1(k)/(\mathcal{E}_1(k) + (e^{a/2} - e^{-a/2})^2)$.

The third statement is based on the application of the transfer matrix in the diagonal direction (cf. Fig. 4). More explicitly, to produce the spectral representation one may start by considering a partially rotated rectangular region, whose main axes are associated with the coordinate system ($x_1+x_2, x_1-x_2, x_3, \dots, x_d$). The finite-volume Hamiltonian is taken with the correspondingly modified periodic boundary conditions which produce cyclicity in these directions. As stated in (5.13), for $\beta < \beta_c(\rho)$ the change does not affect the two-point function’s infinite volume limit.

In this case, there are two transfer matrices $T$ and $T^*$ corresponding to adding one layer of even (resp. odd) vertices, i.e. vertices with $x_1+x_2$ even (resp. odd). The argument by which monotonicity was proven above for the Cartesian directions applies to the two-point function's restriction to the sub-lattice of even vertices since the proof would involve the matrix $TT^*$, which is positive.

Then, if $\hat{S}_{\rho,\beta}^{(\mathrm{mod})}$ is given by (5.19), one finds

$${\widehat{S}}_{\rho ,\beta }^{(\mathrm{m}\mathrm{o}\mathrm{d})}(p)=\sum _{x\in {\mathrm{\Lambda }}_L}{e}^{ip\cdot x}{S}_{\rho ,\beta }(0,x)\sum _{k=0,1}{e}^{i\pi (x_1+x_2)k}=2\sum _{\begin{array}{c}x\in {\mathrm{\Lambda }}_L\\ x_1+x_2\text{ even}\end{array}}{e}^{ip\cdot x}{S}_{\rho ,\beta }(0,x)\mathrm{.}(5.21)\backslash hat\{S\}\_\{\backslash rho,\; \backslash beta\}^\{(\backslash mathrm\{mod\})\}(p)\; =\; \backslash sum\_\{x\; \backslash in\; \backslash Lambda\_L\}\; e^\{ip\; \backslash cdot\; x\}\; S\_\{\backslash rho,\; \backslash beta\}(0,x)\; \backslash sum\_\{k=0,1\}\; e^\{i\backslash pi(x\_1+x\_2)k\}\; =\; 2\; \backslash sum\_\{\backslash substack\{x\; \backslash in\; \backslash Lambda\_L\; \backslash \backslash \; x\_1+x\_2\; \backslash text\{\; even\}\}\}\; e^\{ip\; \backslash cdot\; x\}\; S\_\{\backslash rho,\; \backslash beta\}(0,x).\; \backslash quad\; (5.21)$$

Thus, the third monotonicity statement follows by a direct adaptation of the proof of the first one. $\square$

Corollary 5.5 For a n.n.f. model on $\mathbb{Z}^d$ ($d \ge 1$) at any $\beta < \beta_c(\rho)$, the two-point function satisfies, for all $p \in [-\pi/2, \pi/2]^d$,

$${\widehat{S}}_{\rho ,\beta }(\mathrm{\parallel }p{\mathrm{\parallel }}_{\mathrm{\infty }},0_{\perp })\ge {\widehat{S}}_{\rho ,\beta }(p)\ge {\widehat{S}}_{\rho ,\beta }(\mathrm{\parallel }p{\mathrm{\parallel }}_1,0_{\perp })-\frac{C}{\beta },(5.22)\backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(\backslash |p\backslash |\_\{\backslash infty\},\; 0\_\{\backslash perp\})\; \backslash geq\; \backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(p)\; \backslash geq\; \backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(\backslash |p\backslash |\_\{1\},\; 0\_\{\backslash perp\})\; -\; \backslash frac\{C\}\{\backslash beta\},\; \backslash quad\; (5.22)$$

with $C$ depending on the dimension only.

The restriction to $p \in [-\pi/2, \pi/2]^d$ guarantees that the second term of (5.19) can be bounded by $C/\beta$ (this bounds corresponds to the $-C/\beta$ term on the right-hand side of (5.22)), as explained below Proposition 5.4. ---PAGE_BREAK---

Figure 4: The split of $Z^2$ into even and odd sub-lattices and their stratification into intertwined diagonal hyperplanes. The partition function of $(-L, L]^2$ with rotated-periodic boundary condition can be written as $Z_{2L} = \text{tr}(T_2 T_1)^L$, with $T_j$ a pair of conjugate mapping between the Hilbert spaces of the even and the odd hyperplanes. The product $T_2 T_1 = T_1^* T_1$ provides the even subgraph's transfer matrix in one of this graph's principal directions.

Proof The inequality follows from Proposition 5.4 through the monotonicity lines used for (5.3). $\square$

The previous bound combined with the second statement in Proposition 5.4 yield an interesting consequence for the behaviour of the susceptibility truncated at a distance L, which we define as

$${\chi }_L(\rho ,\beta ):=\sum _{x\in {\mathrm{\Lambda }}_L}{S}_{\rho ,\beta }(x)\mathrm{.}(5.23)\backslash chi\_L(\backslash rho,\; \backslash beta)\; :=\; \backslash sum\_\{x\; \backslash in\; \backslash Lambda\_L\}\; S\_\{\backslash rho,\; \backslash beta\}(x).\; \backslash qquad\; (5.23)$$

Theorem 5.6 (Sliding-scale Infrared Bound) There exists a constant $C = C(d) > 0$ such that for every n.n.f. model on $\mathbb{Z}^d$ ($d > 2$), every $\beta \le \beta_c(\rho)$ and $L \ge \ell \ge 1$,

$$\frac{{\chi }_L(\rho ,\beta )}{L^2}\le \frac{C{\chi }_{\mathrm{\ell }}(\rho ,\beta )}{\beta {\mathrm{\ell }}^2}\mathrm{.}(5.24)\backslash frac\{\backslash chi\_L(\backslash rho,\; \backslash beta)\}\{L^2\}\; \backslash le\; \backslash frac\{C\; \backslash chi\_\backslash ell(\backslash rho,\; \backslash beta)\}\{\backslash beta\; \backslash ell^2\}.\; \backslash qquad\; (5.24)$$

The case $\ell = 1$ is in essence similar to the Infrared Bound (5.9), as is explained below, so that (5.24) may be viewed as a sliding-scale version of this inequality. One may also note that (5.24) is a sharp improvement (replacing the exponent $d$ by 2) on the more naive application of the Messager-Miracle-Sole inequality giving that for every $L \ge \ell \ge 1$,

$$\frac{{\chi }_L(\rho ,\beta )}{L^d}\le \frac{{\chi }_{\mathrm{\ell }}(\rho ,\beta )}{{\mathrm{\ell }}^d}\mathrm{.}(5.25)\backslash frac\{\backslash chi\_L(\backslash rho,\; \backslash beta)\}\{L^d\}\; \backslash le\; \backslash frac\{\backslash chi\_\backslash ell(\backslash rho,\; \backslash beta)\}\{\backslash ell^d\}.\; \backslash qquad\; (5.25)$$

Proof Let us first note that it suffices to prove the claim for all $\beta < \beta_c(\rho)$, with a uniform constant $C$. Its extension to the critical point can be deduced from the continuity

$$S_{\rho ,\beta }(x)=\underset{\beta \to {\beta }_c(\rho )}{\lim }{S}_{{\beta }_c(\rho )}(x)(5.26)S\_\{\backslash rho,\backslash beta\}(x)\; =\; \backslash lim\_\{\backslash beta\; \backslash to\; \backslash beta\_c(\backslash rho)\}\; S\_\{\backslash beta\_c(\backslash rho)\}(x)\; \backslash qquad\; (5.26)$$

(which follows from the main result of [4]). This observation allows us to apply the monotonicity results discussed above. ---PAGE_BREAK---

Below, the constants $C_i$ are to be understood as dependent on $d$ only. Consider the smeared version of $\chi_L(\rho, \beta)$ defined by

$${\tilde{\chi }}_L(\rho ,\beta ):=\sum _{x\in {\mathbb{Z}}^d}{e}^{-(\mathrm{\parallel }x{\mathrm{\parallel }}_2\mathrm{/}L{)}^2}{S}_{\rho ,\beta }(x)\mathrm{.}(5.27)\backslash tilde\{\backslash chi\}\_L(\backslash rho,\; \backslash beta)\; :=\; \backslash sum\_\{x\; \backslash in\; \backslash mathbb\{Z\}^d\}\; e^\{-(\backslash |x\backslash |\_2/L)^2\}\; S\_\{\backslash rho,\backslash beta\}(x).\; \backslash quad\; (5.27)$$

with $|p|2^2 := \sum{i=1}^d p_i^2$. The MMS monotonicity statement (5.4) implies that

$$e^{-d{\chi }_L(\rho ,\beta )}\le {\tilde{\chi }}_L(\rho ,\beta )\le C_1{\chi }_L(\rho ,\beta )(5.28)e^\{-d\backslash chi\_L(\backslash rho,\; \backslash beta)\}\; \backslash le\; \backslash tilde\{\backslash chi\}\_L(\backslash rho,\; \backslash beta)\; \backslash le\; C\_1\backslash chi\_L(\backslash rho,\; \backslash beta)\; \backslash quad\; (5.28)$$

for every $L$, so that it suffices to prove that for every $L \ge \ell \ge 1$,

$$\frac{{\tilde{\chi }}_L(\rho ,\beta )}{L^2}\le C_2\frac{{\tilde{\chi }}_{\mathrm{\ell }}(\rho ,\beta )}{{\mathrm{\ell }}^2}\mathrm{.}(5.29)\backslash frac\{\backslash tilde\{\backslash chi\}\_L(\backslash rho,\; \backslash beta)\}\{L^2\}\; \backslash le\; C\_2\; \backslash frac\{\backslash tilde\{\backslash chi\}\_\backslash ell(\backslash rho,\; \backslash beta)\}\{\backslash ell^2\}.\; \backslash quad\; (5.29)$$

We will work in Fourier space, and use the identity

$${\tilde{\chi }}_L(\rho ,\beta )\asymp L^d{\int }_{[-\pi ,\pi {]}^d}{e}^{-\mathrm{\parallel }p{\mathrm{\parallel }}^2}{\widehat{S}}_{\rho ,\beta }(p)dp,(5.30)\backslash tilde\{\backslash chi\}\_L(\backslash rho,\; \backslash beta)\; \backslash asymp\; L^d\; \backslash int\_\{[-\backslash pi,\; \backslash pi]^d\}\; e^\{-\backslash |p\backslash |^2\}\; \backslash hat\{S\}\_\{\backslash rho,\; \backslash beta\}(p)\; dp,\; \backslash quad\; (5.30)$$

where $f \asymp g$ means $cg \le f \le Cg$ with $c, C$ independent of everything else (we use that the Fourier transform of the Gaussian on the lattice is a Jacobi theta-function within multiplicative constants of $e^{-|p|^2}$ on $[-\pi, \pi]^d$).

Now, let

$$A:=\{p\in [-\frac{\pi \mathrm{\ell }}{L},\frac{\pi \mathrm{\ell }}{L}{]}^d:\mathrm{\mid }{p}_1\mathrm{\mid }=\mathrm{\parallel }p{\mathrm{\parallel }}_{\mathrm{\infty }}\}\mathrm{.}(5.31)A\; :=\; \backslash \{p\; \backslash in\; [-\backslash frac\{\backslash pi\backslash ell\}\{L\},\; \backslash frac\{\backslash pi\backslash ell\}\{L\}]^d\; :\; |p\_1|\; =\; \backslash |p\backslash |\_\backslash infty\backslash \}.\; \backslash quad\; (5.31)$$

Using the symmetries of $\hat{S}_{\rho,\beta}$ and the decay of Corollary 5.5, we find that

$${\int }_{[-\pi ,\pi {]}^d}{e}^{-\mathrm{\parallel }p{\mathrm{\parallel }}^2{L}^2}{\widehat{S}}_{\rho ,\beta }(p)dp\le (d+C_3{e}^{-{\mathrm{\ell }}^2}){\int }_A{e}^{-\mathrm{\parallel }p{\mathrm{\parallel }}^2{L}^2}{\widehat{S}}_{\rho ,\beta }(p)dp\mathrm{.}(5.32)\backslash int\_\{[-\backslash pi,\; \backslash pi]^d\}\; e^\{-\backslash |p\backslash |^2\; L^2\}\; \backslash hat\{S\}\_\{\backslash rho,\; \backslash beta\}(p)\; dp\; \backslash le\; (d\; +\; C\_3\; e^\{-\backslash ell^2\})\; \backslash int\_A\; e^\{-\backslash |p\backslash |^2\; L^2\}\; \backslash hat\{S\}\_\{\backslash rho,\; \backslash beta\}(p)\; dp.\; \backslash quad\; (5.32)$$

Since $|p_1| = |p|_\infty$ for $p \in A$ and $|p|_\infty \ge |p|_1/d$, the second property of Proposition 5.4 and Corollary 5.5 give that

$${\widehat{S}}_{\rho ,\beta }(p)\le {\widehat{S}}_{\rho ,\beta }(\mathrm{\parallel }p{\mathrm{\parallel }}_{\mathrm{\infty }},0_{\perp })\le (d^L{)}^2{\widehat{S}}_{\rho ,\beta }(\frac{L}{\mathrm{\ell }}\mathrm{\parallel }p{\mathrm{\parallel }}_1,0_{\perp })\le (d^L{)}^2({\widehat{S}}_{\rho ,\beta }(\frac{L}{\mathrm{\ell }}p)+C\mathrm{/}\beta )\mathrm{.}(5.33)\backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(p)\; \backslash le\; \backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(\backslash |p\backslash |\_\{\backslash infty\},\; 0\_\{\backslash perp\})\; \backslash le\; (d^L)^2\; \backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(\backslash frac\{L\}\{\backslash ell\}\backslash |p\backslash |\_1,\; 0\_\{\backslash perp\})\; \backslash le\; (d^L)^2\; (\backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(\backslash frac\{L\}\{\backslash ell\}p)\; +\; C/\backslash beta).\; \backslash quad\; (5.33)$$

Using this inequality and making the change of variable $p \mapsto q = L/\ell p$ gives

$${\int }_A\exp [-\mathrm{\parallel }p{\mathrm{\parallel }}^2{L}^2]{\widehat{S}}_{\rho ,\beta }(p)dp\le C_4(\frac{\mathrm{\ell }}{L}{)}^{d-2}({\int }_{[-\pi ,\pi {]}^d}\exp [-\mathrm{\parallel }q{\mathrm{\parallel }}^2{L}^2]{\widehat{S}}_{\rho ,\beta }(q)dq+C_5\mathrm{/}\beta ),(5.34)\backslash int\_A\; \backslash exp[-\backslash |p\backslash |^2\; L^2]\; \backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(p)\; dp\; \backslash le\; C\_4\; (\backslash frac\{\backslash ell\}\{L\})^\{d-2\}\; \backslash left(\; \backslash int\_\{[-\backslash pi,\; \backslash pi]^d\}\; \backslash exp[-\backslash |q\backslash |^2\; L^2]\; \backslash hat\{S\}\_\{\backslash rho,\backslash beta\}(q)\; dq\; +\; C\_5/\backslash beta\; \backslash right),\; \backslash quad\; (5.34)$$

which after plugging in (5.32) and taking the Fourier transform implies that

$${\tilde{\chi }}_L(\rho ,\beta )\le C_6(\frac{\mathrm{\ell }}{L}{)}^{d-2}({\tilde{\chi }}_{\mathrm{\ell }}(\rho ,\beta )+C_5\mathrm{/}\beta )\mathrm{.}(5.35)\backslash tilde\{\backslash chi\}\_L(\backslash rho,\; \backslash beta)\; \backslash le\; C\_6\; (\backslash frac\{\backslash ell\}\{L\})^\{d-2\}\; (\backslash tilde\{\backslash chi\}\_\backslash ell(\backslash rho,\; \backslash beta)\; +\; C\_5/\backslash beta).\; \backslash quad\; (5.35)$$

The inequality (5.29) follows from the fact that $\tilde{\chi}_\ell(\rho, \beta) \ge 1$, so that the constant $C_5/\beta$ can be removed by changing $C_6$ into a larger constant $C_7/\beta$. $\square$

Inequality (5.4) and then the sliding-scale Infrared Bound with $L = |x|$ and $\ell = 1$ (5.24) implies that for every $x \in \mathbb{Z}^d$,

$$S_{\rho ,\beta }(x)\le \frac{C_1}{\mathrm{\mid }x{\mathrm{\mid }}^d}\sum _{y\in \text{Ann}(d\mathrm{\mid }x\mathrm{\mid },2d\mathrm{\mid }x\mathrm{\mid })}{S}_{\rho ,\beta }(y)\le \frac{C_1}{\mathrm{\mid }x{\mathrm{\mid }}^d}{\chi }_{2d\mathrm{\mid }x\mathrm{\mid }}(\rho ,\beta )\le \frac{C_2⟨{\tau }_0^2{⟩}_{\rho ,\beta }}{\mathrm{\mid }x{\mathrm{\mid }}^{d-2}}\mathrm{.}(5.36)S\_\{\backslash rho,\backslash beta\}(x)\; \backslash le\; \backslash frac\{C\_1\}\{|x|^d\}\; \backslash sum\_\{y\; \backslash in\; \backslash text\{Ann\}(d|x|,\; 2d|x|)\}\; S\_\{\backslash rho,\backslash beta\}(y)\; \backslash le\; \backslash frac\{C\_1\}\{|x|^d\}\; \backslash chi\_\{2d|x|\}(\backslash rho,\; \backslash beta)\; \backslash le\; \backslash frac\{C\_2\; \backslash langle\; \backslash tau\_0^2\; \backslash rangle\_\{\backslash rho,\backslash beta\}\}\{|x|^\{d-2\}\}.\; \backslash quad\; (5.36)$$

The factor $\langle \tau_0^2 \rangle_\beta$ in the upper bound may seem pointless for the Ising model where it is simply equal to 1, but it becomes very important when studying unbounded spins, as in Section 7, where it is essential for a dimensionless improved tree diagram bound.

It may be noted that the combination of (5.36) with (5.14) leads to the more standard formulation [18, 19] of the Infrared Bound in $x$-space:

$$S_{\rho ,\beta }(x)\le \frac{C}{\beta \mathrm{\mid }J\mathrm{\mid }\mathrm{\mid }x{\mathrm{\mid }}^{d-2}}\mathrm{.}(5.37)S\_\{\backslash rho,\backslash beta\}(x)\; \backslash le\; \backslash frac\{C\}\{\backslash beta\; |J||x|^\{d-2\}\}.\; \backslash quad\; (5.37)$$ ---PAGE_BREAK---

5.4 A lower bound

The above upper bound will next be supplemented by a power-law lower bound on the two point function at $\beta_c$. Conceptually, it originates in the observation that it the correlations drop on some scale by a fast enough power law then on larger scales they decay exponentially fast. An early version of this principle can be found in Hammersley's analysis of percolation [28]. A general statement was presented in Dobrushin's analysis of the constructive criteria for the high temperature phase. For Ising systems a simple version of such statement can be deduced from the following observation.

Lemma 5.7 For every ferromagnetic model in the GS class on $\mathbb{Z}^d$ ($d \ge 1$) with coupling constants that are invariant under translations, every finite $0 \in \Lambda \subset \mathbb{Z}^d$ and every $y \notin \Lambda$,

$$S_{\rho ,\beta }(y)\le \sum _{\begin{array}{c}u\in \mathrm{\Lambda }\\ v\mathrm{\notin }\mathrm{\Lambda }\end{array}}{S}_{\rho ,\beta }(u)\beta J_{u,v}{S}_{\rho ,\beta }(y-v)\mathrm{.}(5.38)S\_\{\backslash rho,\backslash beta\}(y)\; \backslash le\; \backslash sum\_\{\backslash substack\{u\; \backslash in\; \backslash Lambda\; \backslash \backslash \; v\; \backslash notin\; \backslash Lambda\}\}\; S\_\{\backslash rho,\backslash beta\}(u)\; \backslash beta\; J\_\{u,v\}\; S\_\{\backslash rho,\backslash beta\}(y-v).\; \backslash quad\; (5.38)$$

This statement is a mild extension of Simon's inequality which was originally formulated for the n.n.f. Ising models [44]. Being spin-dimension balanced, it is valid also for the Griffiths-Simon class of variables and more general pair interactions⁵.

The MMS monotonicity allows us to extract the following point-wise implication, which will be used below

Corollary 5.8 (Lower bound on $S_{\rho,\beta}$) For a n.n.f. model in the GS class on $\mathbb{Z}^d$ ($d \ge 1$), there exists $c = c(d) > 0$ such that for every $\beta \le \beta_c(\rho)$ and $x \in \mathbb{Z}^d$,

$$S_{\rho ,\beta }(x)\ge \frac{c}{\beta \mathrm{\mid }J\mathrm{\mid }\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}^{d-1}}\exp (-\frac{d\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}+1}{\xi (\rho ,\beta )})\mathrm{.}(5.39)S\_\{\backslash rho,\backslash beta\}(x)\; \backslash ge\; \backslash frac\{c\}\{\backslash beta|J|\; \backslash |x\backslash |\_\{\backslash infty\}^\{d-1\}\}\; \backslash exp\backslash left(-\backslash frac\{d\backslash |x\backslash |\_\{\backslash infty\}\; +\; 1\}\{\backslash xi(\backslash rho,\; \backslash beta)\}\backslash right).\; \backslash quad\; (5.39)$$

Proof Let us introduce

$$Y_{\rho ,\beta }(\mathrm{\Lambda }):=\sum _{\begin{array}{c}u\in \mathrm{\Lambda }\\ v\mathrm{\notin }\mathrm{\Lambda }\end{array}}{S}_{\rho ,\beta }(u)\beta J_{u,v}\mathrm{.}(5.40)Y\_\{\backslash rho,\backslash beta\}(\backslash Lambda)\; :=\; \backslash sum\_\{\backslash substack\{u\; \backslash in\; \backslash Lambda\; \backslash \backslash \; v\; \backslash notin\; \backslash Lambda\}\}\; S\_\{\backslash rho,\backslash beta\}(u)\; \backslash beta\; J\_\{u,v\}.\; \backslash quad\; (5.40)$$

Set $L := d|x|_{\infty}$. Applying (5.38) with $\Lambda_L$ and $y = ne_1$, and iterating it $\lfloor \frac{n}{L+1} \rfloor$ times (i.e. as many as possible without reducing the last factor to a distance shorter than $L$), we get

$$\beta \mathrm{\mid }J\mathrm{\mid }{S}_{\rho ,\beta }(n{e}_1)\le Y_{\rho ,\beta }({\mathrm{\Lambda }}_L{)}^{\lceil \frac{n}{L+1}\rceil }\mathrm{.}(5.41)\backslash beta\; |J|\; S\_\{\backslash rho,\backslash beta\}(ne\_1)\; \backslash le\; Y\_\{\backslash rho,\backslash beta\}(\backslash Lambda\_L)^\{\backslash lceil\; \backslash frac\{n\}\{L+1\}\; \backslash rceil\}.\; \backslash quad\; (5.41)$$

Since $\lim_n S_{\rho,\beta}(ne_1)^{1/n} = e^{-1/\xi(\rho,\beta)}$, we deduce that

$$e^{-1\mathrm{/}\xi (\rho ,\beta )}\le Y_{\rho ,\beta }({\mathrm{\Lambda }}_L{)}^{\frac{1}{L+1}}\mathrm{.}(5.42)e^\{-1/\backslash xi(\backslash rho,\; \backslash beta)\}\; \backslash le\; Y\_\{\backslash rho,\; \backslash beta\}(\backslash Lambda\_L)^\{\backslash frac\{1\}\{L+1\}\}.\; \backslash quad\; (5.42)$$

On the other hand, by (5.4), for each $x \in \mathbb{Z}^d$, $S_{\rho,\beta}(u) \le S_{\rho,\beta}(x)$ for all $u \in \partial\Lambda_L$, and hence

$$\frac{Y_{\rho ,\beta }({\mathrm{\Lambda }}_L)}{\mathrm{\mid }\mathrm{\partial }{\mathrm{\Lambda }}_L\mathrm{\mid }}\le \beta \mathrm{\mid }J\mathrm{\mid }{S}_{\rho ,\beta }(x)\mathrm{.}(5.43)\backslash frac\{Y\_\{\backslash rho,\backslash beta\}(\backslash Lambda\_L)\}\{|\backslash partial\backslash Lambda\_L|\}\; \backslash le\; \backslash beta\; |J|\; S\_\{\backslash rho,\backslash beta\}(x).\; \backslash quad\; (5.43)$$

The substitution of (5.42) in (5.43) yields the claimed lower bound (5.39). $\square$

⁵The factor $S_{\rho,\beta}(u)$ in (5.38) can also be replaced by the finite volume expectation $\langle \tau_0 \tau_u \rangle_\Lambda$, as in Lieb's improvement of Simon's inequality [36]. Both versions have an easy proof through a simple application of the switching lemma, in its mildly improved form. ---PAGE_BREAK---

5.5 Regularity of the two-point function’s gradient

Proposition 5.9 (gradient estimate) There exists C = C(d) > 0 such that for every n.n.f. model in the GS class, every $\beta \le \beta_c(\rho)$, every $x \in \mathbb{Z}^d$ and every $1 \le i \le d$,

$$\mathrm{\mid }{S}_{\rho ,\beta }(x\pm {\mathbf{e}}_i)-S_{\rho ,\beta }(x)\mathrm{\mid }\le \frac{F(\mathrm{\mid }x\mathrm{\mid })}{\mathrm{\mid }x\mathrm{\mid }}{S}_{\rho ,\beta }(x),(5.44)|S\_\{\backslash rho,\backslash beta\}(x\; \backslash pm\; \backslash mathbf\{e\}\_i)\; -\; S\_\{\backslash rho,\backslash beta\}(x)|\; \backslash leq\; \backslash frac\{F(|x|)\}\{|x|\}\; S\_\{\backslash rho,\backslash beta\}(x),\; \backslash quad\; (5.44)$$

where

$$F(n):=C\frac{S_{\rho ,\beta }(dn{\mathbf{e}}_1)}{S_{\rho ,\beta }(n{\mathbf{e}}_1)}\log \left(\frac{2S_{\rho ,\beta }(\frac{n}{2}{\mathbf{e}}_1)}{S_{\rho ,\beta }(n{\mathbf{e}}_1)}\right)\mathrm{.}(5.45)F(n)\; :=\; C\; \backslash frac\{S\_\{\backslash rho,\; \backslash beta\}(dn\backslash mathbf\{e\}\_1)\}\{S\_\{\backslash rho,\; \backslash beta\}(n\backslash mathbf\{e\}\_1)\}\; \backslash log\; \backslash left(\; \backslash frac\{2S\_\{\backslash rho,\; \backslash beta\}(\backslash frac\{n\}\{2\}\backslash mathbf\{e\}\_1)\}\{S\_\{\backslash rho,\; \backslash beta\}(n\backslash mathbf\{e\}\_1)\}\; \backslash right).\; \backslash quad\; (5.45)$$

The previous proposition is particularly interesting when $S_{\rho,\beta}(2d\mathbf{e}1) \ge c_0 S{\rho,\beta}(\frac{n}{2}\mathbf{e}_1)$, in which case we obtain the existence of a constant $C_0 = C_0(c_0, d) > 0$ such that for every $x \in \partial\Lambda_n$ and $1 \le i \le d$,

$$\begin{array}{cccc}& \mathrm{\mid }{S}_{\rho ,\beta }(x\pm {\mathbf{e}}_i)-S_{\rho ,\beta }(x)\mathrm{\mid }\le \frac{C_0}{\mathrm{\mid }x\mathrm{\mid }}{S}_{\rho ,\beta }(x)\mathrm{.}& & \text{(5.46)}\end{array}|S\_\{\backslash rho,\backslash beta\}(x\; \backslash pm\; \backslash mathbf\{e\}\_i)\; -\; S\_\{\backslash rho,\backslash beta\}(x)|\; \backslash leq\; \backslash frac\{C\_0\}\{|x|\}\; S\_\{\backslash rho,\backslash beta\}(x).\; \backslash tag\{5.46\}$$

Proof Without loss of generality, we may assume that $x = (|x|, x_\perp)$. We first assume that $i=1$. Introduce the three sequences $u_n := S_{\rho,\beta}(ne_1)$, $v_n := S_{\rho,\beta}((n,x_\perp))$ and $w_n := u_n + v_n$. The spectral representation applied to the function $v$ being the sum of the Dirac functions at $0_\perp$ and $x_\perp$ implies the existence of a finite measure $\mu_{x_\perp,\beta}$ such that

$$\begin{array}{cccc}& w_n={\int }_0^{\mathrm{\infty }}{e}^{-na}d{\mu }_{x_{\perp },\beta }(a)\mathrm{.}& & \text{(5.47)}\end{array}w\_n\; =\; \backslash int\_0^\backslash infty\; e^\{-na\}\; d\backslash mu\_\{x\_\backslash perp,\; \backslash beta\}(a).\; \backslash tag\{5.47\}$$

Cauchy-Schwarz gives $w_n^2 \le w_{n-1}w_{n+1}$, which when iterated between $n$ and $n/2$ (assume $n$ is even, the odd case is similar) leads to

$$\frac{w_{n+1}}{w_n}\ge {\left(\frac{w_n}{w_{n\mathrm{/}2}}\right)}^{2\mathrm{/}n}\ge 1-\frac{2}{n}\log \left(\frac{w_{n\mathrm{/}2}}{w_n}\right)\mathrm{.}(5.48)\backslash frac\{w\_\{n+1\}\}\{w\_n\}\; \backslash geq\; \backslash left(\backslash frac\{w\_n\}\{w\_\{n/2\}\}\backslash right)^\{2/n\}\; \backslash geq\; 1\; -\; \backslash frac\{2\}\{n\}\; \backslash log\; \backslash left(\backslash frac\{w\_\{n/2\}\}\{w\_n\}\backslash right).\; \backslash quad\; (5.48)$$

We now use that $u_{n/2} \ge v_{n/2}$, $u_n \ge v_n$, and $u_n \ge u_{n+1}$ which are all consequences of the Messager-Miracle-Sole inequality. Together with trivial algebraic manipulations, we get

$$v_{n+1}\ge v_n-\frac{4\log (2u_{n\mathrm{/}2}\mathrm{/}{u}_n)}{n}{u}_n\mathrm{.}(5.49)v\_\{n+1\}\; \backslash ge\; v\_n\; -\; \backslash frac\{4\backslash log(2u\_\{n/2\}/u\_n)\}\{n\}\; u\_n.\; \backslash quad\; (5.49)$$

The bound we are seeking corresponds to $n = |x|$.

To get the result for $i \neq 1$, use the Messager-Miracle-Sole inequality applied twice to get that

$$\mathrm{\mid }{S}_{\rho ,\beta }(x\pm {\mathbf{e}}_i)-S_{\rho ,\beta }(x)\mathrm{\mid }\le S_{\rho ,\beta }(x-d{\mathbf{e}}_1)-S_{\rho ,\beta }(x+d{\mathbf{e}}_1),(5.50)|S\_\{\backslash rho,\; \backslash beta\}(x\; \backslash pm\; \backslash mathbf\{e\}\_i)\; -\; S\_\{\backslash rho,\; \backslash beta\}(x)|\; \backslash leq\; S\_\{\backslash rho,\; \backslash beta\}(x\; -\; d\backslash mathbf\{e\}\_1)\; -\; S\_\{\backslash rho,\; \backslash beta\}(x\; +\; d\backslash mathbf\{e\}\_1),\; \backslash quad\; (5.50)$$

and then refer to the previous case to conclude (one obtains the result for $n = |x| - d$, but the proof can be easily adapted to get the result for $n = |x|$). $\square$

Remark 5.10 When $x = ne_1$ and $i = 1$, running through the lines of the previous proof shows that one can take $F(n) = 2\log(S_{\rho,\beta}(ne_1)/S_{\rho,\beta}(\frac{n}{2}e_1))$ which is bounded by $(2+o(1))\log n$ thanks to the lower bound (5.39) and the Infrared Bound (5.37). We therefore get that for every $n \le \xi(\rho, \beta)$,

$$S_{\rho ,\beta }(n{e}_1)-S_{\rho ,\beta }((n+1)e_1)\le (2+o(1))\frac{\log n}{n}{S}_{\rho ,\beta }(n{e}_1)\mathrm{.}(5.51)S\_\{\backslash rho,\; \backslash beta\}(ne\_1)\; -\; S\_\{\backslash rho,\; \backslash beta\}((n+1)e\_1)\; \backslash le\; (2+o(1))\; \backslash frac\{\backslash log\; n\}\{n\}\; S\_\{\backslash rho,\; \backslash beta\}(ne\_1).\; \backslash quad\; (5.51)$$

It would be of interest to remove the log n factor, as this would enable a proof that $S_{\rho,\beta}(ne_1)$ does not drop too fast between different scales. ---PAGE_BREAK---

5.6 Regular scales

Using the dyadic distance scales, we shall now introduce the notion of regular scales, which in essence means that on the given scale the two-point function has the properties which in the conditional proof of Section 4, were available under the assumption (4.2).

Definition 5.11 Fix $c, C > 0$. An annular region $\text{Ann}(n/2, 4n)$ is said to be regular if the following four properties are satisfied:

$$P1\text{for every }x,y\in \text{Ann}(n\mathrm{/}2,4n),S_{\rho ,\beta }(y)\le C{S}_{\rho ,\beta }(x);P1\; \backslash quad\; \backslash text\{for\; every\; \}\; x,\; y\; \backslash in\; \backslash text\{Ann\}(n/2,\; 4n),\; S\_\{\backslash rho,\backslash beta\}(y)\; \backslash le\; C\; S\_\{\backslash rho,\backslash beta\}(x);$$

$$P2\text{for every }x,y\in \text{Ann}(n\mathrm{/}2,4n),\mathrm{\mid }{S}_{\rho ,\beta }(x)-S_{\rho ,\beta }(y)\mathrm{\mid }\le \frac{C\mathrm{\mid }x-y\mathrm{\mid }}{\mathrm{\mid }x\mathrm{\mid }}{S}_{\rho ,\beta }(x);P2\; \backslash quad\; \backslash text\{for\; every\; \}\; x,\; y\; \backslash in\; \backslash text\{Ann\}(n/2,\; 4n),\; |S\_\{\backslash rho,\backslash beta\}(x)\; -\; S\_\{\backslash rho,\backslash beta\}(y)|\; \backslash le\; \backslash frac\{C|x-y|\}\{|x|\}\; S\_\{\backslash rho,\backslash beta\}(x);$$

$$P3\text{for every }x\in {\mathrm{\Lambda }}_n\text{ and }y\mathrm{\notin }{\mathrm{\Lambda }}_{Cn},S_{\rho ,\beta }(y)\le \frac{1}{2}{S}_{\rho ,\beta }(x);P3\; \backslash quad\; \backslash text\{for\; every\; \}\; x\; \backslash in\; \backslash Lambda\_n\; \backslash text\{\; and\; \}\; y\; \backslash notin\; \backslash Lambda\_\{Cn\},\; S\_\{\backslash rho,\backslash beta\}(y)\; \backslash le\; \backslash frac\{1\}\{2\}\; S\_\{\backslash rho,\backslash beta\}(x);$$

$$P4{\chi }_{2n}(\rho ,\beta )\ge (1+c){\chi }_n(\rho ,\beta )\mathrm{.}P4\; \backslash quad\; \backslash chi\_\{2n\}(\backslash rho,\; \backslash beta)\; \backslash ge\; (1+c)\backslash chi\_n(\backslash rho,\; \backslash beta).$$

A scale $k$ is said to be regular if the above holds for $n = 2^k$, and a vertex $x \in \mathbb{Z}^d$ will be said to be in a regular scale if it belongs to an annulus with the above properties.

One may note that $P1$ follows trivially from $P2$ but we still choose to state the two properties independently (the proof would work with weaker versions of $P2$ so one can imagine cases where the notion of regular scale could be used with a different version of $P2$ not implying $P1$).

Under the power-law assumption (4.2) of Section 4 every scale is regular at criticality. However, for now we do not have an unconditional proof of that. For an unconditional proof of our main results, this gap will be addressed through the following statement, which is the main result of this section.

Theorem 5.12 (Abundance of regular scales) Fix $d > 2$ and $\alpha > 2$. There exist $c = c(d) > 0$ and $C = C(d) > 0$ such that for every n.n.f. model in the GS class and every $n^\alpha \le N \le \xi(\rho, \beta)$, there are at least $c \log_2(N/n)$ regular scales $k$ with $n \le 2^k \le N$.

Proof The lower bound (5.8) for $S_{\rho,\beta}$ and the Infrared Bound (5.37) imply that

$${\chi }_N(\rho ,\beta )\ge c_{0}N\ge c_0(N\mathrm{/}n{)}^{(\alpha -2)\mathrm{/}(\alpha -1)}{n}^2\ge c_1(N\mathrm{/}n{)}^{(\alpha -2)\mathrm{/}(\alpha -1)}{\chi }_n(\rho ,\beta )\mathrm{.}(5.52)\backslash chi\_N(\backslash rho,\; \backslash beta)\; \backslash ge\; c\_0\; N\; \backslash ge\; c\_0\; (N/n)^\{(\backslash alpha-2)/(\backslash alpha-1)\}\; n^2\; \backslash ge\; c\_1\; (N/n)^\{(\backslash alpha-2)/(\backslash alpha-1)\}\; \backslash chi\_n(\backslash rho,\; \backslash beta).\; \backslash quad\; (5.52)$$

Using the sliding-scale Infrared Bound (5.25), there exist $r, c_2 > 0$ (independent of $n, N$) such that there are at least $c_2 \log_2(N/n)$ scales $m = 2^k$ between $n$ and $N$ such that

$${\chi }_{rm}(\rho ,\beta )\ge {\chi }_{4dm}(\rho ,\beta )+{\chi }_m(\rho ,\beta )\mathrm{.}(5.53)\backslash chi\_\{rm\}(\backslash rho,\; \backslash beta)\; \backslash ge\; \backslash chi\_\{4dm\}(\backslash rho,\; \backslash beta)\; +\; \backslash chi\_m(\backslash rho,\; \backslash beta).\; \backslash quad\; (5.53)$$

Let us verify that the different properties of regular scales are satisfied for such an $m$. Applying (5.4) in the first inequality, the assumption (5.53) in the second, and (5.4) in the third, one has

$$\mathrm{\mid }\text{Ann}(4dm,rm)\mathrm{\mid }{S}_{\rho ,\beta }(4dm{e}_1)\ge {\chi }_{rm}(\rho ,\beta )-{\chi }_{4dm}(\rho ,\beta )\ge {\chi }_m(\rho ,\beta )\ge \mathrm{\mid }{\mathrm{\Lambda }}_{m\mathrm{/}(4d)}\mathrm{\mid }{S}_{\rho ,\beta }(\frac{1}{4}m{e}_1)\mathrm{.}(5.54)|\backslash text\{Ann\}(4dm,\; rm)|\; S\_\{\backslash rho,\backslash beta\}(4dme\_1)\; \backslash ge\; \backslash chi\_\{rm\}(\backslash rho,\; \backslash beta)\; -\; \backslash chi\_\{4dm\}(\backslash rho,\; \backslash beta)\; \backslash ge\; \backslash chi\_m(\backslash rho,\; \backslash beta)\; \backslash ge\; |\backslash Lambda\_\{m/(4d)\}|\; S\_\{\backslash rho,\backslash beta\}(\backslash frac\{1\}\{4\}me\_1).\; \backslash quad\; (5.54)$$

This implies that $S_{\rho,\beta}(4dme_1) \ge c_0 S_{\rho,\beta}(\frac{1}{4}me_1)$, which immediately gives $P1$ by (5.4) for $S_{\rho,\beta}$ and $P2$ by the gradient estimate given by Proposition 5.9. Furthermore, the fact that $S_{\rho,\beta}(x) \ge S_{\rho,\beta}(4dme_1) \ge \frac{c_3}{m^d}\chi_m(\rho, \beta)$ for every $x \in \text{Ann}(m, 2m)$ implies $P4$. To prove P3, ---PAGE_BREAK---

observe that for every $R$, the previous displayed inequality together with the sliding-scale Infrared Bound (5.24) give that for every $y \notin \Lambda_{dRm}$ and $x \in \Lambda_m$,

$$\mathrm{\mid }{\mathrm{\Lambda }}_{Rm}\mathrm{\mid }{S}_{\rho ,\beta }(y)\le {\chi }_{Rm}(\rho ,\beta )\le C_4{R}^2{\chi }_m(\rho ,\beta )\le C_5{R}^2{m}^d{S}_{\rho ,\beta }(x),(5.55)|\backslash Lambda\_\{Rm\}|S\_\{\backslash rho,\backslash beta\}(y)\; \backslash le\; \backslash chi\_\{Rm\}(\backslash rho,\; \backslash beta)\; \backslash le\; C\_4\; R^2\; \backslash chi\_m(\backslash rho,\; \backslash beta)\; \backslash le\; C\_5\; R^2\; m^d\; S\_\{\backslash rho,\backslash beta\}(x),\; \backslash quad\; (5.55)$$

which implies the claim for $C$ and $c$ respectively large and small enough using here the assumption that $d > 2$. $\square$

6 Unconditional proofs of the Ising's results

In this section, we prove our results for every $\beta \le \beta_c$ without making the power-law assumption of Section 4. We emphasize that unlike the introductory discussion of that section, the proofs given below are unconditional. The discussion is also not restricted to the critical point itself and covers more general approaches of the scaling limits, from the side $\beta \le \beta_c$ (hence the correlation length will be mentioned in several places). However, at this stage the discussion is still restricted to the n.n.f. Ising model.

6.1 Unconditional proofs of the intersection-clustering bound and Theorem 1.3 for the Ising model

The notation remains as in Section 4. The endgame in this section will be the unconditional proof of the intersection-clustering bound that we restate below in the right level of generality. The main modification is that the sequence $\mathcal{L}$ of integers $l_k$ will be chosen dynamically, adjusting it to the behaviour of the two-point function. More precisely, recall the definition of the bubble diagram $B_L(\beta)$ truncated at a distance $L$. Fix $D \gg 1$ and define recursively a (possibly finite) sequence $\mathcal{L}$ of integers $l_k = l_k(\beta, D)$ by the formula $l_0 = 0$ and

$$l_{k+1}=\inf \{\mathrm{\ell }:B_{\mathrm{\ell }}(\beta )\ge D{B}_{{\mathrm{\ell }}_k}(\beta )\}\mathrm{.}(6.1)l\_\{k+1\}\; =\; \backslash inf\backslash \{\backslash ell\; :\; B\_\backslash ell(\backslash beta)\; \backslash ge\; D\; B\_\{\backslash ell\_k\}(\backslash beta)\backslash \}.\; \backslash qquad\; (6.1)$$

By the Infrared Bound (5.37), $B_L - B_\ell \le C_0 \log(L/\ell)$ (in dimension $d=4$) from which it is a simple exercise to deduce that under the above definition

$$D^k\le B_{{\mathrm{\ell }}_k}(\beta )\le C{D}^k(6.2)D^k\; \backslash le\; B\_\{\backslash ell\_k\}(\backslash beta)\; \backslash le\; CD^k\; \backslash qquad\; (6.2)$$

for every $k$ and some large constant $C$ independent of $k$.

Proposition 6.1 (clustering bound) For $d=4$ and $D$ large enough, there exists $\delta = \delta(D) > 0$ such that for every $\beta \le \beta_c$, every $K > 3$ with $\ell_K \le \xi(\beta)$, and every $u,x,y,z,t \in \mathbb{Z}^4$ with mutual distances between $x,y,z,t$ larger than $2\ell_K$,

Before proving this proposition, let us explain how it implies the improved tree diagram bound.

Proof of Theorem 1.3 Choose $D$ large enough that the previous proposition holds true. We follow the same lines as in Section 4.1, simply noting that since $B_{\ell_k}(\beta) \le CD^k$, we may choose $K \ge c \log B_L(\beta)$ with $2\ell_K \le L$, where $c$ is independent of $L$ and $\beta$, so that (4.13) implies the improved tree diagram bound inequality. $\square$ ---PAGE_BREAK---

The main modification we need for an unconditional proof of the intersection-clustering bound lies in the derivation of the intersection and mixing properties. The former is similar to Lemma 4.4, but restricted to sources that lie in regular scales. We restate it here in a slightly modified form.

Recall that $I_k$ is the event that there exist unique clusters of $\text{Ann}(\ell_k, \ell_{k+1})$ in $\mathbf{n}_1 + \mathbf{n}_3$ and $\mathbf{n}_2 + \mathbf{n}_4$ crossing the annulus from the inner boundary to the outer boundary and that these two clusters are intersecting.

Lemma 6.2 (Intersection property) Fix $d = 4$. There exists $c > 0$ such that for every $\beta \le \beta_c$, every $k$, and every $y \notin \Lambda_{2\ell_{k+1}}$ in a regular scale,

$$\begin{array}{cccc}& {\mathbf{P}}_{\beta }^{0y,0y,\mathrm{\varnothing },\mathrm{\varnothing }}[I_k]\ge c\mathrm{.}& & \text{(6.4)}\end{array}\backslash mathbf\{P\}\_\{\backslash beta\}^\{0y,0y,\backslash emptyset,\backslash emptyset\}[I\_k]\; \backslash geq\; c.\; \backslash tag\{6.4\}$$

Proof Restricting our attention to the case of y belonging to a regular scale enables us to use properties P1 and P2 of the regularity assumption on the scale. With this additional assumption, we follow the same proof as the one of the conditional version (Lemma 4.4). Introduce the intermediary integers $n \le m \le M \le N$ satisfying

$$\begin{array}{cccc}& {\mathrm{\ell }}_k^4\ge n\ge {\mathrm{\ell }}_k^{3+ϵ},n^4\ge m\ge n^{3+ϵ},M^4\ge N\ge M^{3+ϵ},N^4\ge {\mathrm{\ell }}_{k+1}\ge N^{3+ϵ}\mathrm{.}& & \text{(6.5)}\end{array}\backslash ell\_k^4\; \backslash ge\; n\; \backslash ge\; \backslash ell\_k^\{3+\backslash epsilon\},\; \backslash quad\; n^4\; \backslash ge\; m\; \backslash ge\; n^\{3+\backslash epsilon\},\; \backslash quad\; M^4\; \backslash ge\; N\; \backslash ge\; M^\{3+\backslash epsilon\},\; \backslash quad\; N^4\; \backslash ge\; \backslash ell\_\{k+1\}\; \backslash ge\; N^\{3+\backslash epsilon\}.\; \backslash tag\{6.5\}$$

For the second moment method on $\mathcal{M}$, the first and second moments take the following forms

\begin{equation} \begin{aligned} \mathbf{E}_{\beta}^{0y,0y,\emptyset,\emptyset}[|\mathcal{M}|] &\ge c_1(B_M(\beta) - B_{m-1}(\beta)) \ge c_2 B_{\ell_{k+1}}(\beta), \\ \mathbf{E}_{\beta}^{0y,0y,\emptyset,\emptyset}[|\mathcal{M}|^2] &\le c_3(B_M(\beta) - B_{m-1}(\beta)) B_{2M}(\beta) \le c_3 B_{\ell_{k+1}}(\beta)^2, \end{aligned} \tag{6.6}\tag{6.7} \end{equation}

where in the second inequality of the first line, we used that $D$ is large enough and Lemma 6.3 below to get that

$$B_M(\beta )\ge \frac{B_{{\mathrm{\ell }}_{k+1}}(\beta )}{1+15C}\text{and}{B}_{m-1}(\beta )\le (1+15C)B_{{\mathrm{\ell }}_k}(\beta )\le \frac{1+15C}{D}{B}_{{\mathrm{\ell }}_{k+1}}(\beta )\mathrm{.}B\_M(\backslash beta)\; \backslash ge\; \backslash frac\{B\_\{\backslash ell\_\{k+1\}\}(\backslash beta)\}\{1\; +\; 15C\}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; B\_\{m-1\}(\backslash beta)\; \backslash le\; (1\; +\; 15C)B\_\{\backslash ell\_k\}(\backslash beta)\; \backslash le\; \backslash frac\{1\; +\; 15C\}\{D\}\; B\_\{\backslash ell\_\{k+1\}\}(\backslash beta).$$

For the bound on the probabilities of the events $F_1, \dots, F_4$ defined as in Section 4.2, recall that the vertices $x$ and $z$ there are in our case both equal to $y$ that belongs to a regular scale. Using Property 2 of the regularity of scales, the bounds in (4.21) and (4.22) follow readily from the Infrared Bound (5.37). $\square$

In the previous proof, we used the following statement.

Lemma 6.3 For $d=4$, there exists $C > 0$ such that for every $\beta \le \beta_c$ and every $\ell \le L \le \xi(\beta)$,

$$\begin{array}{cccc}& B_L(\beta )\le (1+C\frac{\log (L\mathrm{/}\mathrm{\ell })}{\log \mathrm{\ell }})B_{\mathrm{\ell }}(\beta )\mathrm{.}& & \text{(6.8)}\end{array}B\_L(\backslash beta)\; \backslash le\; \backslash left(1\; +\; C\; \backslash frac\{\backslash log(L/\backslash ell)\}\{\backslash log\; \backslash ell\}\backslash right)\; B\_\backslash ell(\backslash beta).\; \backslash tag\{6.8\}$$

Proof For every $n \le N$ for which $n = 2^k$ with $k$ regular, we have that (recall the definition of $\chi_n(\beta)$ from the previous section)

---PAGE_BREAK---

where in the first inequality we used (5.4), in the second the sliding-scaled Infrared Bound (5.24), in the third Property P4 of the regularity of $n$, and in the last Cauchy-Schwarz.

Now, there are $\log_2(L/\ell)$ scales between $\ell$ and $L$, and at least $\frac{1}{C}\log_2\ell$ regular scales between 1 and $\ell$ by abundance of regular scales (Theorem 5.12). Since the sums of squared correlations on any of the former contribute less to $B_L(\beta) - B_\ell(\beta)$ than any of the latter to $B_\ell(\beta)$, we deduce that

$$B_L(\beta )\le (1+C\frac{{\log }_2(L\mathrm{/}\mathrm{\ell })}{{\log }_2\mathrm{\ell }})B_{\mathrm{\ell }}(\beta )\mathrm{.}(6.10)B\_L(\backslash beta)\; \backslash leq\; \backslash left(1\; +\; C\; \backslash frac\{\backslash log\_2(L/\backslash ell)\}\{\backslash log\_2\; \backslash ell\}\backslash right)\; B\_\backslash ell(\backslash beta).\; \backslash quad\; (6.10)$$

Next comes the unconditional mixing property.

Theorem 6.4 (random currents' mixing property) For $d \geq 4$, there exist $\alpha, c > 0$ such that for every $t \leq s$, every $\beta \leq \beta_c$, every $n^\alpha \leq N \leq \xi(\beta)$, every $x_i \in \Lambda_n$ and $y_i \notin \Lambda_N$ ($i \leq t$), and every events E and F depending on the restriction of $(\mathbf{n}_1, \dots, \mathbf{n}_s)$ to edges within $\Lambda_n$ and outside of $\Lambda_N$ respectively,

$$\mid {\mathbf{P}}_{\beta }^{x_1{y}_1,\dots ,x_t{y}_t,\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[E\cap F]-{\mathbf{P}}_{\beta }^{x_1^{\mathrm{\prime }}{y}_1^{\mathrm{\prime }},\dots ,x_t^{\mathrm{\prime }}{y}_t^{\mathrm{\prime }},\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[E]{\mathbf{P}}_{\beta }^{x_1{y}_1,\dots ,x_t{y}_t,\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[F]\mid \le s(\log \frac{N}{n}{)}^{-c}\mathrm{.}(6.11)\backslash left|\; \backslash mathbf\{P\}\_\{\backslash beta\}^\{x\_1\; y\_1,\; \backslash dots,\; x\_t\; y\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [E\; \backslash cap\; F]\; -\; \backslash mathbf\{P\}\_\{\backslash beta\}^\{x\text{'}\_1\; y\text{'}\_1,\; \backslash dots,\; x\text{'}\_t\; y\text{'}\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [E]\; \backslash mathbf\{P\}\_\{\backslash beta\}^\{x\_1\; y\_1,\; \backslash dots,\; x\_t\; y\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [F]\; \backslash right|\; \backslash leq\; s\; (\backslash log\; \backslash frac\{N\}\{n\})^\{-c\}.\; \backslash quad\; (6.11)$$

Furthermore, for every $x'_1, \dots, x'_t \in \Lambda_n$ and $y'_1, \dots, y'_t \notin \Lambda_N$, we have that

$$\mid {\mathbf{P}}_{\beta }^{x_1{y}_1,\dots ,x_t{y}_t,\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[E]-{\mathbf{P}}_{\beta }^{x_1^{\mathrm{\prime }}{y}_1^{\mathrm{\prime }},\dots ,x_t^{\mathrm{\prime }}{y}_t^{\mathrm{\prime }},\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[E]\mid \le s(\log \frac{N}{n}{)}^{-c},(6.12)\backslash left|\backslash mathbf\{P\}\_\{\backslash beta\}^\{x\_1\; y\_1,\; \backslash dots,\; x\_t\; y\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [E]\; -\; \backslash mathbf\{P\}\_\{\backslash beta\}^\{x\text{'}\_1\; y\text{'}\_1,\; \backslash dots,\; x\text{'}\_t\; y\text{'}\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [E]\backslash right|\; \backslash leq\; s\; (\backslash log\; \backslash frac\{N\}\{n\})^\{-c\},\; \backslash quad\; (6.12)$$

$$\mid {\mathbf{P}}_{\beta }^{x_1{y}_1,\dots ,x_t{y}_t,\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[F]-{\mathbf{P}}_{\beta }^{x_1^{\mathrm{\prime }}{y}_1^{\mathrm{\prime }},\dots ,x_t^{\mathrm{\prime }}{y}_t^{\mathrm{\prime }},\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[F]\mid \le s(\log \frac{N}{n}{)}^{-c}\mathrm{.}(6.13)\backslash left|\backslash mathbf\{P\}\_\{\backslash beta\}^\{x\_1\; y\_1,\; \backslash ldots,\; x\_t\; y\_t,\; \backslash emptyset,\; \backslash ldots,\; \backslash emptyset\}\; [F]\; -\; \backslash mathbf\{P\}\_\{\backslash beta\}^\{x\text{'}\_1\; y\text{'}\_1,\; \backslash ldots,\; x\text{'}\_t\; y\text{'}\_t,\; \backslash emptyset,\; \backslash ldots,\; \backslash emptyset\}\; [F]\backslash right|\; \backslash leq\; s\; (\backslash log\; \backslash frac\{N\}\{n\})^\{-c\}.\; \backslash quad\; (6.13)$$

We postpone the proof to Section 6.2 below. Before showing how Theorem 6.4 is used in the proof of the improved tree diagram bound, let us make an interlude and comment on this statement.

Discussion The relation (6.11) is an assertion of approximate independence between events at far distances, and (6.12)–(6.13) expresses a degree of independence of the probability of an event from the precise placement of the sources when these are far from the event in question. This result should be of interest on its own, and possibly have other applications, since mixing properties efficiently replace independence in statistical mechanics.

The main difficulty of the theorem concerns currents with a source inside $\Lambda_n$ and a source outside $\Lambda_N$ (i.e. the first $t$ ones). In this case, the currents are constrained to have a path linking the two, and that may be a conduit for information, and correlation, between $\Lambda_n$ and the exterior of $\Lambda_N$. To appreciate the point it may be of help to compare the situation with Bernoulli percolation: there the mixing property without sources is a triviality (by the variables’ independence); while an analogue of the mixing property with sources $x$ and $y$ would concern Bernoulli percolation conditioned on having a path from $x$ to $y$. Proving convergence at criticality, for $x$ set as the origin and $y$ tending to infinity, of these conditioned measures is a notoriously hard problem. It would in particular imply the existence of the so-called Incipient Infinite Cluster (IIC), and the definition of the IIC was justified in 2D [32] and in high dimension [49], but it is still open in dimensions $3 \leq d \leq 10$. When the number of sources is even inside $\Lambda_n$, things become much simpler and one may in fact prove a quantitative ratio weak mixing using mixing properties for (sub)-critical random-cluster measures with cluster-weight 2 provided by [4]. ---PAGE_BREAK---

Theorem 6.4 has an extension to three dimensions using [4], but there it becomes non-quantitative (the careful reader will notice that the condition $d > 3$ is coming from the exponent appearing in the proof of (6.36) in Lemma 6.7 in the next section). More precisely, one may prove that in dimension $d = 3$, for every $n, s$ and $\varepsilon$, there exists a constant $N$ sufficiently large that the previous theorem holds with an error $\varepsilon$ instead of $s(\log \frac{N}{n})^{-c}$. This has a particularly interesting application: one may construct the IIC in dimension $d = 3$ for this model, since the random-cluster model with cluster weight $q = 2$ conditioned on having a path from $x$ to $y$ can be obtained as the random current model with sources $x$ and $y$ together with an additional independent sprinkling (see [5]). This represents a non-trivial result for critical 3D Ising. More generally, we believe that the previous mixing result may be a key tool in the rigorous description of the critical behaviour of the Ising model in three dimensions.

This concludes the interlude, and we return now to the proof of the intersection-clustering bound.

Proof of Proposition 6.1 We follow the same argument as in the proof of the conditional version (Proposition 4.3) and borrow the notation from the corresponding proof at the end of Section 4.2. We fix $\alpha > 2$ large enough that the mixing property Theorem 6.4 holds true. Using Lemma 6.3, we may choose $D = D(\alpha)$ such that $\ell_{k+1} \ge \ell_k^\alpha$.

The proof is exactly identical to the proof of Proposition 4.3, with the exception of the bound on $\mathbf{P}^{0x,0z,\emptyset,\emptyset}[A_S]$ and the fact that we restrict ourselves to subsets $S$ of even integers in ${1, \dots, K-3}$. In order to obtain this result, first observe that since we assumed $\ell_K \le \xi(\beta)$, by Theorem 5.12 there exists $y \in \text{Ann}(\ell_{K-1}, \ell_K)$ in a regular scale. Since the event $A_S$ depends on the currents inside $\Lambda_{\ell_{K-2}}$ (since $S$ does not contain integers strictly larger than $K-3$), and that $\ell_{K-1} \ge \ell_{K-2}^\alpha$, the mixing property (Theorem 6.4) shows that

$${\mathbf{P}}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[A_S]\le {\mathbf{P}}^{0y,0y,\mathrm{\varnothing },\mathrm{\varnothing }}[A_S]+\frac{C}{\sqrt{\log {\mathrm{\ell }}_{K-1}}}\le {\mathbf{P}}^{0y,0y,\mathrm{\varnothing },\mathrm{\varnothing }}[A_S]+2^{-\delta K}\mathrm{.}(6.14)\backslash mathbf\{P\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[A\_S]\; \backslash le\; \backslash mathbf\{P\}^\{0y,0y,\backslash emptyset,\backslash emptyset\}[A\_S]\; +\; \backslash frac\{C\}\{\backslash sqrt\{\backslash log\; \backslash ell\_\{K-1\}\}\}\; \backslash le\; \backslash mathbf\{P\}^\{0y,0y,\backslash emptyset,\backslash emptyset\}[A\_S]\; +\; 2^\{-\backslash delta\; K\}.\; \backslash quad\; (6.14)$$

To derive the first bound on the right-hand side, we apply the mixing property repeatedly (Theorem 6.4) and the intersection property (Lemma 6.2) exactly as in the conditional proof. For the second inequality, we lower bound $\ell_{K-1}$ using $B_{\ell_{K-1}}(\beta) \ge D^{K-1}$ and the Infrared Bound (5.37). $\square$

6.2 The mixing property: proof of Theorem 6.4

As we saw, the mixing property is in the core of the proof of our main result. The strategy of the proof was explained in Section 4.2 when we proved mixing for one current under the power-law assumption. In this section we again define a random variable $N$ which is approximately 1 and is a weighted sum over ($t$-tuple of) vertices connected to the origin. The main difficulty will come from the fact that since we do not fully control the spin-spin correlations, we will need to define $N$ in a smarter fashion. Also, whereas in Section 4.2 we treated the case of a single current ($s=1$), here we generalize to multiple currents.

Fix $\beta \le \beta_c$ and drop it from the notation. Also fix $s \ge t \ge 1$ and $n^\alpha \le N \le \xi(\beta)$. Below, constants $c_i$ and $C_i$ are independent of the choices of $s,t,\beta,n,N$ satisfying the properties above. We introduce the integers $m$ and $M$ such that $m/n = (N/n)^{1/3}$ and $N/M = (N/n)^{1/3}$ (we omit the details of the rounding operation).

For $\mathbf{x} = (x_1, \dots, x_t)$ and $\mathbf{y} = (y_1, \dots, y_t)$, we will use the following shortcut notation

$${\mathbf{P}}^{\mathbf{x}\mathbf{y}}:={\mathbf{P}}^{x_1{y}_1,\dots ,x_t{y}_t,\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}\text{and}{\mathbf{P}}^{\mathbf{x}\mathbf{y}}\otimes {\mathbf{P}}^{\mathrm{\varnothing }},(6.15)\backslash mathbf\{P\}^\{\backslash mathbf\{x\}\backslash mathbf\{y\}\}\; :=\; \backslash mathbf\{P\}^\{x\_1\; y\_1,\; \backslash dots,\; x\_t\; y\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash mathbf\{P\}^\{\backslash mathbf\{x\}\backslash mathbf\{y\}\}\; \backslash otimes\; \backslash mathbf\{P\}^\{\backslash emptyset\},\; \backslash quad\; (6.15)$$ ---PAGE_BREAK---

where the second measure is the law of the random variable $(\mathbf{n}_1, \dots, \mathbf{n}_s, \mathbf{n}'_1, \dots, \mathbf{n}'_s)$, where $(\mathbf{n}'_1, \dots, \mathbf{n}'_s)$ is an independent family of sourceless currents.

To define $\mathbf{N}$, first introduce for every vertex $y \notin \Lambda_{2dm}$, the set (see Fig. 5)

$$A_y(m):=\{u\in \text{Ann}(m,2m):\mathrm{\forall }x\in {\mathrm{\Lambda }}_{m\mathrm{/}d},⟨{\sigma }_x{\sigma }_y⟩\le (1+\frac{C\mathrm{\mid }x-u\mathrm{\mid }}{\mathrm{\mid }y\mathrm{\mid }})⟨{\sigma }_u{\sigma }_y⟩\},(6.16)A\_y(m)\; :=\; \backslash \{u\; \backslash in\; \backslash text\{Ann\}(m,\; 2m)\; :\; \backslash forall\; x\; \backslash in\; \backslash Lambda\_\{m/d\},\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash rangle\; \backslash le\; \backslash left(1\; +\; \backslash frac\{C|x-u|\}\{|y|\}\backslash right)\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_y\; \backslash rangle\backslash \},\; \backslash quad\; (6.16)$$

where $C$ is given by the definition of good scales.

Remark 6.5 When $y$ is in a regular scale, then $A_y(m)$ is equal to $\text{Ann}(m, 2m)$ by Prop- erty P2 of regular scales. The reason why we consider $A_y(m)$ instead of the full annulus $\text{Ann}(m, 2m)$ is technical: since $y$ will not a priori be assumed to belong to a regular scale (in fact $|y|$ may be much larger than $\xi(\beta)$ when $\beta < \beta_c$), we will use (for (6.26) and (6.37) below) the inequality between $\langle \sigma_x \sigma_y \rangle$ and $\langle \sigma_u \sigma_y \rangle$ in several bounds. Now, if $y_1 = |y|$, then

$$A_y(m)\supset \{z\in {\mathbb{Z}}^d:m\le z_1\le 2m\text{ and }0\le z_j\le m\mathrm{/}d\text{ for }j>1\}(6.17)A\_y(m)\; \backslash supset\; \backslash \{z\; \backslash in\; \backslash mathbb\{Z\}^d\; :\; m\; \backslash le\; z\_1\; \backslash le\; 2m\; \backslash text\{\; and\; \}\; 0\; \backslash le\; z\_j\; \backslash le\; m/d\; \backslash text\{\; for\; \}\; j\; >\; 1\backslash \}\; \backslash quad\; (6.17)$$

as the Messager-Miracle-Sole inequality implies⁶ that $\langle \sigma_z \sigma_y \rangle \ge \langle \sigma_x \sigma_y \rangle$ for every $x \in \Lambda_{m/d}$.

From now on, fix a set $\mathcal{H}$ of regular scales $k$ with $m \le 2^k \le M/2$ satisfying that distinct $k, k' \in \mathcal{H}$ are differing by a multiplicative factor at least $C$ (where the constant $C$ is given by Theorem 5.12). We further assume that $|\mathcal{H}| \ge c_1 \log(N/n)$, where $c_1$ is sufficiently small. The existence of $\mathcal{H}$ is guaranteed by the definition of $m$ and $M$ and the abundance of regular scales given by Theorem 5.12.

Define $\mathbf{N} := \prod_{i=1}^{t} \mathbf{N}_i$, where

$${\mathbf{N}}_i:=\frac{1}{\mathrm{\mid }\mathcal{H}\mathrm{\mid }}\sum _{k\in \mathcal{K}}\frac{1}{A_{x_i,y_i}(2^k)}\sum _{u\in A_{y_i}(2^k)}\mathbb{I}[u\stackrel{{\mathbf{n}}_i+{\mathbf{n}}_i^{\mathrm{\prime }}}{\leftarrow }{x}_i],(6.20)\backslash mathbf\{N\}\_i\; :=\; \backslash frac\{1\}\{|\backslash mathcal\{H\}|\}\; \backslash sum\_\{k\; \backslash in\; \backslash mathcal\{K\}\}\; \backslash frac\{1\}\{A\_\{x\_i,\; y\_i\}(2^k)\}\; \backslash sum\_\{u\; \backslash in\; A\_\{y\_i\}(2^k)\}\; \backslash mathbb\{I\}[u\; \backslash xleftarrow\{\backslash mathbf\{n\}\_i+\backslash mathbf\{n\}\text{'}\_i\}\; x\_i],\; \backslash quad\; (6.20)$$

where $a_{x,y}(u) := \langle\sigma_x\sigma_u\rangle\langle\sigma_u\sigma_y\rangle/\langle\sigma_x\sigma_y\rangle$ and $A_{x,y}(m) := \sum_{u\in A_{y}(m)} a_{x,y}(u)$. The first step of the proof is the following concentration inequality.

Proposition 6.6 (Concentration of N) For every $\alpha > 2$, there exists $C_0 = C_0(\alpha, t) > 0$ such that for every $n$ large enough and $n^\alpha \le N \le \xi(\beta)$,

$${\mathbf{E}}^{\mathbf{x},\mathbf{y},\mathrm{\varnothing }[(\mathbf{N}-1{)}^2]}\le \frac{C_0}{\log (N\mathrm{/}n)}\mathrm{.}(6.21)\backslash mathbf\{E\}^\{\backslash mathbf\{x\},\backslash mathbf\{y\},\backslash emptyset[(\backslash mathbf\{N\}-1)^2]\}\; \backslash leq\; \backslash frac\{C\_0\}\{\backslash log(N/n)\}.\; \backslash quad\; (6.21)$$

Proof We shall apply the telescopic formula

$$\mathbf{N}-1=\prod _{i=1}^t{\mathbf{N}}_i-1=\sum _{i=1}^t({\mathbf{N}}_i-1)\prod _{j>i}{\mathbf{N}}_j\backslash mathbf\{N\}-1\; =\; \backslash prod\_\{i=1\}^\{t\}\; \backslash mathbf\{N\}\_i\; -\; 1\; =\; \backslash sum\_\{i=1\}^\{t\}\; (\backslash mathbf\{N\}\_i\; -\; 1)\; \backslash prod\_\{j>i\}\; \backslash mathbf\{N\}\_j$$

⁶The claim follows directly from the inequality $S_\beta(y) \le S_\beta(x)$ for every $x,y$ such that $x_1 \ge 0$ and $y_1 \ge x_1 + \sum_{j>1} |y_j - x_j|$. In order to prove this inequality, define, for $0 \le i \le d$,

$$\begin{array}{cccc}& v^{(i)}:=(x_1+\sum _{j=i}^d\mathrm{\mid }{y}_j-x_j\mathrm{\mid },x_2,\dots ,x_i,y_{i+1},\dots ,y_d)\mathrm{.}& & \text{(6.18)}\end{array}v^\{(i)\}\; :=\; (x\_1\; +\; \backslash sum\_\{j=i\}^\{d\}\; |y\_j\; -\; x\_j|,\; x\_2,\; \backslash dots,\; x\_i,\; y\_\{i+1\},\; \backslash dots,\; y\_d).\; \backslash tag\{6.18\}$$

Successive applications of the Messager-Miracle-Sole inequality with respect to the sum or the difference (depending on whether $x_i$ is positive or negative) of the first and $i$-th coordinates implies that

$$S_{\beta }(y)\le S_{\beta }(v^{(1)})\le S_{\beta }(v^{(2)})\le \cdots \le S_{\beta }(v^{(d)})=S_{\beta }(x)\mathrm{.}(6.19)S\_\{\backslash beta\}(y)\; \backslash le\; S\_\{\backslash beta\}(v^\{(1)\})\; \backslash le\; S\_\{\backslash beta\}(v^\{(2)\})\; \backslash le\; \backslash cdots\; \backslash le\; S\_\{\backslash beta\}(v^\{(d)\})\; =\; S\_\{\backslash beta\}(x).\; \backslash quad\; (6.19)$$ ---PAGE_BREAK---

with the last product interpreted as 1 for $i = t$. Hence, by the Cauchy-Schwarz inequality and the currents' independence,

$${\mathbf{E}}^{xy,\mathrm{\varnothing }}[(\mathbf{N}-1{)}^2]\le t\sum _{i=1}^t{\mathbf{E}}^{xy,\mathrm{\varnothing }}[({\mathbf{N}}_i-1{)}^2]\prod _{j>i}{\mathbf{E}}^{xy,\mathrm{\varnothing }}[{\mathbf{N}}_j^2]\mathrm{.}(6.22)\backslash mathbf\{E\}^\{xy,\backslash emptyset\}[(\backslash mathbf\{N\}-1)^2]\; \backslash leq\; t\; \backslash sum\_\{i=1\}^\{t\}\; \backslash mathbf\{E\}^\{xy,\backslash emptyset\}[(\backslash mathbf\{N\}\_i-1)^2]\; \backslash prod\_\{j>i\}\; \backslash mathbf\{E\}^\{xy,\backslash emptyset\}[\backslash mathbf\{N\}\_j^2].\; \backslash quad\; (6.22)$$

It therefore suffices to show that there exists a constant $C_1 > 0$ such that for every $i \le t$,

$${\mathbf{E}}^{xy,\mathrm{\varnothing }}[({\mathbf{N}}_i-1{)}^2]\le \frac{C_1}{\log (N\mathrm{/}n)}\mathrm{.}(6.23)\backslash mathbf\{E\}^\{xy,\backslash emptyset\}[(\backslash mathbf\{N\}\_i\; -\; 1)^2]\; \backslash le\; \backslash frac\{C\_1\}\{\backslash log(N/n)\}.\; \backslash quad\; (6.23)$$

To lighten the notation, and since the random variable $\mathbf{N}_i$ depends only on $\mathbf{n}_i$ and $\mathbf{n}'_i$, we omit the index in $x_i$, $y_i$, $\mathbf{n}_i$, $\mathbf{n}'_i$ and write instead just $x, y, \mathbf{n}, \mathbf{n}'$. We keep the index in $\mathbf{N}_i$ to avoid confusion with $\mathbf{N}$ which is the product of these random variables.

The proof of (6.23) is also based on a computation of the first and second moments of $\mathbf{N}_i$. For the first moment, the switching lemma and the definition of $\mathbf{N}_i$ imply that $\mathbf{E}^{xy,\emptyset}[\mathbf{N}_i] = 1$. From the lower bound on $|\mathcal{K}|$, to bound the second moment it therefore suffices to show that

$${\mathbf{E}}^{xy,\mathrm{\varnothing }}[{\mathbf{N}}_i^2]\le 1+\frac{C_2}{\mathrm{\mid }\mathcal{K}\mathrm{\mid }},(6.24)\backslash mathbf\{E\}^\{xy,\backslash emptyset\}[\backslash mathbf\{N\}\_i^2]\; \backslash le\; 1\; +\; \backslash frac\{C\_2\}\{|\backslash mathcal\{K\}|\},\; \backslash quad\; (6.24)$$

which follows from the inequality, for every $\ell \ge k$ in $\mathcal{K}$,

$$\sum _{\begin{array}{c}u\in A_y(2^k)\\ v\in A_y(2^{\mathrm{\ell }})\end{array}}{\mathbf{P}}^{xy,\mathrm{\varnothing }}[u,vx]\le A_{x,y}(2^{\mathrm{\ell }})A_{x,y}(2^{\mathrm{\ell }})(1+C_3{2}^{-(\mathrm{\ell }-k)})\mathrm{.}(6.25)\backslash sum\_\{\backslash substack\{u\; \backslash in\; A\_y(2^k)\; \backslash \backslash \; v\; \backslash in\; A\_y(2^\backslash ell)\}\}\; \backslash mathbf\{P\}^\{xy,\backslash emptyset\}[u,\; v\; \backslash underset\{\backslash ell\}\{\backslash overset\{\backslash ell\}\{n+n\text{'}\}\}\; x]\; \backslash le\; A\_\{x,y\}(2^\backslash ell)\; A\_\{x,y\}(2^\backslash ell)(1+C\_3\; 2^\{-(\backslash ell-k)\}).\; \backslash quad\; (6.25)$$

Case $\ell > k$. We find by (A.11) that

$${\mathbf{P}}^{xy,\mathrm{\varnothing }}[u,vx]\le a_{x,y}(u)a_{x,y}(v)(\frac{⟨{\sigma }_x{\sigma }_y⟩⟨{\sigma }_u{\sigma }_v⟩}{⟨{\sigma }_u{\sigma }_y⟩⟨{\sigma }_x{\sigma }_v⟩}+\frac{⟨{\sigma }_x{\sigma }_y⟩⟨{\sigma }_u{\sigma }_v⟩}{⟨{\sigma }_v{\sigma }_y⟩⟨{\sigma }_x{\sigma }_u⟩})\mathrm{.}(6.26)\backslash mathbf\{P\}^\{xy,\backslash emptyset\}[u,v\; \backslash stackrel\{n+n\text{'}\}\{\backslash rightleftarrows\}\; x]\; \backslash le\; a\_\{x,y\}(u)a\_\{x,y\}(v)\; \backslash left(\; \backslash frac\{\backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_v\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_u\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_v\; \backslash rangle\}\; +\; \backslash frac\{\backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_v\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_v\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_u\; \backslash rangle\}\; \backslash right).\; \backslash quad\; (6.26)$$

Now, since $u \in A_y(2^\ell)$, $\langle \sigma_x \sigma_y \rangle \le (1+C^{\frac{|u-x|}{|y|}})\langle \sigma_u \sigma_y \rangle$. Furthermore, since $\ell$ is a regular scale, Property P2 of regular scales implies that $\langle \sigma_u \sigma_v \rangle \le (1+C^{\frac{|u-x|}{|v|}})\langle \sigma_x \sigma_v \rangle$. We deduce that

$$\frac{⟨{\sigma }_x{\sigma }_y⟩⟨{\sigma }_u{\sigma }_v⟩}{⟨{\sigma }_u{\sigma }_y⟩⟨{\sigma }_x{\sigma }_v⟩}\le 1+C_0{2}^{-(\mathrm{\ell }-k)}\mathrm{.}(6.27)\backslash frac\{\backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_v\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_u\; \backslash sigma\_y\; \backslash rangle\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_v\; \backslash rangle\}\; \backslash le\; 1\; +\; C\_0\; 2^\{-(\backslash ell-k)\}.\; \backslash quad\; (6.27)$$

Similarly, since $v \in A_y(2^\ell)$, $\langle \sigma_x \sigma_y \rangle \le (1+C^{\frac{|v-x|}{|y|}})\langle \sigma_v \sigma_y \rangle$. Property P3 for the $\ell-k$ regular scales in $\mathcal{K}$ between $k$ and $\ell$ implies that

$$\frac{⟨{\sigma }_x{\sigma }_y⟩⟨{\sigma }_u{\sigma }_v⟩}{⟨{\sigma }_v{\sigma }_y⟩⟨{\sigma }_x{\sigma }_u⟩}\le C_1{2}^{-(\mathrm{\ell }-k)}\mathrm{.}(6.28)\backslash frac\{\backslash langle\backslash sigma\_x\backslash sigma\_y\backslash rangle\backslash langle\backslash sigma\_u\backslash sigma\_v\backslash rangle\}\{\backslash langle\backslash sigma\_v\backslash sigma\_y\backslash rangle\backslash langle\backslash sigma\_x\backslash sigma\_u\backslash rangle\}\; \backslash le\; C\_1\; 2^\{-(\backslash ell-k)\}.\; \backslash quad\; (6.28)$$

Plugging (6.27)-(6.28) into (6.26) and summing over $u \in A_y(2^k)$ and $v \in A_y(2^\ell)$ gives (6.25). ---PAGE_BREAK---

Case $\ell = k$. Assume that $\langle \sigma_u \sigma_y \rangle \le \langle \sigma_v \sigma_y \rangle$. Use (A.11) to write

$${\mathbf{P}}^{xy,\mathrm{\varnothing }}[u,vx]\le ⟨{\sigma }_v{\sigma }_u⟩(\frac{⟨{\sigma }_x{\sigma }_u⟩}{⟨{\sigma }_x{\sigma }_v⟩}+\frac{⟨{\sigma }_u{\sigma }_y⟩}{⟨{\sigma }_v{\sigma }_y⟩})a_{x,y}(v)\mathrm{.}(6.29)\backslash mathbf\{P\}^\{xy,\backslash emptyset\}[u,\; v\; \backslash underset\{n\; \backslash to\; n\text{'}\}\{\backslash stackrel\{n+n\text{'}\}\{\backslash rightleftarrows\}\}\; x]\; \backslash leq\; \backslash langle\; \backslash sigma\_v\; \backslash sigma\_u\; \backslash rangle\; \backslash left(\; \backslash frac\{\backslash langle\; \backslash sigma\_x\; \backslash sigma\_u\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_x\; \backslash sigma\_v\; \backslash rangle\}\; +\; \backslash frac\{\backslash langle\; \backslash sigma\_u\; \backslash sigma\_y\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_v\; \backslash sigma\_y\; \backslash rangle\}\; \backslash right)\; a\_\{x,y\}(v).\; \backslash quad\; (6.29)$$

By Property P1 of regular scales, the first term under parenthesis is bounded by a constant. The second one is bounded by 1 by assumption. Now, for each $v \in A_y(2^k)$,

where the first inequality is trivial, the second one is true by Property P4, the third by Remark 6.5 (when $y$ is regular then it is a direct consequence of P4, and when it is not one can use (6.17) and the Messager-Miracle-Sole inequality), and the fourth inequality follows from Property P1 of regular scales (to replace $\langle \sigma_0 \sigma_u \rangle$ by $\langle \sigma_x \sigma_u \rangle$) and the fact that since $u \in A_y(2^k)$, $\langle \sigma_x \sigma_y \rangle \le (1+C|u-x||y|)\langle \sigma_u \sigma_y \rangle \le C_5\langle \sigma_u \sigma_y \rangle$.

We deduce that

$$\sum _{u,v\in A_y(2^k)}{\mathbf{P}}^{xy,\mathrm{\varnothing }}[u,v\stackrel{n_1+n_2}{\to }x]\le 2-\sum _{\begin{array}{c}u,v\in A_y(2^k)\\ ⟨{\sigma }_u{\sigma }_y⟩\le ⟨{\sigma }_v{\sigma }_y⟩\end{array}}{\mathbf{P}}^{xy,\mathrm{\varnothing }}[u,v\stackrel{n_1+n_2}{\to }x]\le C_6{A}_{x,y}(2^k{)}^2\mathrm{.}(6.30)\backslash sum\_\{u,v\; \backslash in\; A\_y(2^k)\}\; \backslash mathbf\{P\}^\{xy,\backslash emptyset\}[u,\; v\; \backslash xrightarrow\{n\_1+n\_2\}\; x]\; \backslash le\; 2\; -\; \backslash sum\_\{\backslash substack\{u,v\; \backslash in\; A\_y(2^k)\; \backslash \backslash \; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_y\; \backslash rangle\; \backslash le\; \backslash langle\; \backslash sigma\_v\; \backslash sigma\_y\; \backslash rangle\}\}\; \backslash mathbf\{P\}^\{xy,\backslash emptyset\}[u,\; v\; \backslash xrightarrow\{n\_1+n\_2\}\; x]\; \backslash le\; C\_6\; A\_\{x,y\}(2^k)^2.\; \backslash quad\; (6.30)$$

For a proof of Theorem 6.4 we fix $\alpha > 2$ (which will be taken large enough later). Applying the Cauchy-Schwarz inequality gives

$$\mathrm{\mid }{\mathbf{P}}^{xy}[E\cap F]-{\mathbf{E}}^{xy,\mathrm{\varnothing }}[{\mathbf{N}}_{\mathbb{I}((n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_s)\in E\cap F)}]\mathrm{\mid }\le \sqrt{{\mathbf{E}}^{xy,\mathrm{\varnothing }}[(\mathbf{N}-1{)}^2]}\le \frac{C_1}{\sqrt{\log (N\mathrm{/}n)}}\mathrm{.}(6.31)|\backslash mathbf\{P\}^\{xy\}[E\; \backslash cap\; F]\; -\; \backslash mathbf\{E\}^\{xy,\backslash emptyset\}[\backslash mathbf\{N\}\_\{\backslash mathbb\{I\}((n\_1,\; ...,\; n\_s)\; \backslash in\; E\; \backslash cap\; F)\}]|\; \backslash leq\; \backslash sqrt\{\backslash mathbf\{E\}^\{xy,\backslash emptyset\}[(\backslash mathbf\{N\}\; -\; 1)^2]\}\; \backslash leq\; \backslash frac\{C\_1\}\{\backslash sqrt\{\backslash log(N/n)\}\}.\; \backslash quad\; (6.31)$$

Now, for u = (u₁, ..., u_t) with uᵢ ∈ Ann(m, M) for every i, let G(u₁, ..., u_t) be the event that for every i ≤ s, there exists kᵢ ≤ nᵢ + n′ᵢ such that kᵢ = 0 on Λₙ, kᵢ = nᵢ + n′ᵢ outside Λₙ, and ∂kᵢ is equal to {uᵢ, yᵢ} if i ≤ t and Ø if t < i ≤ s. The switching principle implies as in Section 5.3 that

Also, as before, we have the trivial identity

$${\mathbf{P}}^{\mathbf{x}\mathbf{u},\mathbf{u}\mathbf{y}}[({\mathbf{n}}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathbf{n}}_s)\in E,({\mathbf{n}}_1^{\mathrm{\prime }},\mathrm{.}\mathrm{.}\mathrm{.},{\mathbf{n}}_s^{\mathrm{\prime }})\in F]={\mathbf{P}}^{\mathbf{x}\mathbf{u}}[E]{\mathbf{P}}^{\mathbf{u}\mathbf{y}}[F]\mathrm{.}(6.33)\backslash mathbf\{P\}^\{\backslash mathbf\{xu\},\backslash mathbf\{uy\}\}[(\backslash mathbf\{n\}\_1,\; ...,\; \backslash mathbf\{n\}\_s)\; \backslash in\; E,\; (\backslash mathbf\{n\}\text{'}\_1,\; ...,\; \backslash mathbf\{n\}\text{'}\_s)\; \backslash in\; F]\; =\; \backslash mathbf\{P\}^\{\backslash mathbf\{xu\}\}[E]\; \backslash mathbf\{P\}^\{\backslash mathbf\{uy\}\}[F].\; \backslash quad\; (6.33)$$

We now pause the argument to establish the following lemma.

Lemma 6.7 For $d \ge 4$, there exist $\epsilon > 0$ and $\alpha_0 = \alpha_0(\epsilon) > 0$ large enough such that for every $n^{\alpha_0} \le N \le \xi(\beta)$ and for every $\mathbf{u}$ with $u_i \in A_{y_i}(2^{k_i})$ for some $k_i$ with $m \le 2^{k_i} \le M/2$ for every $1 \le i \le t$,

$$\begin{array}{cccc}& {(\prod _{i=1}^t{a}_{x_i,y_i}(u_i))}^{-1}{P}^{\mathbf{x}y,\mathrm{\varnothing }}[u_i\underset{n_i+n_i^{\mathrm{\prime }}}{\overset{n_i+n_i^{\mathrm{\prime }}}{\to }}{x}_i,\mathrm{\forall }i\le t,G(u_1,\dots ,u_t{)}^c]=P^{\mathbf{x}u,\mathbf{u}\mathbf{y}}[G(u_1,\dots ,u_t{)}^c]\le s{\left(\frac{n}{N}\right)}^{ϵ}\mathrm{.}& & \text{(6.34)}\end{array}\backslash left(\backslash prod\_\{i=1\}^\{t\}\; a\_\{x\_i,\; y\_i\}(u\_i)\backslash right)^\{-1\}\; P^\{\backslash mathbf\{x\}y,\backslash emptyset\}\backslash left[u\_i\backslash xrightarrow[n\_i+n\text{'}\_i]\{n\_i+n\text{'}\_i\}\; x\_i,\; \backslash forall\; i\backslash le\; t,\; G(u\_1,\backslash ldots,u\_t)^c\backslash right]\; =\; P^\{\backslash mathbf\{x\}u,\backslash mathbf\{uy\}\}[G(u\_1,\backslash ldots,u\_t)^c]\; \backslash \backslash \; \backslash le\; s\backslash left(\backslash frac\{n\}\{N\}\backslash right)^\{\backslash epsilon\}.\; \backslash tag\{6.34\}$$ ---PAGE_BREAK---

Figure 5: The currents $\mathbf{n}_i$ (red) and $\mathbf{n}'_i$ (blue). Since the sources of $\mathbf{n}_i$, i.e. $x_i$ and $u_i$, are both in $\Lambda_M$, a reasoning similar to the proof of uniqueness in the intersection property (first control the backbone, proving that it does not cross the annulus $\text{Ann}(M, R)$, and then the remaining sourceless current) enables us to conclude that the probability that the current contains a crossing of $\text{Ann}(M, N)$ is small. Similarly, since the sources $u_i$ and $y_i$ of $\mathbf{n}'_i$ lie both outside of $\Lambda_m$, we can prove that the probability that $\mathbf{n}'_i$ crosses $\text{Ann}(n, m)$ is small. An extra care is needed for establishing the latter since $y$ is not assumed to be regular. To circumvent this problem, we consider only intersection sites $u_i$ in one of the boxes $\Lambda_k(y_i)$, which are depicted here in gray.

Proof Fix $\varepsilon > 0$ sufficiently small (we will see below how small it should be). The first identity follows from the switching lemma so we focus on the second one. Let $G_i$ be the event that the current k_i exists. This event clearly contains (see Fig. 5) the event that $\text{Ann}(M, N)$ is not crossed by a cluster in n_i, and $\text{Ann}(n, m)$ is not crossed by a cluster in n'_i, since in such case k_i can be defined as the sum of n_i restricted to the clusters intersecting $\Lambda_N^c$ (this current has no sources) and n'_i restricted to the clusters intersecting $\Lambda_m^c$ (this current has sources $u_i$ and $y_i$). We focus on the probability of this event for $i \le t$, the case $t < i \le s$ being even simpler since there are no sources.

We bound the probability of n_i crossing Ann(M, N) by splitting Ann(M, N) in two annuli Ann(M, R) and Ann(R, N) with R = √MN, then estimating the probability that the backbone of n_i crosses the inner annulus, and then the probability that the remaining current crosses the outer annulus. More precisely, the chain rule for backbones [3] gives that for α₀ = α₀(ε) > 0 large enough and N ≥ n^α₀, $${\mathbf{P}}^{\text{xy}}[\mathrm{\Gamma }({\mathbf{n}}_i)\text{ crosses Ann}(M,R)]\le \sum _{v\in \mathrm{\partial }{\mathrm{\Lambda }}_R}\frac{⟨{\sigma }_{x_i}{\sigma }_v⟩⟨{\sigma }_v{\sigma }_{u_i}⟩}{⟨{\sigma }_{x_i}{\sigma }_{u_i}⟩}\le C_2{R}^3\frac{M^3}{R^4}\le (n\mathrm{/}N{)}^{\epsilon },(6.35)\backslash mathbf\{P\}^\{\backslash text\{xy\}\}[\backslash Gamma(\backslash mathbf\{n\}\_i)\; \backslash text\{\; crosses\; \}\; \backslash text\{Ann\}(M,\; R)]\; \backslash le\; \backslash sum\_\{v\; \backslash in\; \backslash partial\; \backslash Lambda\_R\}\; \backslash frac\{\backslash langle\; \backslash sigma\_\{x\_i\}\; \backslash sigma\_v\; \backslash rangle\; \backslash langle\; \backslash sigma\_v\; \backslash sigma\_\{u\_i\}\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_\{x\_i\}\; \backslash sigma\_\{u\_i\}\; \backslash rangle\}\; \backslash le\; C\_2\; R^3\; \backslash frac\{M^3\}\{R^4\}\; \backslash le\; (n/N)^\{\backslash varepsilon\},\; \backslash quad\; (6.35)$$ where the lower bound (5.39) to bound the denominator and the Infrared Bound (5.37) for the numerator. Then, observe that the remaining current n_i ∖ Γ(n_i) is sourceless. Adding an additional sourceless current and using the switching lemma and Griffiths inequality [22] (very much like in the bound (4.22) in the proof of Lemma 4.4) gives

$${\mathbf{P}}^{\text{xy}}[{\mathbf{n}}_i\setminus \mathrm{\Gamma }({\mathbf{n}}_i)\text{ crosses Ann}(R,N)]\le \sum _{\begin{array}{c}v\in \mathrm{\partial }{\mathrm{\Lambda }}_R\\ w\in \mathrm{\partial }{\mathrm{\Lambda }}_N\end{array}}⟨{\sigma }_v{\sigma }_w{⟩}^2\le C_3{R}^3{N}^3(R\mathrm{/}N{)}^4\le (n\mathrm{/}N{)}^{ϵ},(6.36)\backslash mathbf\{P\}^\{\backslash text\{xy\}\}[\backslash mathbf\{n\}\_i\; \backslash setminus\; \backslash Gamma(\backslash mathbf\{n\}\_i)\; \backslash text\{\; crosses\; \}\; \backslash text\{Ann\}(R,\; N)]\; \backslash leq\; \backslash sum\_\{\backslash substack\{v\; \backslash in\; \backslash partial\; \backslash Lambda\_R\; \backslash \backslash \; w\; \backslash in\; \backslash partial\; \backslash Lambda\_N\}\}\; \backslash langle\; \backslash sigma\_v\; \backslash sigma\_w\; \backslash rangle^2\; \backslash leq\; C\_3\; R^3\; N^3\; (R/N)^4\; \backslash leq\; (n/N)^\{\backslash epsilon\},\; \backslash quad\; (6.36)$$ where we used the Infrared-Bound 5.37, and in the last one the definition of R and the fact that N ≥ n^α₀ for α₀ large enough. ---PAGE_BREAK---

When dealing with the probability of $\mathbf{n}'_i$ crossing $\text{Ann}(n, m)$, fix $r = \sqrt{nm}$ and apply the same reasoning with the annuli $\text{Ann}(n, r)$ and $\text{Ann}(r, m)$. The equivalent of (6.36) is the same as before, but one must be more careful about the bound on the probability of the event dealing with the backbone:

$${\mathbf{P}}^{\mathbf{x}\mathbf{y}}[\mathrm{\Gamma }({\mathbf{n}}_i^{\mathrm{\prime }})\text{ crosses Ann}(r,m)]\le C_4\sum _{v\in \mathrm{\partial }{\mathrm{\Lambda }}_r}\frac{⟨{\sigma }_{u_i}{\sigma }_v⟩⟨{\sigma }_v{\sigma }_{y_i}⟩}{⟨{\sigma }_{u_i}{\sigma }_{y_i}⟩}\le C_5{r}^3\mathrm{/}{m}^2\le (n\mathrm{/}N{)}^{ϵ},(6.37)\backslash mathbf\{P\}^\{\backslash mathbf\{x\}\backslash mathbf\{y\}\}[\backslash Gamma(\backslash mathbf\{n\}\text{'}\_i)\; \backslash text\{\; crosses\; \}\; \backslash text\{Ann\}(r,m)]\; \backslash le\; C\_4\; \backslash sum\_\{v\; \backslash in\; \backslash partial\; \backslash Lambda\_r\}\; \backslash frac\{\backslash langle\; \backslash sigma\_\{u\_i\}\; \backslash sigma\_v\; \backslash rangle\; \backslash langle\; \backslash sigma\_v\; \backslash sigma\_\{y\_i\}\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_\{u\_i\}\; \backslash sigma\_\{y\_i\}\; \backslash rangle\}\; \backslash le\; C\_5\; r^3/m^2\; \backslash le\; (n/N)^\{\backslash epsilon\},\; \backslash quad\; (6.37)$$

where we used the Infrared Bound (5.37) and our assumption that $u_i$ belongs to one of the $A_k(y_i)$ (to show that $\langle \sigma_v \sigma_{y_i} \rangle \le C_4 \langle \sigma_{u_i} \sigma_{y_i} \rangle$).

$\Delta$

Invoking the above lemma we now return to the proof of Theorem 6.4. Introduce the coefficients $\delta(\mathbf{u}, \mathbf{x}, \mathbf{y})$ equal to

$$\delta (\mathbf{u},\mathbf{x},\mathbf{y}):=\prod _{i=1}^t\frac{a_{x_i,y_i}(u_i)}{\mathrm{\mid }\mathcal{H}\mathrm{\mid }{A}_{x_i,y_i}(2^{k_i})}(6.38)\backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\},\; \backslash mathbf\{y\})\; :=\; \backslash prod\_\{i=1\}^\{t\}\; \backslash frac\{a\_\{x\_i,\; y\_i\}(u\_i)\}\{|\backslash mathcal\{H\}|\; A\_\{x\_i,\; y\_i\}(2^\{k\_i\})\}\; \backslash qquad\; (6.38)$$

for u such that for every i ≤ t, uᵢ ∈ A_yᵢ(2^kᵢ) for some kᵢ, and equal to 0 for other u. Gathering (6.31)–(6.33) as well as Lemma 6.7, and observing that the sum on (u₁, ..., uₜ) of δ(u, x, y) is 1, we obtain that

$$\begin{array}{cccc}& \mathrm{\mid }{\mathbf{P}}^{\mathbf{x}\mathbf{y}}[E\cap F]-\sum _{\mathbf{u}}\delta (\mathbf{u},\mathbf{x},\mathbf{y}){\mathbf{P}}^{\mathbf{x}\mathbf{u}}[E]{\mathbf{P}}^{\mathbf{u}\mathbf{y}}[F]\mathrm{\mid }\le \frac{C_{5}s}{\sqrt{\log (N\mathrm{/}n)}}+2C_{6}s(n\mathrm{/}N{)}^{ϵ}\le \frac{C_{7}s}{\sqrt{\log (N\mathrm{/}n)}},& & \text{(6.39)}\end{array}|\; \backslash mathbf\{P\}^\{\backslash mathbf\{x\}\backslash mathbf\{y\}\}[E\; \backslash cap\; F]\; -\; \backslash sum\_\{\backslash mathbf\{u\}\}\; \backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\},\; \backslash mathbf\{y\})\; \backslash mathbf\{P\}^\{\backslash mathbf\{x\}\backslash mathbf\{u\}\}[E]\; \backslash mathbf\{P\}^\{\backslash mathbf\{u\}\backslash mathbf\{y\}\}[F]\; |\; \backslash leq\; \backslash frac\{C\_5\; s\}\{\backslash sqrt\{\backslash log(N/n)\}\}\; +\; 2C\_6\; s\; (n/N)^\{\backslash epsilon\}\; \backslash leq\; \backslash frac\{C\_7\; s\}\{\backslash sqrt\{\backslash log(N/n)\}\},\; \backslash tag\{6.39\}$$

provided that $N \ge n^{\alpha_0}$ where $\alpha_0$ is given by the previous lemma.

To conclude the proof is now a matter of elementary algebraic manipulations. We begin by proving (6.12) when all the $y_i, y'_i$ for $i \le t$ belong to regular scales (not necessarily the same ones). In this case, apply twice (once for y and once for y') the previous inequality for our event E and the event on the outside being the full event to find

|\mathbf{P}^{\mathbf{x}\mathbf{y}}[E] - \mathbf{P}^{\mathbf{x}\mathbf{y}}'[E]| \leq |\sum_{\mathbf{u}} (\delta(\mathbf{u}, \mathbf{x}, \mathbf{y}) - \delta(\mathbf{u}, \mathbf{x}, \mathbf{y}')) \mathbf{P}^{\mathbf{x}\mathbf{u}}[E]| + \frac{2C_7s}{\sqrt{\log(N/n)}}. \quad (6.40)

Since all the $y_i, y'i$ are in regular scales, Remark 6.5 implies that $A{y_i}(2^{k_i}) = A_{y'_i}(2^{k_i}) =$ $\text{Ann}(2^{k_i}, 2^{k_i+1})$. Furthermore, Property 2 of regular scales implies⁷ that

$$\mathrm{\mid }\delta (\mathbf{u},\mathbf{x},\mathbf{y})-\delta (\mathbf{u},\mathbf{x},{\mathbf{y}}^{\mathrm{\prime }})\mathrm{\mid }\le C_{8}s\frac{M}{N}\delta (\mathbf{u},\mathbf{x},\mathbf{y})\le C_{9}s(n\mathrm{/}N{)}^{1\mathrm{/}3}\delta (\mathbf{u},\mathbf{x},\mathbf{y})\mathrm{.}(6.41)|\backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\},\; \backslash mathbf\{y\})\; -\; \backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\},\; \backslash mathbf\{y\}\text{'})|\; \backslash le\; C\_8\; s\; \backslash frac\{M\}\{N\}\; \backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\},\; \backslash mathbf\{y\})\; \backslash le\; C\_9\; s\; (n/N)^\{1/3\}\; \backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\},\; \backslash mathbf\{y\}).\; \backslash quad\; (6.41)$$

Therefore, (6.12) follows readily (with a large constant $C_{10}$) in this case. The same argument works for the second identity (6.13) for every x, x' and y, noticing that for every regular u for which the coefficients are non-zero,

$$\mathrm{\mid }\delta (\mathbf{u},{\mathbf{x}}^{\mathrm{\prime }},\mathbf{y})-\delta (\mathbf{u},\mathbf{x},\mathbf{y})\mathrm{\mid }\le C_{10}s\frac{m}{n}\delta (\mathbf{u},\mathbf{x},\mathbf{y})\le C_{11}s(n\mathrm{/}N{)}^{1\mathrm{/}3}\delta (\mathbf{u},\mathbf{x},\mathbf{y})\mathrm{.}(6.42)|\backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\}\text{'},\; \backslash mathbf\{y\})\; -\; \backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\},\; \backslash mathbf\{y\})|\; \backslash le\; C\_\{10\}\; s\; \backslash frac\{m\}\{n\}\; \backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\},\; \backslash mathbf\{y\})\; \backslash le\; C\_\{11\}\; s\; (n/N)^\{1/3\}\; \backslash delta(\backslash mathbf\{u\},\; \backslash mathbf\{x\},\; \backslash mathbf\{y\}).\; \backslash quad\; (6.42)$$

⁷Note that in this case $\delta(\mathbf{u}, \mathbf{x}, \mathbf{y})$ and $\delta(\mathbf{u}, \mathbf{x}, \mathbf{y}')$ are both close to

---PAGE_BREAK---

Now, assume further that $N \ge n^{3\alpha_0}$. Consider $\mathbf{z} = (z_1, \dots, z_t)$ with $z_i$ in a regular scale for every $i \le t$ and $z_i \in \text{Ann}(m, M)$. We also pick $\mathbf{y}$ on which we do not assume anything. The fact that the $\delta(\mathbf{u}, \mathbf{x}, \mathbf{y})$ sum to 1 implies that

where in the second line we have used (6.40) for u and z for n and m which is justified since m/n = (N/n)¹/³ ≥ nª₀.

Finally, we repeat the reasoning above. For $N \ge n^{9\alpha_0}$, the proof of (6.11) is obtained by applying (6.39) for $E$, $F$, $N$ and $n$, the previous inequality for $E$, $m$ and $n$ (which is justified since $m/n = (N/n)^{1/3} \ge n^{3\alpha_0}$), and then again (6.39) for $F$, $N$ and $n$. This concludes the proof with $\alpha = 9\alpha_0$.

6.3 Proof of Proposition 1.4

As mentioned in the introduction, Newman [38] showed, using the model’s Lee-Yang property, that gaussianity of the limit is implied by the asymptotic vanishing of the scaling limit of $|U_4^\beta|$. A more quantitative proof is available through the inequality, valid for every $n \ge 2$ and derived using the switching lemma in [1, Proposition 12.1],

Combined with the improved tree diagram bound (1.24), this inequality has the following consequence: for a continuous function $f$ which vanishes outside $[-r, r]^4$, and for $\beta \le \beta_c$ and $L \le \xi(\beta)$,

$$\mathrm{\mid }⟨T_{f,L}(\sigma {)}^{2n}{⟩}_{\beta }-\frac{(2n)!}{2^{n}n!}⟨T_{f,L}(\sigma {)}^2{⟩}_{\beta }^n\mathrm{\mid }\le \frac{3}{2}(2n{)}^4\mathrm{\mid }⟨T_{f,L}(\sigma {)}^{2n-4}{⟩}_{\beta }\mathrm{\mid }f{\mathrm{\mid }}_{\mathrm{\infty }}^{4}S(L,r,\beta ),(6.45)|\backslash langle\; T\_\{f,L\}(\backslash sigma)^\{2n\}\; \backslash rangle\_\backslash beta\; -\; \backslash frac\{(2n)!\}\{2^n\; n!\}\; \backslash langle\; T\_\{f,L\}(\backslash sigma)^2\; \backslash rangle\_\backslash beta^n|\; \backslash le\; \backslash frac\{3\}\{2\}\; (2n)^4\; |\backslash langle\; T\_\{f,L\}(\backslash sigma)^\{2n-4\}\; \backslash rangle\_\backslash beta\; |f|\_\backslash infty^4\; S(L,r,\backslash beta),\; \backslash quad\; (6.45)$$

where

$$S(L,r,\beta ):=\sum _{\begin{array}{c}x\in {\mathbb{Z}}_L^d\\ x_1,\dots ,x_4\in {\mathrm{\Lambda }}_{r}L\end{array}}{2}^{\frac{n}{2}}\frac{⟨{\sigma }_x{\sigma }_{x_1}{⟩}_{\beta }⟨{\sigma }_x{\sigma }_{x_2}{⟩}_{\beta }⟨{\sigma }_x{\sigma }_{x_3}{⟩}_{\beta }⟨{\sigma }_x{\sigma }_{x_4}{⟩}_{\beta }}{\sum _L(\beta {)}^2\cdot B_L(x_1,\dots ,x_4)(\beta {)}^c},(6.46)S(L,\; r,\; \backslash beta)\; :=\; \backslash sum\_\{\backslash substack\{x\; \backslash in\; \backslash mathbb\{Z\}\_L^d\; \backslash \backslash \; x\_1,\; \backslash dots,\; x\_4\; \backslash in\; \backslash Lambda\_r\; L\}\}\; 2^\{\backslash frac\{n\}\{2\}\}\; \backslash frac\{\backslash langle\; \backslash sigma\_x\; \backslash sigma\_\{x\_1\}\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_\{x\_2\}\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_\{x\_3\}\; \backslash rangle\_\backslash beta\; \backslash langle\; \backslash sigma\_x\; \backslash sigma\_\{x\_4\}\; \backslash rangle\_\backslash beta\}\{\backslash sum\_L\; (\backslash beta)^2\; \backslash cdot\; B\_L(x\_1,\; \backslash dots,\; x\_4)\; (\backslash beta)^c\},\; \backslash quad\; (6.46)$$

where $L(x_1, \dots, x_4)$ is the minimal distance between the $x_i$. Also note that by flip symmetry we find that $\langle T_{f,L}(\sigma)^{2n+1} \rangle_\beta = 0$. Multiplying (6.45) by $\frac{z^{2n}}{(2n)!}$ and summing over $n$, we obtain

$$\mid {⟨\exp [z{T}_{f,L}(\sigma )]⟩}_{\beta }-\exp \left[\frac{z^2}{2}{⟨T_{f,L}(\sigma {)}^2⟩}_{\beta }\right]\mid \le \exp \left[\frac{z^2}{2}{⟨T_{f,L}(\sigma {)}^2⟩}_{\beta }\right]C_1{z}^4\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}^{4}S(L,r,\beta )\mathrm{.}(6.47)\backslash left|\; \backslash left\backslash langle\; \backslash exp[zT\_\{f,L\}(\backslash sigma)]\; \backslash right\backslash rangle\_\{\backslash beta\}\; -\; \backslash exp\backslash left[\backslash frac\{z^2\}\{2\}\backslash left\backslash langle\; T\_\{f,L\}(\backslash sigma)^2\; \backslash right\backslash rangle\_\{\backslash beta\}\backslash right]\; \backslash right|\; \backslash leq\; \backslash exp\backslash left[\backslash frac\{z^2\}\{2\}\backslash left\backslash langle\; T\_\{f,L\}(\backslash sigma)^2\; \backslash right\backslash rangle\_\{\backslash beta\}\backslash right]\; C\_1\; z^4\; \backslash |f\backslash |\_\{\backslash infty\}^4\; S(L,r,\backslash beta).\; \backslash quad\; (6.47)$$

Proposition 1.4 therefore follows from the bound

$$S(L,r,\beta )\le C_2{r}^{12}{\left(\frac{\log \log L}{\log L}\right)}^{c}S(L,\; r,\; \backslash beta)\; \backslash le\; C\_2\; r^\{12\}\; \backslash left(\; \backslash frac\{\backslash log\; \backslash log\; L\}\{\backslash log\; L\}\; \backslash right)^c$$

that we prove next. Note that (6.47) is easy to obtain under the power-law assumption (4.2). Without this assumption, the derivation requires a few lines of (not particularly ---PAGE_BREAK---

interesting) computations to address the fact that we do not a priori know that $B_L(\beta)$ grows logarithmically. Roughly speaking, the argument below uses the idea that in the case in which $B_L(\beta)$ is small, then $\Sigma_L(\beta)$ is much smaller than $L^2$, so that both combine in such a way that the previous bound is always valid for $c > 0$ sufficiently small. We now provide the proof of (6.48). Fix $0 < a < 1$ (any choice would do) and split the sum into four sums

Bound on (1) We focus on this term and give more details since it is in fact the main contributor. By Lemma 6.3,

$$B_L(x_1,\dots ,x_4)(\beta )\ge \frac{1}{C_3}{B}_L(\beta )\mathrm{.}(6.50)B\_L(x\_1,\; \backslash dots,\; x\_4)(\backslash beta)\; \backslash geq\; \backslash frac\{1\}\{C\_3\}\; B\_L(\backslash beta).\; \backslash quad\; (6.50)$$

Summing over the sites in $\Lambda_{rL}$, we get that

$$(1)\le C_3\frac{\mathrm{\mid }{\mathrm{\Lambda }}_{rL}\mathrm{\mid }{\chi }_{2rL}(\beta {)}^4}{{\mathrm{\Sigma }}_L(\beta {)}^2{B}_L(\beta {)}^c}\le C_4{r}^{12}{\left(\frac{{\chi }_L(\beta {)}^2}{L^4{B}_L(\beta )}\right)}^c,(6.51)(1)\; \backslash le\; C\_3\; \backslash frac\{|\backslash Lambda\_\{rL\}\; |\; \backslash chi\_\{2rL\}\; (\backslash beta)^4\}\{\backslash Sigma\_L(\backslash beta)^2\; B\_L(\backslash beta)^c\}\; \backslash le\; C\_4\; r^\{12\}\; \backslash left(\; \backslash frac\{\backslash chi\_L(\backslash beta)^2\}\{L^4\; B\_L(\backslash beta)\}\; \backslash right)^c,\; \backslash quad\; (6.51)$$

where in the second inequality we used the sliding-scale Infrared Bound (5.24) to bound $\chi_{2drL}(\beta)$ in terms of $\chi_L(\beta) \le C_5 L^{-4} \Sigma_L(\beta)$ and the Infrared Bound (5.37) to write

$${\chi }_L(\beta )\le C_6{L}^2\mathrm{.}(6.52)\backslash chi\_L(\backslash beta)\; \backslash le\; C\_6\; L^2.\; \backslash quad\; (6.52)$$

Applying Cauchy-Schwarz for the first inequality below, then bounding the two terms in the middle by (6.52) and Lemma 6.3 correspondingly, we find that

$$\frac{{\chi }_L(\beta {)}^2}{L^4{B}_L(\beta )}\le 2\frac{{\chi }_{L\mathrm{/}\log L}(\beta {)}^2}{L^4}+C_7\frac{B_L(\beta )-B_{L\mathrm{/}\log L}(\beta )}{B_L(\beta )}\le C_8\frac{\log \log L}{\log L}\mathrm{.}(6.53)\backslash frac\{\backslash chi\_L(\backslash beta)^2\}\{L^4\; B\_L(\backslash beta)\}\; \backslash le\; 2\; \backslash frac\{\backslash chi\_\{L/\backslash log\; L\}(\backslash beta)^2\}\{L^4\}\; +\; C\_7\; \backslash frac\{B\_L(\backslash beta)\; -\; B\_\{L/\backslash log\; L\}(\backslash beta)\}\{B\_L(\backslash beta)\}\; \backslash le\; C\_8\; \backslash frac\{\backslash log\; \backslash log\; L\}\{\backslash log\; L\}.\; \backslash quad\; (6.53)$$

Plugging this estimate in (6.51) gives

$$(1)\le C_9{r}^{12}{\left(\frac{\log \log L}{\log L}\right)}^c\mathrm{.}(6.54)(1)\; \backslash le\; C\_9\; r^\{12\}\; \backslash left(\; \backslash frac\{\backslash log\; \backslash log\; L\}\{\backslash log\; L\}\; \backslash right)^c\; .\; \backslash quad\; (6.54)$$

Bound on (2) Combine (5.4) and the sliding-scale Infrared Bound (5.24) to get that for $i = 1, \dots, 4$,

$$⟨{\sigma }_x{\sigma }_{x_i}{⟩}_{\beta }\le \frac{C_{10}}{\mathrm{\mid }x{\mathrm{\mid }}^4}{\chi }_{\mathrm{\mid }x\mathrm{\mid }\mathrm{/}d}(\beta )\le \frac{C_{11}}{L^2\mathrm{\mid }x{\mathrm{\mid }}^2}{\chi }_L(\beta )\mathrm{.}(6.55)\backslash langle\; \backslash sigma\_x\; \backslash sigma\_\{x\_i\}\; \backslash rangle\_\backslash beta\; \backslash le\; \backslash frac\{C\_\{10\}\}\{|x|^4\}\; \backslash chi\_\{|x|/d\}(\backslash beta)\; \backslash le\; \backslash frac\{C\_\{11\}\}\{L^2|x|^2\}\; \backslash chi\_L(\backslash beta).\; \backslash quad\; (6.55)$$

Summing over the sites gives the same bound as in (6.51) so that the reasoning in (1) gives

$$(2)\le C_{12}{r}^{12}{\left(\frac{\log \log L}{\log L}\right)}^c\mathrm{.}(6.56)(2)\; \backslash le\; C\_\{12\}\; r^\{12\}\; \backslash left(\; \backslash frac\{\backslash log\; \backslash log\; L\}\{\backslash log\; L\}\; \backslash right)^c\; .\; \backslash quad\; (6.56)$$

Bound on (3) This term is much smaller than the previous two due to the constraint that two sites must be close to each other. In fact, we will not even need the improved part of the tree diagram bound and will simply use that $B_L(x_1, \dots, x_4)(\beta) \ge 1$. Then, we use the Infrared Bound (5.37) to bound the terms $\langle \sigma_x \sigma_{x_i} \rangle_\beta$ and $\langle \sigma_x \sigma_{x_j} \rangle_\beta$ for ---PAGE_BREAK---

which $x_i$ and $x_j$ are at a distance exactly $L(x_1, \dots, x_4)$. Summing over the other two sites $x_k$ and $x_l$ gives a contribution bounded by $\chi_{2rL}(\beta)^2 \le C_{13}L^{-8}r^4\Sigma_L(\beta)^2$ by the sliding-scale Infrared Bound (5.24). Summing over $x$ and then $x_i$ and $x_j$ gives that

$$(3)\le \frac{C_{14}{r}^4\log (Lr)}{L^{4-4a}}\mathrm{.}(6.57)(3)\; \backslash le\; \backslash frac\{C\_\{14\}r^4\; \backslash log(Lr)\}\{L^\{4-4a\}\}.\; \backslash qquad\; (6.57)$$

Bound on (4) This sum is even simpler to bound than (3). Again, we simply use $B_L(x_1, \dots, x_4)(\beta) \ge 1$, bound two of the terms $\langle \sigma_x \sigma_{x_i} \rangle_\beta$ using (6.55), and the other two using the Infrared Bound (5.37). Summing over the vertices and using the constraint that two of the sites must be close to each other gives the bound

$$(4)\le \frac{C_{15}{r}^8}{L^{4-4a}}\mathrm{.}(6.58)(4)\; \backslash le\; \backslash frac\{C\_\{15\}r^8\}\{L^\{4-4a\}\}.\; \backslash qquad\; (6.58)$$

In conclusion, all the sums (1)–(4) are sufficiently small (recall that by definition $r \ge 1$) and the claim is derived.

Remark 6.8 For $\beta < \beta_c$, applying (6.46) with $r=1$ and $L = \xi(\beta)$ gives the following bound on the renormalized coupling constant $g(\beta)$:

$$g(\beta ):=\frac{1}{\xi (\beta {)}^4\chi (\beta {)}^2}\sum _{x,y,z\in {\mathbb{Z}}^d}\mathrm{\mid }{U}_4^{\beta }(0,x,y,z)\mathrm{\mid }\le \log {\left(\frac{1}{\mathrm{\mid }\beta -{\beta }_c\mathrm{\mid }}\right)}^{-c},(6.59)g(\backslash beta)\; :=\; \backslash frac\{1\}\{\backslash xi(\backslash beta)^4\; \backslash chi(\backslash beta)^2\}\; \backslash sum\_\{x,y,z\; \backslash in\; \backslash mathbb\{Z\}^d\}\; |U\_4^\backslash beta(0,x,y,z)|\; \backslash le\; \backslash log\backslash left(\backslash frac\{1\}\{|\backslash beta\; -\; \backslash beta\_c|\}\backslash right)^\{-c\},\; \backslash qquad\; (6.59)$$

where we used that

$$\xi (\beta {)}^2\ge c_0{\chi }_{\xi (\beta )}(\beta )\ge c_1\chi (\beta )\ge c_2\mathrm{/}({\beta }_c-\beta )\mathrm{.}\backslash xi(\backslash beta)^2\; \backslash ge\; c\_0\; \backslash chi\_\{\backslash xi(\backslash beta)\}(\backslash beta)\; \backslash ge\; c\_1\; \backslash chi(\backslash beta)\; \backslash ge\; c\_2\; /\; (\backslash beta\_c\; -\; \backslash beta).$$

(The first inequality follows from the infrared bound, the second is a classical bound obtained first by Sokal [47], and the third is a mean-field lower bound on $\chi(\beta)$ [1]). In field theory this quantity is often referred to as the (dimensionless) renormalized coupling constant. In [29] it was proved that for lattice $\phi_4^4$ measures of small enough $\lambda$ it converges to 0 at the rate $1/\log(\frac{1}{|\beta-\beta_c|})$. Such behaviour is expected to be true, in dimension $d=4$, also for the n.n.f. Ising model.

7 Generalization to models in the Griffiths-Simon class

In this section we extend the results to nearest-neighbor ferromagnetic models in the GS class. An important observation is that the results from the previous section also extend. Note, however, that $\rho$ can have unbounded support, so that to be of relevance the relations of interest need to be expressed in spin-dimension balanced forms. Once this is done, many of the basic diagrammatic bounds which are available for the Ising model extend to the GS class essentially by linearity, and then to the GS class by continuity. Below, we carefully present the generalizations.

In the whole section, $U_4^{\rho,\beta}$ denotes the 4-point Ursell function of the $\tau$ variables, and $B_L(\rho, \beta)$ the bubble diagram truncated at a distance $L$. We also reuse the notation $\xi(\rho, \beta)$ and $\beta_c(\rho)$ introduced in Section 5.

7.1 An improved tree diagram bound for models in the GS class

For bounds which are not homogeneous in the spin dimension, one needs to pay attention to the fact that $\tau$ is neither dimensionless nor bounded, and prepare the extension by reformulating the Ising relations in a spin-dimensionless form. ---PAGE_BREAK---

For example, the basic tree diagram bound (1.22) has four Ising spins on the left side and four pairs on the right. An extension of the inequality to GS models can be reached by site-splitting the terms in which an Ising spin is repeated, using the inequality (5.38) (which has a simple proof by means of the switching lemma) $^8$. The resulting diagrammatic bounds may at first glance appear as slightly more complicated than the one for the Ising case, but it has the advantage of being dimensionally balanced. That is a required condition for a bound to hold uniformly throughout the GS class of models. Additional consideration is needed for the factors by which the tree diagram bound of [1] is improved here. Taking care of that we get the following extension of the result, which also covers the $\phi^4$ lattice models.

Theorem 7.1 (Improved tree diagram bound for the GS class) There exist $C, c > 0$ such that for every n.n.f. model in the GS class on $\mathbb{Z}^4$, every $\beta \le \beta_c(\rho)$, $L \le \xi(\rho, \beta)$ and every $x,y,z,t \in \mathbb{Z}^d$ at distances larger than $L$ of each other,

$$\mathrm{\mid }{U}_4^{\rho ,\beta }(x,y,z,t)\mathrm{\mid }\le C{\left(\frac{B_0(\rho ,\beta )}{B_L(\rho ,\beta )}\right)}^c\sum _u\sum _{u^{\mathrm{\prime }},u^{\mathrm{\prime }\mathrm{\prime }}}⟨{\tau }_x{\tau }_u{⟩}_{\rho ,\beta }\beta J_{u,u^{\mathrm{\prime }}}⟨{\tau }_{u^{\mathrm{\prime }}}{\tau }_y{⟩}_{\rho ,\beta }⟨{\tau }_z{\tau }_u{⟩}_{\rho ,\beta }\beta J_{u,u^{\mathrm{\prime }\mathrm{\prime }}}⟨{\tau }_{u^{\mathrm{\prime }\mathrm{\prime }}}{\tau }_t{⟩}_{\rho ,\beta }\mathrm{.}(7.1)|U\_4^\{\backslash rho,\backslash beta\}(x,y,z,t)|\; \backslash le\; C\; \backslash left(\; \backslash frac\{B\_0(\backslash rho,\; \backslash beta)\}\{B\_L(\backslash rho,\; \backslash beta)\}\; \backslash right)^c\; \backslash sum\_u\; \backslash sum\_\{u\text{'},u\text{'}\text{'}\}\; \backslash langle\; \backslash tau\_x\; \backslash tau\_u\; \backslash rangle\_\{\backslash rho,\backslash beta\}\; \backslash beta\; J\_\{u,u\text{'}\}\; \backslash langle\; \backslash tau\_\{u\text{'}\}\; \backslash tau\_y\; \backslash rangle\_\{\backslash rho,\backslash beta\}\; \backslash langle\; \backslash tau\_z\; \backslash tau\_u\; \backslash rangle\_\{\backslash rho,\backslash beta\}\; \backslash beta\; J\_\{u,u\text{'}\text{'}\}\; \backslash langle\; \backslash tau\_\{u\text{'}\text{'}\}\; \backslash tau\_t\; \backslash rangle\_\{\backslash rho,\backslash beta\}.\; \backslash quad\; (7.1)$$

Before diving into the proof, note that the improved tree diagram bound implies, as it did for the Ising model, the following quantitative bound on the convergence to gaussian of the scaling limit of the $\tau$ field in four dimensional models with variables in the GS class.

Proposition 7.2 There exist two constants $c, C > 0$ such that for every n.n.f. model in the GS class on $\mathbb{Z}^4$, every $\beta \le \beta_c(\rho)$, $L \le \xi(\rho, \beta)$, every continuous function $f: \mathbb{R}^4 \to \mathbb{R}$ with bounded support and every $z \in \mathbb{R}$,

$$\mid {⟨\exp [z{T}_{f,L}(\tau )-\frac{z^2}{2}⟨T_{f,L}(\tau {)}^2{⟩}_{\rho ,\beta }]⟩}_{\rho ,\beta }-1\mid \le \frac{C\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}^4{r}_f^{12}}{(\log L{)}^c}{z}^4\mathrm{.}(7.2)\backslash left|\; \backslash left\backslash langle\; \backslash exp[zT\_\{f,L\}(\backslash tau)\; -\; \backslash frac\{z^2\}\{2\}\; \backslash langle\; T\_\{f,L\}(\backslash tau)^2\; \backslash rangle\_\{\backslash rho,\backslash beta\}]\; \backslash right\backslash rangle\_\{\backslash rho,\backslash beta\}\; -\; 1\; \backslash right|\; \backslash leq\; \backslash frac\{C\; \backslash |f\backslash |\_\{\backslash infty\}^4\; r\_f^\{12\}\}\{(\backslash log\; L)^c\}\; z^4.\; \backslash quad\; (7.2)$$

We now return to the proof of the improved tree diagram bound, following the path outlined above. The GS class of variables is naturally divided into two kinds. The core consists of those that directly fall under the Definition 2.1. The rest can be obtained as weak limits of the former. Since the constants in (7.1) are uniform, it suffices to prove the result for the former to get it for the latter. We therefore focus on site-measures $\rho$ satisfying Definition 2.1, which can directly be represented as Ising measures on a graph where every vertex is replaced by blocks, as explained in the previous section. In this case, we identify $\langle \cdot \rangle_{\rho,\beta}$ with the Ising measure, and $\tau_x$ with the proper average of Ising's variables. With this identification, we can harvest all the nice inequalities that are given by Ising's theory. In particular, we can use the random current representation.

More explicitly, to generalize the argument used in the Ising's proof, we introduce the measure $\mathbf{P}^{xy}$ defined on the graph $\mathbb{Z}^d \times {1, \dots, N}$ in two steps:

• first, sample two integers $1 \le i, j \le N$ with probability

$$Q_i{Q}_j({\sigma }_{x,i}{\sigma }_{y,j}{)}_{\rho ,\beta }\mathrm{/}({\tau }_x{\tau }_y{)}_{\rho ,\beta },Q\_i\; Q\_j\; (\backslash sigma\_\{x,i\}\; \backslash sigma\_\{y,j\})\_\{\backslash rho,\backslash beta\}\; /\; (\backslash tau\_x\; \backslash tau\_y)\_\{\backslash rho,\backslash beta\},$$

• second, sample a current according to the measure $\mathbf{P}{\rho,\beta}^{(x,i),(y,j)}$ corresponding to the random current representation of the Ising model $\langle \cdot \rangle{\rho,\beta}$.

⁸An alternative method for reducing a diagrammatic expression's spin-dimension is to divide by $\langle\sigma_u^2\rangle_0$. Both methods are of use, and may be compared through (5.15). ---PAGE_BREAK---

The interpretation of this object is that of a random current with two random sources $(x, i) \in \mathcal{B}_x$ and $(y, j) \in \mathcal{B}y$. Also note that the superscript $xy$ will unequivocally denote this type of measures (we will avoid using measures with deterministic sources in this section to prevent confusion) and $\mathbf{P}{\rho,\beta}^{\emptyset}$ which have no sources.

The interest in $\mathbf{P}_{\rho,\beta}^{xy}$ over measures with deterministic sets of sources comes from the fact that the probability that the cluster of the sources intersects a set of the form $\mathcal{B}_u$ can be bounded in terms of correlations of the variables $\tau_x, x \in \mathbb{Z}^d$ (see Proposition A.8).

Proof of Theorem 7.1 As mentioned above, every $\rho$ in the GS class is a weak limit of measures satisfying the first condition of Definition 2.1. We therefore focus on such measures.

Exactly like in the case of the Ising model, the core of the proof of Theorem 7.1 will be the proof of the intersection-clustering property that we now state and whose proof is postponed after the proof of the theorem. Define $\ell_0 = 0$ and $\ell_k = \ell_k(\rho, \beta)$ using the same definition (using $B_L(\rho, \beta)$ this time) as in (6.1). Let $\mathcal{T}_u$ be the set of vertices $v \in \mathbb{Z}^d$ such that $\mathcal{B}_v$ is connected in $\mathbf{n}_1 + \mathbf{n}3$ to a box $\mathcal{B}{u'}$ and in $\mathbf{n}_2 + \mathbf{n}4$ to a box $\mathcal{B}{u''}$, with $u'$ and $u''$ at graph distance at most 2 of $u$. Note that $\mathcal{T}_u$ is now a function of $u$ and that it is defined in terms of “coarse intersections”, i.e. lattice sites $v$ such that both clusters intersect $\mathcal{B}_v$ (but do not necessarily intersect each other).

Proposition 7.3 (intersection-clustering bound for the GS class) For $d=4$ and $D$ large enough, there exists $\delta = \delta(D)$ such that for every model in the GS class, every $\beta \le \beta_c(\rho)$, every $K$ such that $\ell_K \le \xi(\rho, \beta)$ and every $u, u', u'', x, y, z, t \in \mathbb{Z}^d$ with $u'$ and $u''$ neighbors of $u$ and $x, y, z, t$ at mutual distances larger than $2\ell_K$,

$${\mathbf{P}}_{\rho ,\beta }^{ux,uz,u^{\mathrm{\prime }}y,u^{\mathrm{\prime }\mathrm{\prime }}t}[{\mathbf{M}}_u({\mathcal{T}}_u;\mathcal{L},K)\le \delta K]\le 2^{-\delta K}\mathrm{.}(7.3)\backslash mathbf\{P\}\_\{\backslash rho,\backslash beta\}^\{ux,uz,u\text{'}y,u\text{'}\text{'}t\}[\backslash mathbf\{M\}\_u(\backslash mathcal\{T\}\_u;\; \backslash mathcal\{L\},\; K)\; \backslash le\; \backslash delta\; K]\; \backslash le\; 2^\{-\backslash delta\; K\}.\; \backslash quad\; (7.3)$$

Postponing the proof of this estimate, we proceed with the proof of the Theorem. Express $U_4^{\rho,\beta}$ in terms of intersection properties of currents by summing (3.14) over vertices of $\mathcal{B}_x, \dots, \mathcal{B}_z$:

$$\mathrm{\mid }{U}_4^{\rho ,\beta }(x,y,z,t)\mathrm{\mid }\le 2⟨{\tau }_x{\tau }_y{⟩}_{\rho ,\beta }⟨{\tau }_z{\tau }_t{⟩}_{\rho ,\beta }{\mathbf{P}}_{\rho ,\beta }^{xy,zt,\mathrm{\varnothing },\mathrm{\varnothing }}[{\mathbf{C}}_{n_1+n_3}(\mathrm{\partial }{\mathbf{n}}_1)\cap {\mathbf{C}}_{n_2+n_4}(\mathrm{\partial }{\mathbf{n}}_2)\mathrm{\ne }\mathrm{\varnothing }],(7.4)|U\_4^\{\backslash rho,\backslash beta\}(x,y,z,t)|\; \backslash le\; 2\backslash langle\backslash tau\_x\backslash tau\_y\backslash rangle\_\{\backslash rho,\backslash beta\}\backslash langle\backslash tau\_z\backslash tau\_t\backslash rangle\_\{\backslash rho,\backslash beta\}\backslash mathbf\{P\}\_\{\backslash rho,\backslash beta\}^\{xy,zt,\backslash emptyset,\backslash emptyset\}[\backslash mathbf\{C\}\_\{n\_1+n\_3\}(\backslash partial\backslash mathbf\{n\}\_1)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_2+n\_4\}(\backslash partial\backslash mathbf\{n\}\_2)\; \backslash neq\; \backslash emptyset],\; \backslash quad\; (7.4)$$

where $\mathbf{C}{n_1+n_3}(\partial n_1)$ and $\mathbf{C}{n_2+n_4}(\partial n_2)$ refer to the clusters in $\mathbf{n}_1+\mathbf{n}_3$ and $\mathbf{n}_2+\mathbf{n}_4$ of the sources in $\partial n_1$ and $\partial n_2$ respectively (we introduce this notation since the sources are not deterministic anymore).

Define $K \ge c \log[B_L(\rho, \beta)/B_0(\rho, \beta)]$ as in the Ising case. We now implement the same reasoning as for the Ising model, with the twist that we consider coarse intersections. If $\mathbf{C}{n_1+n_3}(\partial n_1)$ and $\mathbf{C}{n_2+n_4}(\partial n_2)$ intersect, then

• either the number of $u \in \mathbb{Z}^d$ such that $\mathbf{C}{n_1+n_3}(\partial n_1)$ and $\mathbf{C}{n_2+n_4}(\partial n_2)$ intersect $\mathcal{B}_u$ is larger than or equal to $2^{\delta K/5}$,

• or there exists $u \in \mathbb{Z}^d$ such that $\mathbf{C}{n_1+n_3}(\partial n_1)$ and $\mathbf{C}{n_2+n_4}(\partial n_2)$ intersect $\mathcal{B}_u$, and $\mathbf{M}_u(\mathcal{T}_u; \mathcal{L}, K) < \delta K$.

Using the Markov inequality and (A.38) on the first line, and Lemma A.7 in the second one, we find (drop $\rho$ and $\beta$ from notation)

---PAGE_BREAK---

Lemma A.7 was invoked here since in the present context $\mathbf{M}_u(\mathcal{T}_u; \mathcal{L}, K)$ is defined in terms of coarse rather than true intersections. The intersection-clustering bound (Proposition 7.3) concludes the proof. $\square$

We now need to prove Proposition 7.3. The proof itself is exactly the same as for Proposition 6.1 (the monotonicity property of (A.2) is not impacted), except for the proofs of the mixing and intersection properties (i.e. statements corresponding to Lemma 6.2 and Theorem 6.4 respectively). Below, we briefly detail the statements and proofs of these results. Let $I_k(0)$ be the event that there exists $v \in \text{Ann}(\ell_k, \ell_{k+1})$ such that $\mathcal{B}_v$ is connected in $\mathbf{n}_1 + \mathbf{n}_3$ and in $\mathbf{n}_2 + \mathbf{n}_4$ to the union of the boxes $\mathcal{B}_w$ with $w$ at a distance at most 2 of 0.

Lemma 7.4 (intersection property for the GS class) There exists $c > 0$ such that for every $\rho$ in the GS class, every $\beta \le \beta_c(\rho)$, every $k$, every neighbour $0'$ of the origin, and every $y \notin \Lambda_{2\ell_{k+1}}$ in a regular scale,

$${\mathbf{P}}_{\rho ,\beta }^{0y,0^{\mathrm{\prime }}y,\mathrm{\varnothing },\mathrm{\varnothing }}[I_k(0)]\ge c\mathrm{.}(7.6)\backslash mathbf\{P\}\_\{\backslash rho,\backslash beta\}^\{0y,0\text{'}y,\backslash emptyset,\backslash emptyset\}[I\_k(0)]\; \backslash ge\; c.\; \backslash quad\; (7.6)$$

Proof Reuse the notions included in the proofs of the intersection property in previous sections. Let

$$\mathcal{M}:=\sum _{v\in \text{Ann}(m,M)}\sum _{i,i^{\mathrm{\prime }}=1}^n{Q}_i^2\mathbb{I}[\mathrm{\partial }{\mathbf{n}}_1\stackrel{{\mathbf{n}}_1+{\mathbf{n}}_3}{\leftarrow }(v,i)]Q_{i^{\mathrm{\prime }}}^2\mathbb{I}[\mathrm{\partial }{\mathbf{n}}_1\stackrel{{\mathbf{n}}_1+{\mathbf{n}}_3}{\leftarrow }(v,i^{\mathrm{\prime }})]\mathrm{.}(7.7)\backslash mathcal\{M\}\; :=\; \backslash sum\_\{v\; \backslash in\; \backslash text\{Ann\}(m,M)\}\; \backslash sum\_\{i,i\text{'}=1\}^\{n\}\; Q\_i^2\; \backslash mathbb\{I\}[\backslash partial\; \backslash mathbf\{n\}\_1\; \backslash xleftarrow\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_3\}\; (v,i)]\; Q\_\{i\text{'}\}^2\; \backslash mathbb\{I\}[\backslash partial\; \backslash mathbf\{n\}\_1\; \backslash xleftarrow\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_3\}\; (v,i\text{'})].\; \backslash quad\; (7.7)$$

A computation similar to before gives

$${\mathbf{E}}_{\rho ,\beta }^{0y,0^{\mathrm{\prime }}y,\mathrm{\varnothing },\mathrm{\varnothing }}[\mathrm{\mid }\mathcal{M}\mathrm{\mid }]\ge c_1(B_M(\rho ,\beta )-B_{m-1}(\rho ,\beta ))(7.8)\backslash mathbf\{E\}\_\{\backslash rho,\; \backslash beta\}^\{0y,\; 0\text{'}y,\; \backslash emptyset,\; \backslash emptyset\}[|\backslash mathcal\{M\}|]\; \backslash geq\; c\_1(B\_M(\backslash rho,\; \backslash beta)\; -\; B\_\{m-1\}(\backslash rho,\; \backslash beta))\; \backslash quad\; (7.8)$$

$${\mathbf{E}}_{\rho ,\beta }^{0y,0^{\mathrm{\prime }}y,\mathrm{\varnothing },\mathrm{\varnothing }}[\mathrm{\mid }\mathcal{M}{\mathrm{\mid }}^2]\le C_2{B}_{{\mathrm{\ell }}_{k+1}}(\rho ,\beta {)}^2\mathrm{.}(7.9)\backslash mathbf\{E\}\_\{\backslash rho,\; \backslash beta\}^\{0y,\; 0\text{'}y,\; \backslash emptyset,\; \backslash emptyset\}[|\backslash mathcal\{M\}|^2]\; \backslash le\; C\_2\; B\_\{\backslash ell\_\{k+1\}\}(\backslash rho,\; \backslash beta)^2.\; \backslash quad\; (7.9)$$

Now, in the first line we use the same reasoning as below (6.6). We include it for completeness to see where the division by $B_0(\rho, \beta)$ enters into the game (it is the only place it does). The Infrared Bound (5.36) (note that $\langle \tau_0^2 \rangle_{\rho,\beta} = B_0(\rho, \beta)$) implies that

$$B_M(\rho ,\beta )-B_{m-1}(\rho ,\beta )\ge B_{{\mathrm{\ell }}_{k+1}}(\rho ,\beta )-B_{{\mathrm{\ell }}_k}(\rho ,\beta )-C_3{B}_0(\rho ,\beta )\ge (1-\frac{1+C_3}{D})B_{{\mathrm{\ell }}_{k+1}}(\rho ,\beta )\mathrm{.}(7.10)B\_M(\backslash rho,\; \backslash beta)\; -\; B\_\{m-1\}(\backslash rho,\; \backslash beta)\; \backslash ge\; B\_\{\backslash ell\_\{k+1\}\}(\backslash rho,\; \backslash beta)\; -\; B\_\{\backslash ell\_k\}(\backslash rho,\; \backslash beta)\; -\; C\_3\; B\_0(\backslash rho,\; \backslash beta)\; \backslash ge\; \backslash left(1\; -\; \backslash frac\{1+C\_3\}\{D\}\backslash right)\; B\_\{\backslash ell\_\{k+1\}\}(\backslash rho,\; \backslash beta).\; \backslash quad\; (7.10)$$

Cauchy-Schwarz therefore implies the fact that $\mathcal{M} \neq \emptyset$ with positive probability, which implies in particular the existence of a vertex $v \in \text{Ann}(m,M)$ which is connected in $\mathbf{n}_1 + \mathbf{n}_3$ to $\mathcal{B}_0$ and in $\mathbf{n}_2 + \mathbf{n}_4$ to $\mathcal{B}'_0$.

The second part of the proof bounding the probabilities of $F_1, \dots, F_4$ follows by the same proof as for the Ising model. More precisely, for $F_1$, the chain rule for backbones [3] and a decomposition on the first edge of the backbone with one endpoint in (a block of a vertex in) $\Lambda_{n-1}$ and the other (in a block of a vertex) in $\partial\Lambda_n$, and then the first edge after this between an endpoint (in a block of a vertex) outside $\Lambda_{\ell_k}$ and one in (a block of a vertex in) $\Lambda_{\ell_k}$ implies that

$${\mathbf{P}}_{\rho ,\beta }^{0y,\mathrm{\varnothing }}[F_1]\le \sum _{\begin{array}{c}v\in \mathrm{\partial }{\mathrm{\Lambda }}_n\\ w\in \mathrm{\partial }{\mathrm{\Lambda }}_{{\mathrm{\ell }}_k}\\ v^{\mathrm{\prime }},w^{\mathrm{\prime }}\in {\mathbb{Z}}^d\end{array}}\frac{⟨{\tau }_0{\tau }_v{⟩}_{\rho ,\beta }\beta J_{v^{\mathrm{\prime }},v}⟨{\tau }_v{\tau }_{w^{\mathrm{\prime }}}{⟩}_{\rho ,\beta }\beta J_{w^{\mathrm{\prime }},w}⟨{\tau }_w{\tau }_y{⟩}_{\rho ,\beta }}{⟨{\tau }_0{\tau }_y{⟩}_{\rho ,\beta }}\le C_3{n}^3{\mathrm{\ell }}_k^3{n}^{-4}\le C_4{\mathrm{\ell }}_k^{-ϵ}\mathrm{.}(7.11)\backslash mathbf\{P\}\_\{\backslash rho,\; \backslash beta\}^\{0y,\; \backslash emptyset\}[F\_1]\; \backslash le\; \backslash sum\_\{\backslash substack\{v\; \backslash in\; \backslash partial\backslash Lambda\_n\; \backslash \backslash \; w\; \backslash in\; \backslash partial\backslash Lambda\_\{\backslash ell\_k\}\; \backslash \backslash \; v\text{'},\; w\text{'}\; \backslash in\; \backslash mathbb\{Z\}^d\}\}\; \backslash frac\{\backslash langle\; \backslash tau\_0\; \backslash tau\_v\; \backslash rangle\_\{\backslash rho,\; \backslash beta\}\; \backslash beta\; J\_\{v\text{'},\; v\}\; \backslash langle\; \backslash tau\_v\; \backslash tau\_\{w\text{'}\}\; \backslash rangle\_\{\backslash rho,\; \backslash beta\}\; \backslash beta\; J\_\{w\text{'},\; w\}\; \backslash langle\; \backslash tau\_w\; \backslash tau\_y\; \backslash rangle\_\{\backslash rho,\; \backslash beta\}\}\{\backslash langle\; \backslash tau\_0\; \backslash tau\_y\; \backslash rangle\_\{\backslash rho,\; \backslash beta\}\}\; \backslash le\; C\_3\; n^3\; \backslash ell\_k^3\; n^\{-4\}\; \backslash le\; C\_4\; \backslash ell\_k^\{-\backslash epsilon\}.\; \backslash quad\; (7.11)$$ ---PAGE_BREAK---

This inequality uses Property P2 of regular scales, the lower bound (5.39) on the two-point function, and the Infrared Bound (5.37). For $F_3$, the same reasoning as for Ising, with Proposition A.8 replacing the switching lemma, leads to

$$\begin{array}{cccc}& {\mathbf{P}}_{\rho ,\beta }^{0,x,\mathrm{\varnothing }}[F_3]\le \sum _{\begin{array}{c}v\in \mathrm{\partial }{\mathrm{\Lambda }}_n\\ w\in \mathrm{\partial }{\mathrm{\Lambda }}_m\end{array}}{\mathbf{P}}_{\rho ,\beta }^{\mathrm{\varnothing },\mathrm{\varnothing }}[{\mathcal{B}}_v\stackrel{{\mathrm{n}}_1+{\mathrm{n}}_2}{\leftarrow }{\mathcal{B}}_w]\le \sum _{\begin{array}{c}v\in \mathrm{\partial }{\mathrm{\Lambda }}_n\\ w\in \mathrm{\partial }{\mathrm{\Lambda }}_m\\ v^{\mathrm{\prime }},w^{\mathrm{\prime }}\in {\mathbb{Z}}^d\end{array}}⟨{\tau }_v{\tau }_w⟩\beta J_{w,w^{\mathrm{\prime }}}⟨{\tau }_{w^{\mathrm{\prime }}}{\tau }_{v^{\mathrm{\prime }}}⟩\beta J_{v^{\mathrm{\prime }},v}\le C_5{l}_k^{-ϵ},& & \text{(7.12)}\end{array}\backslash mathbf\{P\}\_\{\backslash rho,\backslash beta\}^\{0,x,\backslash emptyset\}[F\_3]\; \backslash le\; \backslash sum\_\{\backslash substack\{v\; \backslash in\; \backslash partial\; \backslash Lambda\_n\; \backslash \backslash \; w\; \backslash in\; \backslash partial\; \backslash Lambda\_m\}\}\; \backslash mathbf\{P\}\_\{\backslash rho,\backslash beta\}^\{\backslash emptyset,\backslash emptyset\}[\backslash mathcal\{B\}\_v\; \backslash xleftarrow\{\backslash mathrm\{n\}\_1+\backslash mathrm\{n\}\_2\}\; \backslash mathcal\{B\}\_w]\; \backslash le\; \backslash sum\_\{\backslash substack\{v\; \backslash in\; \backslash partial\; \backslash Lambda\_n\; \backslash \backslash \; w\; \backslash in\; \backslash partial\; \backslash Lambda\_m\; \backslash \backslash \; v\text{'},\; w\text{'}\; \backslash in\; \backslash mathbb\{Z\}^d\}\}\; \backslash langle\; \backslash tau\_v\; \backslash tau\_w\; \backslash rangle\; \backslash beta\; J\_\{w,w\text{'}\}\; \backslash langle\; \backslash tau\_\{w\text{'}\}\; \backslash tau\_\{v\text{'}\}\; \backslash rangle\; \backslash beta\; J\_\{v\text{'},v\}\; \backslash le\; C\_5\; l\_k^\{-\backslash epsilon\},\; \backslash tag\{7.12\}$$

where in the last line we used again the Infrared Bound (5.37). $\square$

We now turn to the proof of the mixing property for the measures $P_\beta^{xy}$, which is the exact replica of the Ising statement.

Theorem 7.5 (mixing of random currents for the GS class) For $d \ge 4$, there exist $\alpha, c > 0$ such that for every $\rho$ satisfying Definition 2.1, every $t \le s$, every $\beta \le \beta_c(\rho)$, every $n^\alpha \le N \le \xi(\rho, \beta)$, every $x_i \in \Lambda_n$ and $y_i \notin \Lambda_N$ for every $i \le t$, and every events E and F depending on the restriction of $(\mathbf{n}_1, \dots, \mathbf{n}_s)$ to edges within $\Lambda_n$ and outside of $\Lambda_N$ respectively,

$$\mathrm{\mid }{\mathbf{P}}_{\rho ,\beta }^{x_1{y}_1,\dots ,x_t{y}_t,\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[E\cap F]-{\mathbf{P}}_{\rho ,\beta }^{x_1^{\mathrm{\prime }}{y}_1^{\mathrm{\prime }},\dots ,x_t^{\mathrm{\prime }}{y}_t^{\mathrm{\prime }},\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[E]\mathrm{\mid }{\mathbf{P}}_{\rho ,\beta }^{x_1{y}_1,\dots ,x_t{y}_t,\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[F]\mathrm{\mid }\le s(\log \frac{N}{n}{)}^{-c}\mathrm{.}(7.13)|\; \backslash mathbf\{P\}\_\{\backslash rho,\; \backslash beta\}^\{x\_1\; y\_1,\; \backslash dots,\; x\_t\; y\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [E\; \backslash cap\; F]\; -\; \backslash mathbf\{P\}\_\{\backslash rho,\; \backslash beta\}^\{x\text{'}\_1\; y\text{'}\_1,\; \backslash dots,\; x\text{'}\_t\; y\text{'}\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [E]\; |\; \backslash mathbf\{P\}\_\{\backslash rho,\; \backslash beta\}^\{x\_1\; y\_1,\; \backslash dots,\; x\_t\; y\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [F]\; |\; \backslash le\; s\; (\backslash log\; \backslash frac\{N\}\{n\})^\{-c\}.\; \backslash quad\; (7.13)$$

Furthermore, for every $x'_1, \dots, x'_t \in \Lambda_n$ and $y'_1, \dots, y'_t \notin \Lambda_N$,

$$\mathrm{\mid }{\mathbf{P}}_{\rho ,\beta }^{x_1{y}_1,\dots ,x_t{y}_t,\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[E]-{\mathbf{P}}_{\rho ,\beta }^{x_1^{\mathrm{\prime }}{y}_1^{\mathrm{\prime }},\dots ,x_t^{\mathrm{\prime }}{y}_t^{\mathrm{\prime }},\mathrm{\varnothing },\dots ,\mathrm{\varnothing }}[E]\mathrm{\mid }\le s(\log \frac{N}{n}{)}^{-c},(7.14)|\backslash mathbf\{P\}\_\{\backslash rho,\backslash beta\}^\{x\_1\; y\_1,\; \backslash dots,\; x\_t\; y\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [E]\; -\; \backslash mathbf\{P\}\_\{\backslash rho,\backslash beta\}^\{x\text{'}\_1\; y\text{'}\_1,\; \backslash dots,\; x\text{'}\_t\; y\text{'}\_t,\; \backslash emptyset,\; \backslash dots,\; \backslash emptyset\}\; [E]|\; \backslash le\; s\; (\backslash log\; \backslash frac\{N\}\{n\})^\{-c\},\; \backslash quad\; (7.14)$$

\left| |\mathbf{P}_{\rho,\beta}^{x_1 y_1, \dots, x_t y_t, \emptyset, \dots, \emptyset} [F] - \mathbf{P}_{\rho,\beta}^{x'_1 y'_1, \dots, x'_t y'_t, \emptyset, \dots, \emptyset} [F]| \le s (\log \frac{N}{n})^{-c}. \quad (7.15)

Proof The beginning is the same as for the Ising model, until the definition of the variable $\mathbf{N}_i$ that now becomes

$${\mathbf{N}}_i:=\frac{1}{\mathrm{\mid }\mathcal{H}\mathrm{\mid }}\sum _{k\in \mathcal{K}}\frac{1}{A_{x_i,y_i}(k)}\sum _{u\in A_k(y_i)}{Q}_j^2[\mathbb{I}[(u,j)\stackrel{{\mathbf{n}}_i+{\mathbf{n}}_i^{\mathrm{\prime }}}{\leftarrow }\mathrm{\partial }{\mathbf{n}}_i]],(7.16)\backslash mathbf\{N\}\_i\; :=\; \backslash frac\{1\}\{|\backslash mathcal\{H\}|\}\; \backslash sum\_\{k\; \backslash in\; \backslash mathcal\{K\}\}\; \backslash frac\{1\}\{A\_\{x\_i,\; y\_i\}(k)\}\; \backslash sum\_\{u\; \backslash in\; A\_k(y\_i)\}\; Q\_j^2\; [\backslash mathbb\{I\}[(u,j)\; \backslash xleftarrow\{\backslash mathbf\{n\}\_i\; +\; \backslash mathbf\{n\}\text{'}\_i\}\; \backslash partial\backslash mathbf\{n\}\_i]],\; \backslash quad\; (7.16)$$

where $a_{x,y}(u) := (\tau_x\tau_u)(\tau_u\tau_y)/(\tau_x\tau_y)$ and $A_{x,y}(k) := \sum_{u\in A_k(y_i)} a_{x,y}(u)$. The proof of the concentration inequality follows the same lines as in the Ising case. Indeed, the choice of the weight $Q_j^2$ enables to rewrite the moments of the random variables $\mathbf{N}_i$ in terms of the correlations of the random variables $(\tau_z : z \in \mathbb{Z}^d)$. The rest of the proof is exactly the same, with trivial changes. For instance, in the proof of Lemma 6.7, one must be careful to derive bounds on probabilities involving $\beta|J|$. This is easily doable using Proposition A.8 exactly like in the previous proof. $\square$

A Appendix

A.1 Random currents's partial monotonicity statements

An inconvenient feature of the random current representation is the lack of an FKG-type monotonicity, as the one valid for the Fortuin-Kasteleyn random cluster models (cf. [25]). The addition of a pair of sources may enhance the configuration, e.g. forcing a long line ---PAGE_BREAK---

where such were rare, but in some situations it may facilitate a split in a connecting line, thereby reducing the current's connectivity properties. Nevertheless, some monotonicity properties can still be found, and are used in our analysis.

In this section, we set $\sigma_A$ for the product of the spins in $A$ and write $\mathbf{C}n(S) = \cup{x \in S} \mathbf{C}_n(x)$.

Lemma A.1 Let $A, B, S$ be subsets of $\Lambda$ and $F$ a non-negative function defined over pairs of currents, which is determined by just the values of $(\mathbf{n}_1, \mathbf{n}2)$ along the edges touching the connected cluster $\mathbf{C}{\mathbf{n}_1+\mathbf{n}_2}(S)$ and such that $F(\mathbf{n}_1, \mathbf{n}_2) = 0$ whenever that cluster intersects $B$ and $(\partial\mathbf{n}_1, \partial\mathbf{n}_2) = (A, B)$. Then

$${\mathbf{E}}_{\mathrm{\Lambda },\beta }^{A,B}[F({\mathbf{n}}_1,{\mathbf{n}}_2)]={\mathbf{E}}_{\mathrm{\Lambda },\beta }^{A,\mathrm{\varnothing }}[F({\mathbf{n}}_1,{\mathbf{n}}_2)\frac{⟨{\sigma }_B{⟩}_{\mathrm{\Lambda }\setminus C_{{\mathbf{n}}_1+{\mathbf{n}}_2}(S),\beta }}{⟨{\sigma }_B{⟩}_{\mathrm{\Lambda },\beta }}]\le {\mathbf{E}}_{\mathrm{\Lambda },\beta }^{A,\mathrm{\varnothing }}[F({\mathbf{n}}_1,{\mathbf{n}}_2)]\mathrm{.}(\text{A.1})\backslash mathbf\{E\}\_\{\backslash Lambda,\backslash beta\}^\{A,B\}[F(\backslash mathbf\{n\}\_1,\; \backslash mathbf\{n\}\_2)]\; =\; \backslash mathbf\{E\}\_\{\backslash Lambda,\backslash beta\}^\{A,\backslash emptyset\}[F(\backslash mathbf\{n\}\_1,\; \backslash mathbf\{n\}\_2)\; \backslash frac\{\backslash langle\; \backslash sigma\_B\; \backslash rangle\_\{\backslash Lambda\; \backslash setminus\; C\_\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\}(S),\backslash beta\}\}\{\backslash langle\; \backslash sigma\_B\; \backslash rangle\_\{\backslash Lambda,\backslash beta\}\}]\; \backslash leq\; \backslash mathbf\{E\}\_\{\backslash Lambda,\backslash beta\}^\{A,\backslash emptyset\}[F(\backslash mathbf\{n\}\_1,\; \backslash mathbf\{n\}\_2)].\; \backslash quad\; (\backslash text\{A.1\})$$

Proof The second inequality is a trivial application of Griffiths’ inequality [22]. The first one is proven by a fairly straightforward manipulation involving currents that we now present. We drop $\beta$ from the notation. Fix $T \subset \Lambda$ not intersection $B$ and choose $F$ given by

$$F({\mathbf{n}}_1,{\mathbf{n}}_2):=\mathbb{I}[{\mathbf{C}}_{{\mathbf{n}}_1+{\mathbf{n}}_2}(S)=T]\mathbb{I}[{\mathbf{n}}_1=\mathbf{n}]\mathbb{I}[{\mathbf{n}}_2=\mathbf{m}\text{ on }T](\text{A.2})F(\backslash mathbf\{n\}\_1,\; \backslash mathbf\{n\}\_2)\; :=\; \backslash mathbb\{I\}[\backslash mathbf\{C\}\_\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\}(S)\; =\; T]\; \backslash mathbb\{I\}[\backslash mathbf\{n\}\_1\; =\; \backslash mathbf\{n\}]\; \backslash mathbb\{I\}[\backslash mathbf\{n\}\_2\; =\; \backslash mathbf\{m\}\; \backslash text\{\; on\; \}\; T]\; \backslash quad\; (\backslash text\{A.2\})$$

for n and m currents on Λ and T respectively. For such a choice of function, we find that

where $\mathbf{n}'_2$ is referring to a current on $\Lambda \setminus T$. In the second line, we used that for $F(\mathbf{n}_1, \mathbf{n}_2)$ to be non-zero, $\mathbf{n}_1$ must be equal to $\mathbf{n}$ and $\mathbf{n}_2$ be decomposed into the current $\mathbf{m}$ on $T$ and a current $\mathbf{n}'_2$ outside $T$ (also, $\mathbf{n}2(x,y)$ is equal to zero for every $x \in T$ and $y \notin T$). In the last line, we skipped the steps corresponding to going backward line to line to end up with $\mathbf{E}{\Lambda}^{A,\emptyset}[F(\mathbf{n}_1, \mathbf{n}_2)]$.

The proof follows readily for every function F satisfying the assumptions of the lemma. Also, we obtain the result on Z^d by letting Λ tend to Z^d.

An interesting application of the lemma is the following pair of disentangling bounds. The first inequality appeared in [1, Proposition 5.2], the second is new.

Corollary A.2 For every $\beta > 0$, every four vertices $x, y, z, t \in \mathbb{Z}^d$ and every set $S \subset \mathbb{Z}^d$,

---PAGE_BREAK---

Proof Fix $\beta > 0$, $\Lambda$ finite (the claim will then follow by letting $\Lambda$ tend to $\mathbb{Z}^d$) and drop $\beta$ from the notation. For the first identity, introduce the random variable

$$\mathbf{C}=\mathbf{C}({\mathbf{n}}_1,{\mathbf{n}}_2,{\mathbf{n}}_3):={\mathbf{C}}_{{\mathbf{n}}_3}({\mathbf{C}}_{{\mathbf{n}}_1+{\mathbf{n}}_2}(x))\mathrm{.}(\text{A.6})\backslash mathbf\{C\}\; =\; \backslash mathbf\{C\}(\backslash mathbf\{n\}\_1,\; \backslash mathbf\{n\}\_2,\; \backslash mathbf\{n\}\_3)\; :=\; \backslash mathbf\{C\}\_\{\backslash mathbf\{n\}\_3\}(\backslash mathbf\{C\}\_\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\}(x)).\; \backslash quad\; (\backslash text\{A.6\})$$

Lemma A.1 applied in the first and third lines, Griffiths’ inequality [22], and the trivial inclusion $\mathbf{C}_{\mathbf{n}_1+\mathbf{n}_2}(x) \subset \mathbf{C}$ in the second, give

which gives the first inequality.

The second identity requires two successive applications of Lemma A.1. First, conditioning on $\mathbf{n}_2 + \mathbf{n}4$, the proposition applied to $S := \mathbf{C}{\mathbf{n}_2+\mathbf{n}_4}(0) \cap S$ gives

$${\mathbf{P}}_{\mathrm{\Lambda }}^{0x,0z,\mathrm{\varnothing },0t}[{\mathbf{C}}_{n_1+n_3}(0)\cap {\mathbf{C}}_{n_2+n_4}(0)\cap S\mathrm{\ne }\mathrm{\varnothing }]\le {\mathbf{P}}_{\mathrm{\Lambda }}^{0x,0z,0y,0t}[{\mathbf{C}}_{n_1+n_3}(0)\cap {\mathbf{C}}_{n_2+n_4}(0)\cap S\mathrm{\ne }\mathrm{\varnothing }]\mathrm{.}(\text{A.8})\backslash mathbf\{P\}\_\{\backslash Lambda\}^\{0x,0z,\backslash emptyset,0t\}[\backslash mathbf\{C\}\_\{n\_1+n\_3\}(0)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_2+n\_4\}(0)\; \backslash cap\; S\; \backslash neq\; \backslash emptyset]\; \backslash leq\; \backslash mathbf\{P\}\_\{\backslash Lambda\}^\{0x,0z,0y,0t\}[\backslash mathbf\{C\}\_\{n\_1+n\_3\}(0)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_2+n\_4\}(0)\; \backslash cap\; S\; \backslash neq\; \backslash emptyset].\; \backslash quad\; (\backslash text\{A.8\})$$

Similarly, conditioning on $\mathbf{n}_1 + \mathbf{n}3$, the proposition applied to $S' := \mathbf{C}{\mathbf{n}_1+\mathbf{n}_3}(0) \cap S$ gives

$${\mathbf{P}}_{\mathrm{\Lambda }}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[{\mathbf{C}}_{n_1+n_3}(0)\cap {\mathbf{C}}_{n_2+n_4}(0)\cap S\mathrm{\ne }\mathrm{\varnothing }]\le {\mathbf{P}}_{\mathrm{\Lambda }}^{0x,0z,\mathrm{\varnothing },\mathrm{\varnothing }}[\mathrm{\mid }{\mathbf{C}}_{n_1+n_3}(0)\cap {\mathbf{C}}_{n_2+n_4}(0)\mathrm{\mid }S\mathrm{\ne }\mathrm{\varnothing }],(\text{A.9})\backslash mathbf\{P\}\_\{\backslash Lambda\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[\backslash mathbf\{C\}\_\{n\_1+n\_3\}(0)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_2+n\_4\}(0)\; \backslash cap\; S\; \backslash neq\; \backslash emptyset]\; \backslash leq\; \backslash mathbf\{P\}\_\{\backslash Lambda\}^\{0x,0z,\backslash emptyset,\backslash emptyset\}[|\backslash mathbf\{C\}\_\{n\_1+n\_3\}(0)\; \backslash cap\; \backslash mathbf\{C\}\_\{n\_2+n\_4\}(0)|\; S\; \backslash neq\; \backslash emptyset],\; \backslash quad\; (\backslash text\{A.9\})$$

thus concluding the proof. $\square$

A.2 Multi-point connectivity probabilities

The following two relations facilitate the derivation of estimates guided by the random walk analogy.

Proposition A.3 For every $x, u, v \in \mathbb{Z}^d$, we have that

The equality (A.10) is a direct consequence of the switching lemma and has been used several times in the past. The inequality (A.11) is an important new addition, which is proven below. Its structure suggests a more general k-step random walk type bound, but the present proof does not extend to k > 2. In particular, if a k-step bound could be proven for every k, it would improve the concentration estimate for $\mathcal{N}$ in the proof of mixing from an inverse logarithmic bound to a small polynomial one, which would translate into a similar bound for the mixing property which may be very useful for the study of the critical regime. Note that this would not improve the log correction in our result since the intersection property also requires the $\ell_k$ to grow fast. ---PAGE_BREAK---

Proof Fix $\beta > 0$ and drop it from the notation. We work with finite $\Lambda$ and then take the limit as $\Lambda$ tends to $\mathbb{Z}^d$. In the whole proof, $\leftrightarrow$ denotes the connection in $\mathbf{n}_1 + \mathbf{n}_2$, and $\leftarrow\to$ denotes the absence of connection. As mentioned above, (A.10) follows readily from the switching lemma. To prove (A.11), use the switching lemma to find

$${\mathbf{P}}_{\mathrm{\Lambda }}^{0x,\mathrm{\varnothing }}[u,v\leftrightarrow 0]=\frac{⟨{\sigma }_0{\sigma }_u{⟩}_{\mathrm{\Lambda }}⟨{\sigma }_u{\sigma }_x{⟩}_{\mathrm{\Lambda }}}{⟨{\sigma }_0{\sigma }_x{⟩}_{\mathrm{\Lambda }}}{\mathbf{P}}_{\mathrm{\Lambda }}^{0u,ux}[v\leftrightarrow u]\mathrm{.}(\text{A.12})\backslash mathbf\{P\}\_\{\backslash Lambda\}^\{0x,\backslash emptyset\}[u,v\; \backslash leftrightarrow\; 0]\; =\; \backslash frac\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_u\; \backslash rangle\_\{\backslash Lambda\}\; \backslash langle\; \backslash sigma\_u\; \backslash sigma\_x\; \backslash rangle\_\{\backslash Lambda\}\}\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\_\{\backslash Lambda\}\}\; \backslash mathbf\{P\}\_\{\backslash Lambda\}^\{0u,ux\}[v\; \backslash leftrightarrow\; u].\; \backslash quad\; (\backslash text\{A.12\})$$

Then, our goal is to show that

$${\mathbf{P}}_{\mathrm{\Lambda }}^{0u,ux}[v\leftrightarrow u]\le {\mathbf{P}}_{\mathrm{\Lambda }}^{0u,\mathrm{\varnothing }}[v\leftrightarrow u]+{\mathbf{P}}_{\mathrm{\Lambda }}^{\mathrm{\varnothing },ux}[v\leftrightarrow u]-{\mathbf{P}}_{\mathrm{\Lambda }}^{\mathrm{\varnothing },\mathrm{\varnothing }}[v\leftrightarrow u](\text{A.13})\backslash mathbf\{P\}\_\{\backslash Lambda\}^\{0u,ux\}[v\; \backslash leftrightarrow\; u]\; \backslash leq\; \backslash mathbf\{P\}\_\{\backslash Lambda\}^\{0u,\backslash emptyset\}[v\; \backslash leftrightarrow\; u]\; +\; \backslash mathbf\{P\}\_\{\backslash Lambda\}^\{\backslash emptyset,ux\}[v\; \backslash leftrightarrow\; u]\; -\; \backslash mathbf\{P\}\_\{\backslash Lambda\}^\{\backslash emptyset,\backslash emptyset\}[v\; \backslash leftrightarrow\; u]\; \backslash quad\; (\backslash text\{A.13\})$$

which implies (A.11) readily using (A.10). In order to show (A.13), set $\mathbf{C} = \mathbf{C}_{n_1+n_2}(v)$ and apply Lemma A.1 to $F(\mathbf{n}_1, \mathbf{n}_2) := \mathbb{I}[u \leftrightarrow v]$ to obtain

$${\mathbf{P}}_{\mathrm{\Lambda }}^{0u,ux}[u\leftrightarrow v]={\mathbf{E}}_{\mathrm{\Lambda }}^{0u,\mathrm{\varnothing }}[\mathbb{I}[u\leftrightarrow v]\frac{⟨{\sigma }_0{\sigma }_y{⟩}_{\mathrm{\Lambda }\setminus C}}{⟨{\sigma }_0{\sigma }_y{⟩}_{\mathrm{\Lambda }}}]\mathrm{.}(\text{A.14})\backslash mathbf\{P\}\_\{\backslash Lambda\}^\{0u,ux\}[u\; \backslash leftrightarrow\; v]\; =\; \backslash mathbf\{E\}\_\{\backslash Lambda\}^\{0u,\backslash emptyset\}[\backslash mathbb\{I\}[u\; \backslash leftrightarrow\; v]\; \backslash frac\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_y\; \backslash rangle\_\{\backslash Lambda\; \backslash setminus\; C\}\}\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_y\; \backslash rangle\_\{\backslash Lambda\}\}].\; \backslash quad\; (\backslash text\{A.14\})$$

Next, apply Lemma A.1 to

$$F({\mathbf{n}}_1,{\mathbf{n}}_2):=\mathbb{I}[u\leftrightarrow v](1-\frac{⟨{\sigma }_0{\sigma }_x{⟩}_{\mathrm{\Lambda }\setminus C}}{⟨{\sigma }_0{\sigma }_x{⟩}_{\mathrm{\Lambda }}})\ge 0(\text{A.15})F(\backslash mathbf\{n\}\_1,\; \backslash mathbf\{n\}\_2)\; :=\; \backslash mathbb\{I\}[u\; \backslash leftrightarrow\; v]\; \backslash left(1\; -\; \backslash frac\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\_\{\backslash Lambda\; \backslash setminus\; C\}\}\{\backslash langle\; \backslash sigma\_0\; \backslash sigma\_x\; \backslash rangle\_\{\backslash Lambda\}\}\backslash right)\; \backslash geq\; 0\; \backslash quad\; (\backslash text\{A.15\})$$

(the inequality is due to Griffiths’ inequality [22]) to obtain (A.13) thanks to the following inequalities

□

Remark A.4 Griffiths’ inequality [22] plugged in (A.14) gives

$${\mathbf{P}}^{0u,ux}[v\leftrightarrow u]\ge {\mathbf{P}}^{0u,\mathrm{\varnothing }}[v\leftrightarrow u]\mathrm{.}(\text{A.17})\backslash mathbf\{P\}^\{0u,ux\}[v\; \backslash leftrightarrow\; u]\; \backslash geq\; \backslash mathbf\{P\}^\{0u,\backslash emptyset\}[v\; \backslash leftrightarrow\; u].\; \backslash quad\; (\backslash text\{A.17\})$$

Remark A.5 The inequalities (A.13) and (A.17) can be extended to every set $S \subset \mathbb{Z}^d$ and every two vertices $x, y \in \mathbb{Z}^d$:

$${\mathbf{P}}_{\beta }^{0x,\mathrm{\varnothing }}[0S]\le {\mathbf{P}}_{\beta }^{0x,0y}[0S]\le {\mathbf{P}}_{\beta }^{0x,\mathrm{\varnothing }}[0S]+{\mathbf{P}}_{\beta }^{\mathrm{\varnothing },0y}[0S]-{\mathbf{P}}_{\beta }^{\mathrm{\varnothing },\mathrm{\varnothing }}[0S]\mathrm{.}(\text{A.18})\backslash mathbf\{P\}\_\{\backslash beta\}^\{0x,\backslash emptyset\}[0\; \backslash underset\{n\_1+n\_2\}\{\backslash stackrel\{n\_1+n\_2\}\{\backslash rightleftharpoons\}\}\; S]\; \backslash leq\; \backslash mathbf\{P\}\_\{\backslash beta\}^\{0x,0y\}[0\; \backslash underset\{n\_1+n\_2\}\{\backslash stackrel\{n\_1+n\_2\}\{\backslash rightleftharpoons\}\}\; S]\; \backslash leq\; \backslash mathbf\{P\}\_\{\backslash beta\}^\{0x,\backslash emptyset\}[0\; \backslash underset\{n\_1+n\_2\}\{\backslash stackrel\{n\_1+n\_2\}\{\backslash rightleftharpoons\}\}\; S]\; +\; \backslash mathbf\{P\}\_\{\backslash beta\}^\{\backslash emptyset,0y\}[0\; \backslash underset\{n\_1+n\_2\}\{\backslash stackrel\{n\_1+n\_2\}\{\backslash rightleftharpoons\}\}\; S]\; -\; \backslash mathbf\{P\}\_\{\backslash beta\}^\{\backslash emptyset,\backslash emptyset\}[0\; \backslash underset\{n\_1+n\_2\}\{\backslash stackrel\{n\_1+n\_2\}\{\backslash rightleftharpoons\}\}\; S].\; \backslash quad\; (\backslash text\{A.18\})$$

A.3 The spectral representation

In Section 5.3 we make use of a spectral representation of the correlation function $S(x) := (\tau_0\tau_x)$. Though the statement is well known, cf. [21] and references therein, for completeness of the presentation following is its derivation. For the present purpose it is convenient to present the system’s Hamiltonian as the semi-definite function

$$\begin{array}{cccc}& H=-\sum _{x,y}{J}_{x,y}({\tau }_x-{\tau }_y{)}^2\mathrm{.}& & \text{(A.19)}\end{array}H\; =\; -\; \backslash sum\_\{x,y\}\; J\_\{x,y\}\; (\backslash tau\_x\; -\; \backslash tau\_y)^2\; .\; \backslash tag\{A.19\}$$ ---PAGE_BREAK---

The difference from the expressions used elsewhere in the paper are the diagonal quadratic terms $\tau_x^2$ whose effect on the Gibbs measure can be incorporated by an adjustment in the spins' a-priori distribution (which is doable as we assumed in the first place that the site distribution was satisfying (2.2)).

To avoid burdensome notation, when the domain over which the spins are defined is clear from the context of the discussion we shall use the symbol $\tau = {\tau_x}_x$ to denote the entire collection of spins in that region, and by $\rho_0(d\tau)$ the corresponding product measure.

Proposition A.6 (Spectral Representation) Let $\rho_0$ be a single variable distribution for which the Gibbs states on $\mathbb{Z}^d$ with the n.n.f. Hamiltonian satisfies

Then, for every $0 < \beta < \infty$ and every square-summable $v \in \ell^2(\mathbb{Z}^{d-1})$, there exists a positive measure $\mu_{v,\beta}$ with a total mass satisfying

$$\begin{array}{cccc}& {\mu }_{v,\beta }([0,\mathrm{\infty }))\le \mathrm{\parallel }v{\mathrm{\parallel }}_2^2⟨\mathrm{\mid }{\tau }_0{\mathrm{\mid }}^2{⟩}_{\beta }& & \text{(A.21)}\end{array}\backslash mu\_\{v,\backslash beta\}([0,\; \backslash infty))\; \backslash le\; \backslash |v\backslash |\_2^2\; \backslash langle\; |\backslash tau\_0|^2\; \backslash rangle\_\backslash beta\; \backslash tag\{A.21\}$$

such that for every $n \in \mathbb{Z}$,

$$\begin{array}{cccc}& \sum _{x_{\perp },y_{\perp }\in {\mathbb{Z}}^{d-1}}{v}_{x_{\perp }}\overline{y_{y_{\perp }}}{S}_{\beta }((n,x_{\perp }-y_{\perp }))={\int }_0^{\mathrm{\infty }}{e}^{-a\mathrm{\mid }n\mathrm{\mid }}d{\mu }_{v,\beta }(a)\mathrm{.}& & \text{(A.22)}\end{array}\backslash sum\_\{x\_\backslash perp,\; y\_\backslash perp\; \backslash in\; \backslash mathbb\{Z\}^\{d-1\}\}\; v\_\{x\_\backslash perp\}\; \backslash overline\{y\_\{y\_\backslash perp\}\}\; S\_\backslash beta((n,\; x\_\backslash perp\; -\; y\_\backslash perp))\; =\; \backslash int\_0^\backslash infty\; e^\{-a|n|\}\; d\backslash mu\_\{v,\backslash beta\}(a).\; \backslash tag\{A.22\}$$

For $\beta < \beta_c$ the measure' support is limited to $a \ge 1/\xi(\beta)$ (here $\xi(\beta)$ is the correlation length of the system).

In particular, with $v = \delta_\perp$ the Kronecker function (at the origin) on $\mathbb{Z}^{d-1}$, this yields the following spectral representation for the correlation function along a principal axis

$$\begin{array}{cccc}& S_{\beta }((n,0_{\perp }))={\int }_{1\mathrm{/}\xi (\beta )}^{\mathrm{\infty }}{e}^{-a\mathrm{\mid }n\mathrm{\mid }}d{\mu }_{{\delta }_{\perp },\beta }(a),& & \text{(A.23)}\end{array}S\_\backslash beta((n,\; 0\_\backslash perp))\; =\; \backslash int\_\{1/\backslash xi(\backslash beta)\}^\backslash infty\; e^\{-a|n|\}\; d\backslash mu\_\{\backslash delta\_\backslash perp,\; \backslash beta\}(a),\; \backslash tag\{A.23\}$$

with a measure whose total mass is $\mu_{\delta_\perp, \beta}([0, \infty)) = (\lvert\tau_0\rvert^2)_\beta$.

Proof Throughout the proof $\beta$ is held constant, and to a large extent will be omitted from the notation. It is convenient to first derive the corresponding statements for finite volume versions of the model, in tubular domains with periodic boundary conditions $\mathbb{T}(m, \ell) := (\mathbb{Z}/m\mathbb{Z}) \times (\mathbb{Z}/\ell\mathbb{Z})^{d-1}$ (with the notational convention $\mathbb{Z}/\infty\mathbb{Z} = \mathbb{Z}$). The corresponding finite volume correlation function is naturally denoted $S_{m,\ell;\beta}(x) := \langle \tau_0\tau_x \rangle_{\mathbb{T}(m,\ell)}$.

Let $\mathcal{V}\ell$ be the $\mathbb{C}$-vector space of $L^2(\otimes{x\in(\mathbb{Z}/\ell\mathbb{Z})^{d-1}} \rho(d\tau_x))$ of functions supported on the transversal hyperplane $(\mathbb{Z}/\ell\mathbb{Z})^{d-1}$, over the product measure $\otimes\rho(d\tau_x)$. On $\mathcal{V}_\ell$, let $T_\ell$ be the self adjoint operator whose kernel is given by

$$\begin{array}{cccc}& T_{\mathrm{\ell }}(\tau ,{\tau }^{\mathrm{\prime }}):=\exp \{-\frac{\beta J}{4}\sum _{\begin{array}{c}x,y\subset (\mathbb{Z}\mathrm{/}\mathrm{\ell }\mathbb{Z}{)}^{d-1}\\ \{x,y\}\text{ edge}\end{array}}[({\tau }_x-{\tau }_y{)}^2+({\tau }_x^{\mathrm{\prime }}-{\tau }_y^{\mathrm{\prime }}{)}^2]-\frac{\beta J}{2}\sum _{x\in (\mathbb{Z}\mathrm{/}\mathrm{\ell }\mathbb{Z}{)}^{d-1}}({\tau }_x-{\tau }_x^{\mathrm{\prime }}{)}^2\}\mathrm{.}& & \text{(A.24)}\end{array}T\_\backslash ell(\backslash tau,\; \backslash tau\text{'})\; :=\; \backslash exp\; \backslash left\backslash \{\; -\backslash frac\{\backslash beta\; J\}\{4\}\; \backslash sum\_\{\backslash substack\{x,y\; \backslash subset\; (\backslash mathbb\{Z\}/\backslash ell\backslash mathbb\{Z\})^\{d-1\}\; \backslash \backslash \; \backslash \{x,y\backslash \}\; \backslash text\{\; edge\}\}\}\; [(\tau \_x\; -\; \tau \_y)^2\; +\; (\tau \text{'}\_x\; -\; \tau \text{'}\_y)^2]\; -\; \backslash frac\{\backslash beta\; J\}\{2\}\; \backslash sum\_\{x\; \backslash in\; (\backslash mathbb\{Z\}/\backslash ell\backslash mathbb\{Z\})^\{d-1\}\}\; (\tau \_x\; -\; \tau \text{'}\_x)^2\; \backslash right\backslash \}.\; \backslash tag\{A.24\}$$

This operator serves as the “transfer matrix” in terms of which the partition function can be presented as a trace:

$$\begin{array}{cccc}& Z_{m,\mathrm{\ell }}:=\mathrm{Tr}(T_{\mathrm{\ell }}^m)\mathrm{.}& & \text{(A.25)}\end{array}Z\_\{m,\backslash ell\}\; :=\; \backslash operatorname\{Tr\}(T\_\{\backslash ell\}^\{m\}).\; \backslash tag\{A.25\}$$

To express the correlations functions, let us consider the multiplication operators

$$\begin{array}{cccc}& \tau [v]:=\sum _{x\in (\mathbb{Z}\mathrm{/}\mathrm{\ell }\mathbb{Z}{)}^{d-1}}v(x){\tau }_x& & \text{(A.26)}\end{array}\backslash tau[v]\; :=\; \backslash sum\_\{x\; \backslash in\; (\backslash mathbb\{Z\}/\backslash ell\backslash mathbb\{Z\})^\{d-1\}\}\; v(x)\backslash tau\_x\; \backslash tag\{A.26\}$$ ---PAGE_BREAK---

associated with square summable functions $v : (\mathbb{Z}/\ell\mathbb{Z})^{d-1} \to \mathbb{C}$.

In this notation, the correlation function of spins at sites (which we write as $(n, x_\perp) \in T_{m,\ell}$) satisfy

$$\sum _{x_1,y_1\in (\mathbb{Z}\mathrm{/}\mathrm{\ell }\mathbb{Z}{)}^{d-1}}\overline{v_{y_1}}{v}_{x_1}{S}_{m,\mathrm{\ell };\beta }((n,y_{\perp }-x_{\perp }))=\frac{\mathrm{Tr}(T_{\mathrm{\ell }}^{m-n}\bar{\tau }[v]T_{\mathrm{\ell }}^n\tau [v])}{\mathrm{Tr}(T_{\mathrm{\ell }}^m)}\mathrm{.}(\text{A.27})\backslash sum\_\{x\_1,\; y\_1\; \backslash in\; (\backslash mathbb\{Z\}/\backslash ell\backslash mathbb\{Z\})^\{d-1\}\}\; \backslash overline\{v\_\{y\_1\}\}\; v\_\{x\_1\}\; S\_\{m,\backslash ell;\backslash beta\}((n,\; y\_\backslash perp\; -\; x\_\backslash perp))\; =\; \backslash frac\{\backslash operatorname\{Tr\}(T\_\backslash ell^\{m-n\}\; \backslash bar\{\backslash tau\}[v]\; T\_\backslash ell^n\; \backslash tau[v])\}\{\backslash operatorname\{Tr\}(T\_\backslash ell^m)\}\; .\; \backslash quad\; (\backslash text\{A.27\})$$

We next claim that for any $\ell < \infty$ the operator $T_\ell$ is:

(i) self adjoint and compact (and thus with spectrum which is discrete, except for possible accumulation at 0);

(ii) positive definite;

(iii) non-degenerate at the top of its spectrum, with a strictly positive eigenfunction.

Item (i) is implied by the kernel’s symmetry and the finiteness of its Hilbert-Schmidt norm:

$$\mathrm{T}\mathrm{r}{T}_{\mathrm{\ell }}^{\ast }{T}_{\mathrm{\ell }}=\iint \rho (d\tau )\rho (d{\tau }^{\mathrm{\prime }})\mathrm{\mid }{T}_{\mathrm{\ell }}(\tau ,{\tau }^{\mathrm{\prime }}){\mathrm{\mid }}^2\le 1.(\text{A.28})\backslash mathrm\{Tr\}\; T\_\{\backslash ell\}^\{*\}\; T\_\{\backslash ell\}\; =\; \backslash iint\; \backslash rho(d\backslash tau)\; \backslash rho(d\backslash tau\text{'})\; |T\_\{\backslash ell\}(\backslash tau,\; \backslash tau\text{'})|^2\; \backslash le\; 1.\; \backslash qquad\; (\backslash text\{A.28\})$$

Positivity (ii) can be deduced by the criteria of [19] (see also [10]) applied to the reflection symmetry with respect to the hyperplanes passing through mid-edges. The last assertion (iii) is implied by (i) combined with the kernel’s pointwise positivity (cf. Krein-Rutman theorem [33]).

Rewritten in terms of the spectral representation of $T_\ell$, (A.27) takes the form:

$$\sum _{x_1,y_1\in {\mathbb{Z}}^{d-1}}{v}_{x_1}\overline{v_{y_1}}{S}_{m,l;\beta }((n,x_{\perp }-y_{\perp }))=\frac{\sum _{{\lambda }_1,{\lambda }_2\in \text{Spec}(T_l)}{\lambda }_1^{m-n}{\lambda }_2^n⟨e_{{\lambda }_1}\mathrm{\mid }\bar{\tau }[v]\mathrm{\mid }{e}_{{\lambda }_2}⟩⟨e_{{\lambda }_2}\mathrm{\mid }\tau [v]\mathrm{\mid }{e}_{{\lambda }_1}⟩}{\sum _{\lambda \in \text{Spec}(T_l)}{\lambda }^m},(A.29)\backslash sum\_\{x\_1,\; y\_1\; \backslash in\; \backslash mathbb\{Z\}^\{d-1\}\}\; v\_\{x\_1\}\; \backslash overline\{v\_\{y\_1\}\}\; S\_\{m,l;\backslash beta\}((n,\; x\_\{\backslash perp\}\; -\; y\_\{\backslash perp\}))\; =\; \backslash frac\{\backslash sum\_\{\backslash lambda\_1,\; \backslash lambda\_2\; \backslash in\; \backslash text\{Spec\}(T\_l)\}\; \backslash lambda\_1^\{m-n\}\; \backslash lambda\_2^n\; \backslash langle\; e\_\{\backslash lambda\_1\}\; |\; \backslash bar\{\backslash tau\}[v]\; |\; e\_\{\backslash lambda\_2\}\; \backslash rangle\; \backslash langle\; e\_\{\backslash lambda\_2\}\; |\; \backslash tau[v]\; |\; e\_\{\backslash lambda\_1\}\; \backslash rangle\}\{\backslash sum\_\{\backslash lambda\; \backslash in\; \backslash text\{Spec\}(T\_l)\}\; \backslash lambda^m\},\; \backslash quad\; (A.29)$$

where ${|e_\lambda\rangle}$ is an orthonormal basis of eigenvectors of $T_\ell$. By the structure of the spectrum described above, in the limit $m \to \infty$ only the terms with $\lambda_1 = \lambda_{max}$ and $\lambda = \lambda_{max}$ are of relevance, and one is left with the single sum:

with $e^{-a} = \lambda/\lambda_{\max}$ and $\mu_{v,\beta,\ell}$ the above discrete spectral measure (whose support starts at $\xi(\ell, \beta)$, the inverse rate of decay in $x$ of $S_{\infty,\ell}(x)$).

Next we consider the limit $\ell \to \infty$ at fixed $x_\perp - y_\perp$. It is known, through the FKG inequality, that the correlation function converges pointwise, i.e. for any $(n, x_\perp - y_\perp)$ and $\beta$,

$$S_{\beta }((n,x_{\perp }-y_{\perp }))=\underset{\mathrm{\ell }\to \mathrm{\infty }}{\lim }{S}_{\mathrm{\infty },\mathrm{\ell };\beta }((n,x_{\perp }-y_{\perp }))(\text{A.31})S\_\{\backslash beta\}((n,\; x\_\{\backslash perp\}\; -\; y\_\{\backslash perp\}))\; =\; \backslash lim\_\{\backslash ell\; \backslash to\; \backslash infty\}\; S\_\{\backslash infty,\; \backslash ell;\; \backslash beta\}((n,\; x\_\{\backslash perp\}\; -\; y\_\{\backslash perp\}))\; \backslash quad\; (\backslash text\{A.31\})$$

Through (A.30) this translates into convergence of the moments of $e^{-a}$ under the measures $\mu_{v,\beta,\ell}$. The moment criterion for the convergence of positive measures over bounded intervals (here $[0,1]$) allows to conclude existence of the (weak) limit $\lim_{\ell\to\infty} \mu_{v,\beta,\ell} = \mu_{v,\beta}$ (which need not be a point measure) with which the claimed relation (A.22) holds. $\square$

One may observe that the above result applies to all $\beta$. It may be added that for $\beta < \beta_c$ the spectral measure's support is bounded away from 0. In contrast, for $\beta > \beta_c$ the measures associated with $v$ of non-zero sum include a point mass there, i.e. $\mu_{v,\beta}({0}) \neq 0$, which is the spectral representation of the long range order. ---PAGE_BREAK---

A.4 Intersection properties for random current representation of models in the GS class

We start with the version of the switching lemma that we will use. Below, $\delta_{uv}$ denotes the current equal to 1 on the edge $uv$ and 0 otherwise.

Lemma A.7 (Coarse switching) Let $S,T$ be two disjoint sets of vertices. For every event $E$ depending on the sum of two currents and every $x \neq y$,

$$P_{\beta }^{xy,\mathrm{\varnothing }}[xS,{\mathbf{n}}_1+{\mathbf{n}}_2\in E]\le \beta \sum _{a\in S,b\mathrm{\notin }S}{J}_{a,b}\frac{⟨{\sigma }_x{\sigma }_a⟩⟨{\sigma }_b{\sigma }_y⟩}{⟨{\sigma }_x{\sigma }_y⟩}{P}_{\beta }^{xa,by}[{\mathbf{n}}_1+{\mathbf{n}}_2+{\delta }_{ab}\in E],(A.32)P\_\{\backslash beta\}^\{xy,\backslash emptyset\}[x\; \backslash stackrel\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\}\{\backslash longleftrightarrow\}\; S,\; \backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\; \backslash in\; E]\; \backslash leq\; \backslash beta\; \backslash sum\_\{a\; \backslash in\; S,\; b\; \backslash notin\; S\}\; J\_\{a,b\}\; \backslash frac\{\backslash langle\; \backslash sigma\_x\; \backslash sigma\_a\; \backslash rangle\; \backslash langle\; \backslash sigma\_b\; \backslash sigma\_y\; \backslash rangle\}\{\backslash langle\; \backslash sigma\_x\; \backslash sigma\_y\; \backslash rangle\}\; P\_\{\backslash beta\}^\{xa,by\}[\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2+\backslash delta\_\{ab\}\; \backslash in\; E],\; \backslash quad\; (A.32)$$

$$P_{\beta }^{\mathrm{\varnothing },\mathrm{\varnothing }}[ST,{\mathbf{n}}_1+{\mathbf{n}}_2\in E]\le {\beta }^2\sum _{\begin{array}{c}a\in S,b\mathrm{\notin }S\\ s\in T,t\mathrm{\notin }T\end{array}}{J}_{a,b}{J}_{s,t}⟨{\sigma }_a{\sigma }_s⟩⟨{\sigma }_t{\sigma }_b⟩P_{\beta }^{as,tb}[{\mathbf{n}}_1+{\mathbf{n}}_2+{\delta }_{ab}+{\delta }_{st}\in E]\mathrm{.}(A.33)P\_\{\backslash beta\}^\{\backslash emptyset,\backslash emptyset\}[S\; \backslash stackrel\{\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\}\{\backslash longleftrightarrow\}\; T,\; \backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2\; \backslash in\; E]\; \backslash leq\; \backslash beta^2\; \backslash sum\_\{\backslash substack\{a\; \backslash in\; S,\; b\; \backslash notin\; S\; \backslash \backslash \; s\; \backslash in\; T,\; t\; \backslash notin\; T\}\}\; J\_\{a,b\}\; J\_\{s,t\}\; \backslash langle\; \backslash sigma\_a\; \backslash sigma\_s\; \backslash rangle\; \backslash langle\; \backslash sigma\_t\; \backslash sigma\_b\; \backslash rangle\; P\_\{\backslash beta\}^\{as,tb\}[\backslash mathbf\{n\}\_1+\backslash mathbf\{n\}\_2+\backslash delta\_\{ab\}+\backslash delta\_\{st\}\; \backslash in\; E].\; \backslash quad\; (A.33)$$

Proof We start with the first inequality. Fix $\Lambda$ finite. By multiplying by the quantity $\langle \sigma_x \sigma_y \rangle_\beta 4^{-|\Lambda|} Z(\Lambda, J, \beta)^2$, and then making the change of variable $\mathbf{m} = \mathbf{n}_1 + \mathbf{n}_2$, $\mathbf{n}_2 = \mathbf{n}$, we find that

where in the last line we used that the number of even subgraphs of the multi-graph $\mathcal{M}$ (see for instance definition in [5]) associated with $\mathbf{m}$ is given by $2^{|\mathbf{m}|+k(\mathbf{m})-|\Lambda|}$, where $|\mathbf{m}|$ means the total sum of $\mathbf{m}$, and $k(\mathbf{m})$ is the number of connected components. Now, observe that

$$w_{2\beta }(\mathbf{m})\mathbb{I}[xS]2^{k(\mathbf{m})}\le \sum _{a\mathrm{\notin }S,b\in S}\beta J_{a,b}{w}_{2\beta }(\mathbf{m}-{\delta }_{ab})\mathbb{I}[xb]\mathbb{I}[{\mathbf{m}}_{ab}\ge 1]2^{k(\mathbf{m}-{\delta }_{ab})}\mathrm{.}(A.35)w\_\{2\backslash beta\}(\backslash mathbf\{m\})\backslash mathbb\{I\}[x\; \backslash stackrel\{\backslash mathbf\{m\}\}\{\backslash longleftrightarrow\}\; S]\; 2^\{k(\backslash mathbf\{m\})\}\; \backslash leq\; \backslash sum\_\{a\; \backslash notin\; S,\; b\; \backslash in\; S\}\; \backslash beta\; J\_\{a,b\}\; w\_\{2\backslash beta\}(\backslash mathbf\{m\}-\backslash delta\_\{ab\})\backslash mathbb\{I\}[x\; \backslash stackrel\{\backslash mathbf\{m\}-\backslash delta\_\{ab\}\}\{\backslash longleftrightarrow\}\; b]\backslash mathbb\{I\}[\backslash mathbf\{m\}\_\{ab\}\; \backslash geq\; 1]\; 2^\{k(\backslash mathbf\{m\}-\backslash delta\_\{ab\})\}.\; \backslash quad\; (A.35)$$

Indeed, we are necessarily in one of the following cases: consider the edges $ab$ with $a \in S$, $b \notin S$, and $a$ connected to $y$ in $\mathbf{m}-\delta_{ab}$. Assume that

• there is an edge $ab$ as above with $\mathbf{m}{ab} \ge 2$, in such case $k(\mathbf{m}-\delta{ab}) = k(\mathbf{m})$ and $w_{2\beta}(\mathbf{m}) = \frac{2\beta J_{ab}}{\mathbf{m}{ab}} w{2\beta}(\mathbf{m}-\delta_{ab}) \le \beta J_{a,b} w_{2\beta}(\mathbf{m}-\delta_{ab});$

• there is a loop in the cluster of $x$ in $\mathbf{m}$ which is intersecting the edge-boundary of $S$, in such case there are two edges $ab$ satisfying the property above, with $k(\mathbf{m}-\delta_{ab}) = k(\mathbf{m})$ and $w_{2\beta}(\mathbf{m}) \le 2\beta J_{a,b} w_{2\beta}(\mathbf{m}-\delta_{ab});$

• otherwise, there is only one edge $ab$ with $\mathbf{m}{ab}=1$, in such case $k(\mathbf{m}-\delta{ab}) = k(\mathbf{m})+1$ and $w_{2\beta}(\mathbf{m}) \le 2\beta J_{a,b} w_{2\beta}(\mathbf{m}-\delta_{ab}).$ ---PAGE_BREAK---

Injecting the last displayed inequality in the first one, and then making the change of variable $\mathbf{m}' = \mathbf{m} - \delta_{uv}$, we find that

where in the last line we used the switching lemma. Dividing this relation by the factor $\langle\sigma_x\sigma_y\rangle_{\Lambda,\beta}4^{-|\Lambda|}Z(\Lambda, J, \beta)^2$ and letting $\Lambda$ tend to the full lattice implies the first claim.

The second claim follows from the same reasoning using pairs of edges (ab,st) with $a \in S$, $b \notin S$, $s \in T$ and $t \notin T$ such that $a$ is connected to $s$ in $\mathbf{n}_1 + \mathbf{n}_2$. $\square$

We deduce the following pair of diagrammatic bounds on the connectivity probabilities.

Proposition A.8 For every distinct $x, y, u, v \in \mathbb{Z}^d$

Proof For the first one, sum (A.33) for E being the full event and vertices in $\mathcal{B}_x$ and $\mathcal{B}_y$, and use (A.10). For the second one, do the same with (A.32) instead. $\square$

Acknowledgments The work of M. Aizenman on this project was supported in part by the NSF grant DMS-1613296, and that of H. Duminil-Copin by the NCCR SwissMAP, the Swiss NSF and an IDEX Chair from Paris-Saclay. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 757296). The joint work was advanced through mutual visits to Princeton and Geneva University, sponsored by a Princeton-Unige partnership grant. We thank S. Goswami, A. Raoufi, P.-F. Rodriguez, and F. Severo for stimulating discussions, and M. Oulamara, R. Panis, P. Wildemann, and an anonymous referee for careful reading of the paper.

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