Critical Two-Point Function for Long-Range O(n) Models Below the Upper Critical Dimension

Abstract

We consider the n-component $|\varphi |^4$ lattice spin model ($n \ge 1$) and the weakly self-avoiding walk ($n=0$) on $\mathbb Z^d$, in dimensions $d=1,2,3$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance r as $r^{-(d+\alpha )}$ with $\alpha \in (0,2)$. The upper critical dimension is $d_c=2\alpha$. For $\varepsilon >0$, and $\alpha = \frac{1}{2} (d+\varepsilon )$, the dimension $d=d_c-\varepsilon$ is below the upper critical dimension. For small $\varepsilon$, weak coupling, and all integers $n \ge 0$, we prove that the two-point function at the critical point decays with distance as $r^{-(d-\alpha )}$. This “sticking” of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.

1 Introduction and Main Result

Broadly speaking, the mathematical understanding of critical phenomena for spin systems has progressed in dimension $d=2$, where exact solutions and SLE are important tools; in dimensions $d>4$, where infrared bounds and the lace expansion are useful; and in dimension $d=4$, where renormalisation group (RG) methods have been applied. The physically most important case of $d=3$ is more difficult, and mathematical methods are scarce.

In the physics literature, the $\varepsilon$-expansion was introduced to study non-integer dimensions slightly below $d=4$. An alternate approach is to consider long-range models, which change the upper critical dimension from $d_c=4$ to a lower value $d_c=2\alpha$ with $\alpha \in (0,2)$. By choosing $d=1,2,3$ and $\alpha = \frac{1}{2} (d+\varepsilon )$ with small $\varepsilon$, it is possible to study integer dimension d which is slightly below the upper critical dimension $2\alpha = d+\varepsilon$. In this paper, we consider n-component spins and the weakly self-avoiding walk in this long-range context, and prove that the critical two-point function has mean-field decay $r^{-(d-\alpha )}$ also below the upper critical dimension. Our method involves a RG analysis in the vicinity of a non-Gaussian fixed point.

1.1 Introduction

We consider long-range O(n) models on $\mathbb Z^{d}$ for integers $n \ge 0$ and dimensions $d=1,2,3$. The case $n=0$ is the continuous-time weakly self-avoiding walk, and the case $n \ge 1$ is the n-component $|\varphi |^4$ lattice spin model. For $n=0$ the underlying random walk model takes steps of length r with probabilities decaying as $r^{-(d+\alpha )}$ with $\alpha \in (0,2)$, and for $n \ge 1$ the spin-spin interaction in the Hamiltonian has that same decay. More precisely, the models are based on the fractional Laplacian $(-{\varDelta })^{\alpha /2}$, whose kernel decays at large distance as $r^{-(d+\alpha )}$.

The upper critical dimension is predicted to be $d_c=2\alpha$ for all $n \ge 0$. Thus, for $\alpha < \frac{d}{2}$, mean-field behaviour is predicted; this has been proved for self-avoiding walk, for the Ising model, for the 1-component $\varphi ^4$ model, and for other models [3, 17, 22, 23]. In the physics literature, it is observed that below the upper critical dimension the critical two-point function continues to exhibit the mean-field decay $r^{-(d-\alpha )}$ for $\alpha \in (\frac{d}{2},2-\eta )$, and then crosses over to $r^{-(d-2+\eta )}$ decay for $\alpha \in (2-\eta ,2)$. Here $\eta$ is the exponent for the nearest-neighbour model; for $n=1$ this is $\eta = \frac{1}{4}$ for $d=2$ [35], and a recent estimate for $d=3$ is $\eta = 0.03631(3)$ [18]. The earliest paper to elucidate the critical behaviour of long-range models is [20], with [33] roughly contemporaneous and [29] providing further development. A very recent paper which analyses the crossover for the two-point function in detail for $n=1$ is [8]. At the crossover, when $\alpha =\alpha _*=2-\eta$, a logarithmic correction is predicted, with overall decay $\frac{1}{r^{d-\alpha _*}}\frac{1}{\log r}$ [8, 11]. The relationship with conformal invariance is explained in [28].

Let $n = 0,1,2,\ldots$; $d=1,2,3$; and $\alpha = \frac{1}{2} (d+\varepsilon )$. We use a rigorous RG argument to prove that for small $\varepsilon >0$, the critical two-point function has decay $r^{-(d-\alpha )}$. This proves the “sticking” of the critical exponent at its mean-field value, for $\alpha$ slightly above $\frac{d}{2}$, or equivalently, for d slightly below the upper critical dimension $d_c=2\alpha$. Our proof extends recent results and methods used to study the $\varepsilon$-expansion for the critical exponents for the susceptibility and specific heat of the long-range models [31]. It also relies on results and techniques developed to study related problems for the 4-dimensional nearest-neighbour models [5, 16, 32]. However, our treatment of observables differs from that used in the 4-dimensional case, via our application of a cluster expansion.

Earlier mathematical work which applies RG methods to long-range models includes the construction of global RG trajectories for $n=0$ and $d=3$ [27], and for a continuum version of the $n=1$ model in [1, 12]. These references do not study critical exponents. The exponents for critical correlations in a certain hierarchical version of the model, for $d=3$ and $n=1$, are computed in [2]. For a closely related continuum model with $n=1$ in dimensions $d=2,3$, a proof of the “sticking” of the critical exponent for the critical two-point function was announced in a 2013 lecture [24].

1.2 Fractional Laplacian

The models we study are defined in terms of the fractional Laplacian. We now define the fractional Laplacian and list some of its properties. Further details can be found in [31, Sects. 2–3].

Let $d \ge 1$ and $\alpha \in (0,2)$. We write |x| for the Euclidean norm of $x\in \mathbb Z^d$. Let J be the $\mathbb Z^{d} \times \mathbb Z^{d}$ matrix with $J_{xy}=1$ if $|x-y|=1$, and otherwise $J_{xy}=0$. Let I denote the identity matrix. The lattice Laplacian on $\mathbb Z^{d}$ is ${\varDelta }= J-2dI$. For $k = (k_1,\ldots ,k_d) \in [-\pi ,\pi ]^d$, let

$$\begin{aligned} \lambda (k) = 4 \sum _{j=1}^d \sin ^2 (k_j/2) = 2 \sum _{j=1}^d \left( 1-\cos k_j\right) . \end{aligned}$$

(1.1)

The matrix element $-{\varDelta }_{x,y}$ can be written as the Fourier integral

$$\begin{aligned} -{\varDelta }_{x,y} = \frac{1}{(2\pi )^d}\int _{[-\pi ,\pi ]^d} \lambda (k) e^{ik\cdot (x-y)}dk. \end{aligned}$$

(1.2)

The fractional Laplacian is the matrix $(-{\varDelta })^{\alpha /2}$ defined by

$$\begin{aligned} \left( -{\varDelta }\right) ^{\alpha /2}_{x,y}= \frac{1}{(2\pi )^d}\int _{[-\pi ,\pi ]^d} \lambda (k)^{\alpha /2} e^{ik\cdot ( x-y)}dk. \end{aligned}$$

(1.3)

For $|x-y| \rightarrow \infty$, the fractional Laplacian decays as

$$\begin{aligned} -(-{\varDelta })^{\alpha /2}_{x,y} \asymp |x-y|^{-(d+\alpha )} \end{aligned}$$

(1.4)

(see [31, Lemma 2.1], or [10, Theorem 5.3] for a more precise and more general statement). Here, and in the following, we write $a \asymp b$ to denote the existence of $c>0$ such that $c^{-1} b \le a \le c b$. For $d \ge 1$, $\alpha \in (0,2\wedge d)$, $\bar{m}^2>0$, $m^2 \in [0,\bar{m}^2]$, and $x \ne 0$, the resolvent obeys

$$\begin{aligned} \left( \left( -{\varDelta }\right) ^{\alpha /2}+m^2\right) ^{-1}_{0,x}\le c \frac{1}{|x|^{d-\alpha }} \frac{1}{1+m^4|x|^{2\alpha }}, \end{aligned}$$

(1.5)

with c depending on $d,\alpha ,\bar{m}^2$ (see [31, Lemma 3.2]). For $m^2=0$, an asymptotic formula

$$\begin{aligned} \left( (-{\varDelta })^{\alpha /2}\right) _{0,x}^{-1}\sim c_{d,\alpha } \frac{1}{|x|^{d-\alpha }} \end{aligned}$$

(1.6)

is proven in [9, Theorem 2.4], with precise constant $c_{d,\alpha }$.

Given integers $L,N >1$, let ${\varLambda }= {\varLambda }_N = \mathbb Z^d/L^N\mathbb Z^d$ denote the d-dimensional discrete torus of side length $L^N$. The torus fractional Laplacian is defined by

$$\begin{aligned} \left( -{\varDelta }_{{\varLambda }_N}\right) ^{\alpha /2}_{x,y}= \sum _{z \in \mathbb Z^{d}} (-{\varDelta })^{\alpha /2}_{x,y+zL^N} \quad (x,y\in {\varLambda }_N). \end{aligned}$$

(1.7)

The sum on the right-hand side of (1.7) converges, by (1.4).

1.3 The $|\varphi |^4$ Model

We first define the model on the torus ${\varLambda }={\varLambda }_N$, as usual for spin systems. Let $d \ge 1$ and $\alpha \in (0,2)$. Let $n \ge 1$. The spin field $\varphi$ is a function $\varphi : {\varLambda }\rightarrow \mathbb R^n$, denoted $x \mapsto \varphi _x$, which we may regard as an element $\varphi \in (\mathbb R^n)^{\varLambda }$. The Euclidean norm of $v =(v^1,\ldots ,v^n)\in \mathbb R^n$ is $|v| = [\sum _{i=1}^n (v^i)^2]^{1/2}$, with inner product $v \cdot w = \sum _{i=1}^n v^i w^i$. We extend the action of the fractional Laplacian to act on the spin field component-wise, namely $((-{\varDelta }_{\varLambda })^{\alpha /2} \varphi )_x^i = \sum _{y\in {\varLambda }} (-{\varDelta }_{\varLambda })^{\alpha /2}_{x,y} \varphi ^i_y$.

Given $g>0$ and $\nu \in \mathbb R$, we define the interaction $V: (\mathbb R^n)^{\varLambda }\rightarrow \mathbb R$ by

$$\begin{aligned} V(\varphi ) = \sum _{x\in {\varLambda }} \left( \tfrac{1}{4} g |\varphi _x|^4 + \tfrac{1}{2}\nu |\varphi _x|^2 + \tfrac{1}{2}\varphi _x \cdot ((-{\varDelta }_{\varLambda })^{\alpha /2} \varphi )_x \right) . \end{aligned}$$

(1.8)

The partition function is defined by

$$\begin{aligned} Z_{g,\nu ,N} = \int _{(\mathbb R^n)^{{\varLambda }}} e^{-V(\varphi )} d\varphi , \end{aligned}$$

(1.9)

where $d\varphi$ is the Lebesgue measure on $(\mathbb R^n)^{{\varLambda }}$. The expectation of a random variable $F:(\mathbb R^n)^{{\varLambda }} \rightarrow \mathbb R$ is

$$\begin{aligned} \langle F \rangle _{g,\nu ,N} = \frac{1}{Z_{g,\nu ,N}} \int _{(\mathbb R^n)^{{\varLambda }}} F(\varphi ) e^{-V(\varphi )} d\varphi . \end{aligned}$$

(1.10)

Given lattice points $a,b$, we define the finite- and infinite-volume two-point function by

$$\begin{aligned} G_{a,b, N}(g,\nu ; n)&= \langle {\varphi _a^1 \varphi _b^1}\rangle _{g,\nu , N}= \frac{1}{n} \langle {\varphi _a\cdot \varphi _b}\rangle _{g,\nu , N}, \end{aligned}$$

(1.11)

$$\begin{aligned} G_{a,b}(g,\nu ; n)&= \lim _{N \rightarrow \infty } G_{a,b, N}(g,\nu ; n). \end{aligned}$$

(1.12)

On the left-hand side of (1.12) we have $a,b\in \mathbb Z^{d}$, and on the right-hand side we identify these points with elements of ${\varLambda }_N$ for large N, by regarding the vertices of ${\varLambda }_N$ as a cube in $\mathbb Z^d$ (without boundaries identified) approximately centred at the origin. The susceptibility is defined by

$$\begin{aligned} \chi (g, \nu ; n) = \lim _{N \rightarrow \infty } \sum _{b\in {\varLambda }_N} G_{a,b,N}(g,\nu ;n) \end{aligned}$$

(1.13)

and can be used to identify the critical point of the model. By translation invariance, $\chi$ is independent of a. Existence of the infinite volume limits in (1.12)–(1.13), in our context, is discussed below.

1.4 Weakly Self-Avoiding Walk

Let $d \ge 1$ and $\alpha \in (0,2)$. Let X denote the continuous-time Markov chain with state space $\mathbb Z^{d}$ and infinitesimal generator $Q=-(-{\varDelta }_{\mathbb Z^{d}})^{\alpha /2}$. Verification that Q has the attributes required of a generator is given in [31, Lemma 2.4]. Let P be the probability measure associated with X, and E the corresponding expectation; a subscript a specifies $X(0)=a$. The transition probabilities are given by

(1.14)

The local time of X at x up to time T is the random variable . The self-intersection local time up to time T is the random variable

(1.15)

Given $g>0$, $\nu \in \mathbb R$, and $a,b\in \mathbb Z^{d}$, the continuous-time weakly self-avoiding walk two-point function is defined by the integral

(1.16)

and the susceptibility is defined by

$$\begin{aligned} \chi (g,\nu ;0) = \sum _{b\in \mathbb Z^d} G_{a,b}(g,\nu ;0) = \int _0^\infty E_{a}(e^{-gI_T}) e^{- \nu T} dT. \end{aligned}$$

(1.17)

The labels 0 on the left-hand sides of (1.16)–(1.17) reflect the fact that the weakly self-avoiding walk corresponds to the formal $n=0$ case of the n-component $|\varphi |^4$ model. As in earlier work on the 4-dimensional case, e.g., [31, 32], we treat both cases $n \ge 1$ (spins) and $n=0$ (self-avoiding walk) simultaneously and rigorously, via a supersymmetric spin representation for the weakly self-avoiding walk.

1.5 Susceptibility and Critical Point

Let $d=1,2,3$; $n \ge 0$; L be sufficiently large; $\varepsilon >0$ be sufficiently small; and $\alpha = \frac{1}{2} (d+\varepsilon )$. Let $\tau ^{(\alpha )}$ denote the diagonal element of the Green function, i.e., $\tau ^{(\alpha )} = ((-{\varDelta })^{\alpha /2})^{-1}_{0,0}$. One of the main results of [31] is that there exists $\bar{s}\asymp \varepsilon$ such that, for $g \in [\frac{63}{64}\bar{s},\frac{65}{64}\bar{s}]$, there exist $\nu _c=\nu _c(g;n)= -(n+2) \tau ^{(\alpha )} g(1+O(g))$ and $C>0$ such that for $\nu =\nu _c+t$ with $t \downarrow 0$,

$$\begin{aligned} C^{-1} t^{-(1 + \frac{n+2}{n+8} \frac{\varepsilon }{\alpha } -C\varepsilon ^2)} \le \chi (g,\nu ;n) \le C t^{-(1 + \frac{n+2}{n+8} \frac{\varepsilon }{\alpha } +C\varepsilon ^2)}. \end{aligned}$$

(1.18)

This is a statement that there is a critical point at $\nu =\nu _c$, and that the critical exponent $\gamma$ exists to order $\varepsilon$, with

$$\begin{aligned} \gamma = 1 + \frac{n+2}{n+8} \frac{\varepsilon }{\alpha } +O(\varepsilon ^2) \quad \quad (n \ge 0) . \end{aligned}$$

(1.19)

It is part of the statement that for $n \ge 1$ the susceptibility is given by the infinite-volume limit (1.13), under the above hypotheses. The critical exponent for the specific heat is also computed to order $\varepsilon$ in [31], for $n \ge 1$.

1.6 Main Result

Our main result is the following theorem, which shows that just below the upper critical dimension, the exponent for the critical two-point function “sticks” at its mean-field value (see (1.6)), as predicted by [20]. The theorem applies for all $n \ge 0$, including the case $n=0$ of the weakly self-avoiding walk. The critical value $\nu _c=\nu _c(g;n)$ is the one mentioned in Sect. 1.5. As part of the proof of the theorem, it is shown that for $n \ge 1$ the infinite-volume limit (1.12) exists for $\nu =\nu _c$.

Theorem 1.1

Let $d=1,2,3$; $n \ge 0$; L be sufficiently large; $\varepsilon >0$ be sufficiently small; and $\alpha = \frac{1}{2} (d+\varepsilon )$. For $g \in [\frac{63}{64}\bar{s},\frac{65}{64}\bar{s}]$ the critical two-point function obeys, as $|a-b|\rightarrow \infty$,

$$\begin{aligned} G_{a,b}(g, \nu _c; n) = (1+O(\varepsilon ))((-{\varDelta })^{\alpha /2})^{-1}_{a,b} \asymp \frac{1}{|a-b|^{d-\alpha }}. \end{aligned}$$

(1.20)

Note that Theorem 1.1 identifies the constant in the decay of the interacting two-point function only up to an error of order $\varepsilon$. However, the error is uniformly bounded in a, b, so the power in the decay rate takes its mean-field value, and this is true to all orders in $\varepsilon$.

1.7 Strategy of Proof

The proof is based on a rigorous RG method developed in a series of papers by Bauerschmidt, Brydges and Slade, where the focus is on the nearest-neighbour models in dimension 4. The method is adapted to the long-range setting in [31].

Fix g as in the statement of Theorem 1.1. In [31], given small $m^2>0$, a critical value $\nu _0^c(m^2)$ is constructed, with the property that the critical point $\nu _c$ is given by $\nu _c=\lim _{m^2 \downarrow 0}\nu _0^c(m^2)$. Let

$$\begin{aligned} U_0&= \sum _{x\in {\varLambda }} \left( \tfrac{1}{4} g\vert \varphi _x\vert ^4 + \tfrac{1}{2}\nu _0^c\vert \varphi _x\vert ^2 \right) - \sigma _a\varphi _a^1 -\sigma _b\varphi _b^1 . \end{aligned}$$

(1.21)

For $x=a,b$, let $D_{\sigma _x} = \frac{\partial }{\partial \sigma _x}|_{\sigma _a=\sigma _b=0}$. For $n\ge 1$, the two-point function obeys

$$\begin{aligned} G_{a,b,N}(g,\nu _0^c(m^2)+m^2;n)&= D_{\sigma _a}D_{\sigma _b} \log \mathbb {E}_C e^{-U_0}, \end{aligned}$$

(1.22)

where $\mathbb {E}_C$ denotes Gaussian expectation with covariance $C = ((-{\varDelta }_{\varLambda })^{\frac{\alpha }{2}}+m^2)^{-1}$ ($m^2>0$ ensures existence of the inverse). Thus the two-point function is interpreted as a perturbation of a Gaussian expectation. A similar representation is valid for the weakly self-avoiding walk, using a Gaussian superexpectation.

Perturbation theory is performed inductively in a multi-scale fashion, using a finite-range decomposition $C= C_1+\cdots + C_N$, with $C_j$ of range $\sim L^j$. This is implemented via the Gaussian convolution identity $\mathbb {E}_C\theta = \mathbb {E}_{C_N}\theta \circ \cdots \circ \mathbb {E}_{C_1}\theta$, where $\mathbb {E}_C\theta$ denotes Gaussian convolution. At every step in the induction, we get a representation

$$\begin{aligned} \mathbb {E}_{C_j}\theta \circ \cdots \circ \mathbb {E}_{C_1}\theta e^{-U_0}\approx e^{-U_j}, \end{aligned}$$

(1.23)

where the polynomial

$$\begin{aligned} U_j&=u_j\vert {\varLambda }\vert +\sum _{x\in {\varLambda }} \left( \tfrac{1}{4} g_j\vert \varphi _x\vert ^4 + \tfrac{1}{2}\nu _j \vert \varphi _x\vert ^2 \right) - \lambda _{a,j}\sigma _a\varphi _a^1 -\lambda _{b,j}\sigma _b\varphi _b^1 -\frac{1}{2} (q_{a,j} + q_{b,j})\sigma _a\sigma _b \end{aligned}$$

(1.24)

includes all Euclidean- and O(n)-invariant monomials that are relevant and marginal according to the RG philosophy. The error in this approximation is irrelevant in the RG sense and is controlled uniformly in the volume by parametrising it as a polymer gas. According to (1.23), after the final step of the induction has been performed, we obtain

$$\begin{aligned} G_{a,b,N}(g,\nu _0^c+m^2;n)&= D_{\sigma _a}D_{\sigma _b}\log \mathbb {E}_C e^{-U_0} \approx -D_{\sigma _a}D_{\sigma _b}U_N\vert _{\varphi =0} = \frac{1}{2} (q_{a,N}+q_{b,N}). \end{aligned}$$

(1.25)

To control $q_{x,N}$ ($x=a,b$), we need to study the RG dynamical system

$$\begin{aligned} (g_j,\nu _j,u_j,\lambda _{x,j},q_{x,j})\rightarrow (g_{j+1},\nu _{j+1},u_{j+1},\lambda _{x,j+1},q_{x,j+1}), \end{aligned}$$

(1.26)

and its non-perturbative corrections. The initial condition is $(g_0,\nu _0,u_0,\lambda _{x,0},q_{x,0})=(g,\nu _0^c,0,1,1,0,0)$. (In fact, the coupling constant $u_j$ does not play an important role for the two-point function.) For $d=4$, the dynamical system has a Gaussian fixed point. We use the adaptation of the RG method, as developed in [31], to the long-range setting below the upper critical dimension, where the fixed point is instead non-Gaussian. In [31] only the flow of $g_j,\nu _j,u_j$ was studied and $\lambda _j,q_j$ did not appear, but the flow of $g_j,\nu _j,u_j$ remains identical when these additional coupling constants do appear. For the nearest-neighbour model on $\mathbb Z^4$, the RG method was applied in [5, 32] to prove $|a-b|^{-2}$ decay of the critical two-point function for all $n \ge 0$. We mainly follow the approach of [5, 32]. In particular, our treatment of the flow of $q_{x,j}$ remains the same and yields

$$\begin{aligned} q_{x,j} \approx \lambda _{a,j_{ab}}\lambda _{b,j_{ab}}w_{j;a,b}, \end{aligned}$$

(1.27)

where $w_{j} = \sum _{k=1}^j C_k$, and where $j_{ab} = \lfloor \log _L (2\vert a-b\vert ) \rfloor$ is the coalescence scale defined to ensure that $C_{k;a,b} = 0$ when $k\le j_{ab}$. By definition of $j_{ab}$, the right-hand side of (1.27) is zero for j below the coalescence scale, and this remains true non-perturbatively as well: $q_{x,j}=0$ for scales $j \le j_{ab}$.

The flow of $\lambda _j$ was analysed recursively for the Gaussian RG fixed point in [5, 32], but for the non-Gaussian fixed point in our current setting the recursive analysis cannot be applied due to the non-summability of remainder terms, and a different approach is needed. Let $\bar{D} = \sum _{x\in {\varLambda }} \frac{\partial }{\partial \varphi _x^1}|_{\varphi =0}$ and $\bar{D}^2 = \sum _{x,y\in {\varLambda }} \frac{\partial ^2}{\partial \varphi _x^1 \partial \varphi _y^1}|_{\varphi =0}$. According to (1.23),

$$\begin{aligned} \lambda _{a,j} = e^{u_j\vert {\varLambda }\vert } \bar{D} D_{\sigma _a} e^{-U_j} \approx e^{u_j\vert {\varLambda }\vert } \bar{D} D_{\sigma _a} \mathbb {E}_{w_j}\theta e^{-U_0}. \end{aligned}$$

(1.28)

Let $w_j^{(1)} = \sum _{x\in {\varLambda }} w_{j;a,x}$, which is independent of a. Using Gaussian integration by parts and translation invariance, we show in (4.7) that

$$\begin{aligned} \bar{D} D_{\sigma _a} \mathbb {E}_{w_j}\theta e^{-U_0}&= \mathbb {E}_{w_j} e^{-U_0|_{\sigma =0}} + \frac{1}{\vert {\varLambda }\vert }w_j^{(1)} \bar{D}^2\mathbb {E}_{w_j}\theta e^{-U_0|_{\sigma =0}} . \end{aligned}$$

(1.29)

By using (1.23) to evaluate the two terms in the above right-hand side approximately, we thus obtain

$$\begin{aligned} \lambda _{a,j} \approx 1 + w_j^{(1)}\nu _j. \end{aligned}$$

(1.30)

This relates $\lambda _j$ to the bulk coupling constants $g_j,\nu _j$ whose flow is known from [31]. In particular, it is shown in [31] that $w_j^{(1)}\nu _j = O(\varepsilon )$. All of the above is carried out uniformly in $m^2$, which permits the limit $m^2 \downarrow 0$ to be taken after the infinite-volume limit. Since $\lim _{m^2 \downarrow 0}\lim _{N \rightarrow \infty } w_{N:a,b} = ((-{\varDelta }_{\mathbb Z^{d}})^{\alpha /2})^{-1}_{a,b} \asymp |a-b|^{-(d-\alpha )}$, all this, together with the rigorous versions of (1.27) and (1.25), implies our main result (1.20). The non-perturbative corrections to (1.30) due to the irrelevant error coordinate are controlled using a cluster expansion. This is the main innovation in the proof of Theorem 1.1.

The remainder of the paper is organised as follows. In Sect. 2, we provide some background and definitions needed for the RG method. In Sect. 3, we formulate the RG map and state the main theorem which provides estimates on the RG map; this is an adaptation of the main result of [16] as applied to the long-range model in [31]. The main difference, compared to [31], is the inclusion of observables in the RG map. The flow of the observable coupling constant $\lambda _j$ is analysed in Sect. 4. The flow of the observable coupling constant $q_j$ is then analysed in Sect. 5, where the proof of Theorem 1.1 is completed.

2 Set-Up for RG Method

In this section, we summarise some notation and background for the RG method, needed for the proof of Theorem 1.1. Additional details can be found in [31].

2.1 Formula for Two-Point Function

We begin with a formula for the two-point function that serves as our starting point.

2.1.1 The Case $n \ge 1$

For $n \ge 1$, we define

$$\begin{aligned} \tau _x = \frac{1}{2} |\varphi _x|^2 \quad \quad (n\ge 1). \end{aligned}$$

(2.1)

Given $g>0$, $\nu \in \mathbb R$, $m^2>0$, we set

$$\begin{aligned} g_0 = g , \quad \nu _0 = \nu -m^2. \end{aligned}$$

(2.2)

Given $a,b \in {\varLambda }$, we introduce observable fields $\sigma _a,\sigma _b \in \mathbb R$, and define $V_0$ and $Z_0$ by

(2.3)

with $V_0({\varLambda }) = \sum _{x\in {\varLambda }}V_0(\varphi _x)$.

Given a ${\varLambda }\times {\varLambda }$ covariance matrix w, let $\mathbb {E}_w$ denote the Gaussian expectation with covariance w. Let $C=((-{\varDelta }_{{\varLambda }_N})^{\alpha /2} +m^2)^{-1}$. By shifting part of the $|\varphi |^2$ term into the covariance, the expectation (1.10) can be rewritten as

$$\begin{aligned} \langle F \rangle _{g,\nu ,N}= \frac{\mathbb {E}_C F e^{-V^\varnothing _0({\varLambda })}}{\mathbb {E}_C e^{-V^\varnothing _0({\varLambda })}}, \end{aligned}$$

(2.4)

where $V^\varnothing _0({\varLambda })$ denotes the evaluation of $V_0({\varLambda })$ at $\sigma _a = \sigma _b = 0$. When F is a monomial, it is standard to write this ratio of expectations as a derivative of a logarithmic generating function. Let $D_{\sigma _a}$ denote the operator $\frac{\partial }{\partial \sigma _a}|_{\sigma _a = \sigma _b = 0}$, and similarly for higher derivatives. Then the two-point function is given, for $n \ge 1$, by

$$\begin{aligned} G_{a,b,N}(g,\nu ;n) = \langle \varphi _a^1 \varphi _b^1 \rangle _{g,\nu ,N} = D_{\sigma _a\sigma _b}^2 \log \mathbb {E}_C e^{- V_{0} ({\varLambda })}. \end{aligned}$$

(2.5)

2.1.2 The Case $n =0$

For $n= 0$, as in several previous papers (e.g., [5, 6, 32]) we formulate the weakly self-avoiding walk model as the infinite-volume limit of a supersymmetric version of the $|\varphi |^4$ model. The supersymmetric model involves a complex boson field $(\phi _x,\bar{\phi }_x)_{x \in {\varLambda }}$ and a fermion field given by the 1-forms $\psi _x = \frac{1}{\sqrt{2\pi i}}d\phi _x$, $\bar{\psi }_x = \frac{1}{\sqrt{2\pi i}}d\bar{\phi }_x$. For $n=0$, in place of (2.1), we set

$$\begin{aligned} \tau _x = \phi _x\bar{\phi }_x + \psi _x \wedge \bar{\psi }_x \quad \quad (n=0), \end{aligned}$$

(2.6)

and we replace $\varphi _a^1,\varphi _b^1$ in (2.3) by $\bar{\phi }_a,\phi _b$.

For $n=0$, a formula closely related to (2.5) is given, e.g., in [32, (6.5)], with $\mathbb {E}_C$ in (2.5) replaced by the Gaussian superexpectation. As in [32], our formalism applies to the supersymmetric model with only notational changes, with n interpreted as $n=0$ in formulas such as (1.19), and with the Gaussian expectation replaced by a superexpectation. For notational simplicity, we concentrate throughout the paper on the case $n \ge 1$.

2.2 Progressive Integration

In our version of the RG method, the expectation $\mathbb {E}_C e^{-V_0({\varLambda })}$ of (2.5) is evaluated in a multi-scale fashion, via a finite-range decomposition of the covariance C. We use the same finite-range decomposition $C=C_1+C_2+\cdots +C_{N-1}+C_{N,N}$ of the covariance $C=((-{\varDelta }_{{\varLambda }_N})^{\alpha /2} +m^2)^{-1}$ that is described and analysed in [31, Sect. 3]. A closely related decomposition was first introduced in [25] and subsequently developed in [26]. The covariances $C_j$ are translation invariant, and have the finite-range property

$$\begin{aligned} C_{j;x,y} = 0 \quad \text {if} { \; |x-y| \ge \frac{1}{2} L^j} . \end{aligned}$$

(2.7)

Thus, we may regard $C_j$ either as a covariance on $\mathbb Z^{d}$ or on ${\varLambda }_N$, as long as $N>j$. Viewing the $C_j$ as covariances on $\mathbb Z^{d}$, we also have a decomposition of the infinite-volume covariance given by $((-{\varDelta })^{\alpha /2} + m^2)^{-1} = \sum _{j=1}^\infty C_j$. We leave implicit the dependence of the covariance $C_j$ on $m^2$. According to [31, (3.11)], for $m^2$ bounded, the covariances $C_j$ satisfy the estimates

$$\begin{aligned} |C_{j;x,y}| \le c L^{-(d - \alpha ) (j - 1)} (1 + m^4 L^{2\alpha (j - 1)})^{-1}. \end{aligned}$$

(2.8)

For $n \ge 1$, and for an integrable $F : (\mathbb R^n)^{\varLambda }\rightarrow \mathbb R$, we define the convolution $\mathbb {E}_C\theta F$ by

$$\begin{aligned} (\mathbb {E}_C\theta F)(\varphi ) = \mathbb {E}_C F(\varphi +\zeta ), \end{aligned}$$

(2.9)

where the expectation $\mathbb {E}_C$ on the right-hand side acts on $\zeta$ and leaves $\varphi$ fixed. A similar construction is used for $n=0$ (see, e.g., [6, Sect. 4.1]). By [13, Proposition 2.6], the Gaussian convolution can be evaluated as

$$\begin{aligned} \mathbb {E}_{C}\theta F = \big ( \mathbb {E}_{C_N}\theta \circ \mathbb {E}_{C_{N-1}}\theta \circ \cdots \circ \mathbb {E}_{C_{1}}\theta \big ) F, \end{aligned}$$

(2.10)

with an abuse of notation where $C_N$ means $C_{N,N}$. To compute the expectation $\mathbb {E}_C e^{-V_0({\varLambda })}$ in (2.5), we use (2.10) to evaluate $\mathbb {E}_C \theta e^{-V_0({\varLambda })}$ progressively, as follows. We write $\mathbb {E}_{j} = \mathbb {E}_{C_j}$ and let

$$\begin{aligned} Z_{j+1} = \mathbb {E}_{j+1}\theta Z_j \quad \quad (0 \le j<N), \end{aligned}$$

(2.11)

with $Z_0 = e^{-V_0({\varLambda })}$ as in (2.3). By (2.10), $\mathbb {E}_CZ_0$ is obtained by setting $\varphi =0$ in

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

(2.12)

This leads us to study the recursion $Z_j \mapsto Z_{j+1}$.

2.3 Function Space

The observable fields $\sigma _a,\sigma _b$ are needed only for the purpose of evaluating the second derivative in (2.5). Therefore, dependence on the observable fields which is higher order than quadratic plays no role. We make use of this by defining the function space $\mathcal {N}$ as explained below. We also define the $T_\varphi$ seminorm on $\mathcal {N}$. These definitions are as in, e.g., [14, 32]. We focus on the case $n\ge 1$; the modifications needed for $n=0$ are as in, e.g., [15].

2.3.1 The Space $\mathcal {N}$

Given $p_\mathcal {N}> 0$, let

$$\begin{aligned} \mathcal {N}^\varnothing = C^{p_\mathcal {N}}((\mathbb R^n)^{\varLambda }). \end{aligned}$$

(2.13)

As in [31, Sect. 6.2.1], we fix any $p_\mathcal {N}\ge 10$. For $n = 0$, $\mathcal {N}^\varnothing$ is instead a space of even differential forms with $p_\mathcal {N}$-times differentiable coefficients.

In order to treat functions of the observable fields $\sigma _a,\sigma _b$, we define an extension $\mathcal {N}$ of $\mathcal {N}^\varnothing$ exactly as in [32, Sect. 2.4.1]. Namely, let $\mathcal {N}'$ be the space of real-valued functions of $\varphi ,\sigma _a,\sigma _b$ which are $C^{p_\mathcal {N}}$ in $\varphi$ and $C^\infty$ in $\sigma _a,\sigma _b$. An ideal $\mathcal {I}$ in $\mathcal {N}'$ is formed by those elements of $\mathcal {N}'$ whose formal power series expansion in the observable fields to order $1,\sigma _a, \sigma _b,\sigma _a\sigma _b$ is equal to zero. We define $\mathcal {N}$ as the quotient algebra $\mathcal {N}=\mathcal {N}'/\mathcal {I}$. Then $\mathcal {N}$ has a direct sum decomposition

$$\begin{aligned} \mathcal {N}= \mathcal {N}'/\mathcal {I}= \mathcal {N}^\varnothing \oplus \mathcal {N}^a \oplus \mathcal {N}^b \oplus \mathcal {N}^{ab}, \end{aligned}$$

(2.14)

where elements of $\mathcal {N}^a, \mathcal {N}^b , \mathcal {N}^{ab}$ are given by elements of $\mathcal {N}^\varnothing$ multiplied by $\sigma _a$, by $\sigma _b$, and by $\sigma _a\sigma _b$ respectively. Thus, elements of $\mathcal {N}$ can be identified with polynomials over $\mathcal {N}^\varnothing$ in the observable fields with terms only of order $1,\sigma _a, \sigma _b,\sigma _a\sigma _b$, i.e. $F \in \mathcal {N}$ can be written as

$$\begin{aligned} F = F_\varnothing + \sigma _a F_a + \sigma _b F_b +\sigma _a \sigma _b F_{ab} \end{aligned}$$

(2.15)

with $F_c\in \mathcal {N}^\varnothing$ for each $c\in \{\varnothing , a,b,ab\}$. There are natural projections $\pi _c: \mathcal {N}\rightarrow \mathcal {N}^c$ defined for such F by $\pi _\varnothing F = F_\varnothing$, $\pi _a F = \sigma _a F_a$, $\pi _b F = \sigma _b F_b$, and $\pi _{ab} F = \sigma _a\sigma _b F_{ab}$. We set $\pi _*=1-\pi _\varnothing$. The expectation $\mathbb {E}_C$ acts term-by-term on $F \in \mathcal {N}$, namely $\mathbb {E}_C F = \mathbb {E}_C F_\varnothing + \sigma _a \mathbb {E}_C F_a + \sigma _b \mathbb {E}_C F_b +\sigma _a \sigma _b \mathbb {E}_C F_{ab}$ for F as in (2.15).

2.3.2 Seminorms

A family of seminorms is used to control the size of elements of $\mathcal {N}$. Let ${\varLambda }^*$ denote the set of sequences of any finite length (including length 0), composed of elements of ${\varLambda }\times \{ 1, \ldots , n \}$. Let $\varphi \in (\mathbb R^n)^{\varLambda }$ be a field, and let $F \in \mathcal {N}^\varnothing$. Given ${\vec x} = ((x_1, i_1), \ldots , (x_p, i_p)) \in {\varLambda }^*$, we write $|\vec x| = p$ and let

$$\begin{aligned} F_{\vec x}(\varphi )= \frac{\partial ^p F(\varphi )}{\partial \varphi ^{i_1}_{x_1} \ldots \partial \varphi ^{i_p}_{x_p}}. \end{aligned}$$

(2.16)

A test function g is a mapping $g:{\varLambda }^* \rightarrow \mathbb R$, written $\vec x \mapsto g_{\vec x}$. We define the $\varphi$-pairing of F with a test function g by

$$\begin{aligned} \langle F, g \rangle _\varphi = \sum _{|\vec x| \le p_\mathcal {N}} \frac{1}{|\vec x|!} F_{\vec x}(\varphi ) g_{\vec x}. \end{aligned}$$

(2.17)

Given a parameter $\mathfrak {h}_j > 0$, a scale-dependent norm $\Vert g\Vert _\Phi = \Vert g\Vert _{\Phi _j(\mathfrak {h}_j)}$ is defined on test functions in [31, (6.8)]. The $\Phi =\Phi _j(\mathfrak {h}_j)$ norm controls the size of a test function and its discrete gradients up to order $p_\Phi =4$, but its precise definition is immaterial for the present discussion. With $B_\Phi (1)$ the unit ball in $\Phi$, we define the $T_\varphi = T_{\varphi ,j}(\mathfrak {h}_j)$ seminorm on $\mathcal {N}^\varnothing$ by

$$\begin{aligned} \Vert F\Vert _{T_\varphi }= \sup _{g\in B_\Phi (1)} |\langle F, g\rangle _\varphi |. \end{aligned}$$

(2.18)

Given an additional parameter $\mathfrak {h}_\sigma = \mathfrak {h}_{\sigma ,j}$, we extend this definition to all of $\mathcal {N}$ exactly as in [14], i.e., the seminorm of F of the form (2.15) is defined to be

$$\begin{aligned} \Vert F\Vert _{T_\varphi } = \Vert F_\varnothing \Vert _{T_\varphi }+ \mathfrak {h}_\sigma (\Vert F_a\Vert _{T_\varphi } + \Vert F_b\Vert _{T_\varphi })+ \mathfrak {h}_\sigma ^2 \Vert F_{ab}\Vert _{T_\varphi }. \end{aligned}$$

(2.19)

2.4 Blocks, Polymers and Scales

2.4.1 Blocks and Polymers

The finite-range covariance decomposition is well-suited to a block decomposition of the torus ${\varLambda }_N$ of period $L^N$ into disjoint blocks of side $L^j$, for scales $0 \le j \le N$. This decomposition is an important ingredient in our choice of the coordinates in which we represent the RG map. We now describe it in detail, along with a number of useful related definitions, as in [16].

We partition the torus ${\varLambda }_N$, which has period $L^N$, into disjoint j-blocks of side $L^j$ ($j\le N$). Each j-block is a translate of the block $\{ x \in {\varLambda }: 0 \le x_i < L^j, i = 1, \ldots , d\}$. We denote the collection of j-blocks by $\mathcal {B}_j$.

A j-polymer is any (possibly empty) union of j-blocks, and $\mathcal {P}_j$ denotes the set of j-polymers. Given $X \in \mathcal {P}_j$, we denote by $\mathcal {B}_j(X)$ the set of j-blocks in X, and denote by $\mathcal {P}_j(X)$ the set of j-polymers in X. A nonempty polymer X is connected if for any $x, x' \in X$, there is a sequence $x = x_0, \ldots , x_n = x' \in X$ with $\Vert x_{i+1} - x_i\Vert _\infty = 1$ for $i = 0, \ldots , n - 1$. Let $\mathcal {C}_j$ denote the set of connected j-polymers and, for any $X\in \mathcal {P}_j$, let $\mathrm{Comp}_j(X) \subset \mathcal {C}_j(X)$ be the set of connected components of X. The empty set $\varnothing$ is not in $\mathcal {C}_j$.

We say that two polymers X, Y do not touch if $\hbox {min }\{\Vert x-y\Vert _\infty : x \in X, y \in Y\} >1$. We call a connected polymer $X\in \mathcal {C}_j$ a small set if it consists of at most $2^d$ j-blocks, and write $\mathcal{S}_j$ for the collection of small sets in $\mathcal {C}_j$. The small-set neighbourhood of a polymer $X\in \mathcal {P}_j$ is $X^\Box = \cup _{Y \in \mathcal{S}_j(X): X \cap Y \ne \varnothing }Y$.

For $F_1, F_2 : \mathcal {P}_j \rightarrow \mathcal {N}$, we define the scale-j circle product $F_1 \circ F_2 : \mathcal {P}_j \rightarrow \mathcal {N}$ by

$$\begin{aligned} (F_1 \circ F_2)(Y) = \sum _{X\in \mathcal {P}_j(Y)} F_1(Y{\setminus } X) F_2(X) \quad \quad (Y \in \mathcal {P}_j). \end{aligned}$$

(2.20)

We only consider maps $F:\mathcal {P}_j \rightarrow \mathcal {N}$ with the property that $F(\varnothing )=1$. The identity element for the circle product is the map defined by , if $X \ne \varnothing$.

2.4.2 Mass and Coalescence Scales

Two scales play an important role for the nature of the RG recursion (2.11). We define the mass scale $j_m$ by

$$\begin{aligned} j_m = \lceil f_m \rceil , \quad \quad f_m = 1+ \frac{1}{\alpha } \log _L m^{-2}. \end{aligned}$$

(2.21)

By definition, $j_m$ is the smallest scale for which $m^2L^{\alpha (j_m-1)} \ge 1$. The mass scale is the scale beyond which the mass $m^2$ plays a significant helpful role in the decay of the covariance $C_j$. Indeed, by (2.8) and the elementary inequality (with notation $x_+=\max \{x,0\}$)

$$\begin{aligned} (1 + m^4 L^{2\alpha (j - 1)})^{-1} \le L^{-2\alpha (j - j_m)_+}, \end{aligned}$$

(2.22)

we have

$$\begin{aligned} |C_{j;x,y}|\le c L^{-(d - \alpha )(j - 1) - 2\alpha (j - j_m)_+}. \end{aligned}$$

(2.23)

We also define the coalescence scale $j_{ab}$ by

$$\begin{aligned} j_{ab} = \big \lfloor \log _{L} (2 |a- b|) \big \rfloor . \end{aligned}$$

(2.24)

By definition, $j_{ab}$ is the unique integer such that

$$\begin{aligned} \tfrac{1}{2} L^{j_{ab}} \le |a-b| < \tfrac{1}{2} L^{j_{ab}+1}. \end{aligned}$$

(2.25)

By (2.7), $C_{j;a,b}= 0$ for all $j\le j_{ab}$, and hence

$$\begin{aligned} ((-{\varDelta })^{\alpha /2} +m^2)^{-1}_{a,b} = \sum _{j=1}^\infty C_{j;a,b} = \sum _{j=j_{ab}+1}^\infty C_{j;a,b}. \end{aligned}$$

(2.26)

Ultimately, we take the limit $m^2 \downarrow 0$ before considering large $|a-b|$, so we can and do assume that $j_m > j_{ab}$.

2.5 Localisation Operator Loc

We use the operator $\hbox {Loc}$ defined and analysed in [14], to extract a local polynomial from an element $F\in \mathcal {N}$. For appropriate $X \subset {\varLambda }$, the local polynomial $\hbox {Loc}_X F$ extracts the parts of F that are relevant and marginal in the RG sense.

2.5.1 Local Polynomials

The range of the operator $\hbox {Loc}_X$ is a certain vector space $\mathcal {U}(X)$ of local polynomials in the field. We now define this vector space, taking into account that the elements $F \in \mathcal {N}$ to which $\hbox {Loc}_X$ will be applied obey Euclidean covariance and O(n) invariance on $\mathcal {N}^\varnothing$.

Given bulk coupling constants $g,\nu , u \in \mathbb R$; $a,b \in {\varLambda }$; observable fields $\sigma _a,\sigma _b \in \mathbb R$; and the observable coupling constants $\lambda _a,\lambda _b,q_a,q_b \in \mathbb R$; let

(2.27)

(2.28)

The symbol $\varnothing$ denotes the bulk. (For $n=0$, we can take $u=0$ in $U^\varnothing$ due to supersymmetry; see [7].) For U as in (2.28) and $X \subset {\varLambda }$, we write

$$\begin{aligned} U(X) = \sum _{x \in X}U(\varphi _x) . \end{aligned}$$

(2.29)

Let $\mathcal {U}(X)$ denote the space of polynomials of the form (2.29). Let $\mathcal {V}(X) \subset \mathcal {U}(X)$ be the subspace for which $u = q_a = q_b = 0$. Note that $V_0$ of (2.3) obeys $V_0(X) \in \mathcal {V}(X)$ with $\lambda _{a}=\lambda _{b}=1$.

2.5.2 Definition of Loc

To define $\hbox {Loc}$, we must first define a set of polynomial test functions, as in [14, Sect. 1.3]. Let $p>0$, let $a= (a_1, \ldots , a_p)$ with each $a_r \in \mathbb {N}_0^d$, and let $k=(k_1,\ldots ,k_p)$ with each $k_r \in \{1,\ldots , n\}$. Let ${\varLambda }' \subset {\varLambda }$ be a coordinate patch as defined in [14, Sect. 1.3] (e.g., ${\varLambda }'$ can be any small set as defined in Sect. 2.4). Recall the set ${\varLambda }^*$ of sequences, defined in Sect. 2.3.2. We define a test function $q^{a,k}$, supported on sequences $\vec x = ((x_1, i_1), \ldots , (x_p, i_p)) \in {\varLambda }^*$ with each $x_r \in {\varLambda }'$, by

$$\begin{aligned} q^{a,k}_{\vec x} = \prod _{r=1}^p \delta _{i_r,k_r} x_r^{a_r} = \prod _{r=1}^p \delta _{i_r,k_r} \prod _{l=1}^d x_{r,l}^{a_{r,l}} . \end{aligned}$$

(2.30)

We include the case $p=0$ by interpreting (2.30) as the constant number 1 in this case. The role of the coordinate patch, which cannot “wrap around” the torus, is to permit polynomial test functions such as (2.30) to be well-defined. We define the field dimension $[\varphi ] =\frac{d-\alpha }{2}$. The dimension of the test function $q^{a,k}$ is defined to equal $p[\varphi ] + \sum _{r=1}^p \sum _{l=1}^d a_{r,l}$. Given $d_+\ge 0$, we let $\Pi = \Pi ^{d_+}[{\varLambda }'] \subset \Phi$ denote the span of all test functions $q^{a,k}$ of dimension at most $d_+$.

Let $\mathcal {N}^\varnothing ({\varLambda }')$ denote the space of functionals of the field that only depend on field values at points in ${\varLambda }'$. By [14, Proposition 1.5], for $X \subset {\varLambda }'$, there is a unique operator ${{\mathrm{\mathrm{loc}}}}_X : \mathcal {N}^\varnothing ({\varLambda }') \rightarrow \mathcal {V}(X)$ (independent of the choice of ${\varLambda }'$) such that

$$\begin{aligned} \langle F, g \rangle _0 = \langle {{\mathrm{\mathrm{loc}}}}_X F, g \rangle _0 \text { for all } g\in \Pi , \end{aligned}$$

(2.31)

with the pairing given by (2.17) with $\varphi =0$. To extend ${{\mathrm{\mathrm{loc}}}}$ from $\mathcal {N}^\varnothing$ to $\mathcal {N}$, suppose we are given $d_+^c$ for $c= \varnothing , a, b, ab$. We let $\mathrm{loc}^c_X$ denote the operator $\mathrm{loc}_X$ with $d_+ = d_+^c$ and, for F as in (2.15), define

$$\begin{aligned} \hbox {Loc}_Y F = \mathrm{loc}^\varnothing _X F_\varnothing + \sigma _a \mathrm{loc}^a_X F_a + \sigma _b \mathrm{loc}^b_X F_b + \sigma _a\sigma _b \mathrm{loc}^{ab}_X F_{ab}. \end{aligned}$$

(2.32)

It remains to specify the $d_+^c$.

In [31], for scales below the mass scale the range of the restriction of $\hbox {Loc}$ to $\mathcal {N}^\varnothing$ is specified as the span of $\{1,\tau ,\tau ^2\}$. This corresponds to the choice $d_+^\varnothing = d$ (using symmetry considerations to disregard non-symmetric monomials). Above the mass scale, a different range for $\hbox {Loc}$ is used, namely $d_+^\varnothing =d-\alpha$, due to the enhanced decay of the covariance decomposition (see [31, Sect. 4.2]).

As in [7, Sect. 3.2], we set $d_+^a=d_+^b= [\varphi ]$ when $\hbox {Loc}$ acts at scales strictly less than $j_{ab}$, and set $d_+^a=d_+^b=0$ for larger scales. We always take $d_+^{ab}=0$.

The following elementary lemma will be useful. Let denote the constant test function supported on sequences of length 1 and defined by . Likewise, let denote the constant test function supported on sequences of length 2 and defined by . Note that are each of the form (2.30), with respective dimensions $[\varphi ]$ and $2[\varphi ]$.

Lemma 2.1

Given $x\in {\varLambda }$ and a coordinate patch ${\varLambda }' \ni x$, suppose that $F \in \mathcal {N}^\varnothing ({\varLambda }')$. For $j<N$ and $m=1,2$,

(2.33)

Moreover, if $j < j_{ab}$, then for $c= a, b$,

(2.34)

Proof

The first statement is an immediate consequence of the definition of $\mathrm{Loc}$ together with the fact that the test function has dimension $m[\varphi ] \le d-\alpha \le d_+^\varnothing$ for $m = 1, 2$. The second statement follows similarly using the fact that the dimension $[\varphi ]$ of is equal to $d_+^a = d_+^b$ if $j < j_{ab}$. $\square$

3 RG Map

In the absence of observables, i.e., with $\sigma _a=\sigma _b=0$, the RG map for the long-range models is constructed and bounded in [31] using the main theorem of [16]. The result is given in [31, Theorem 6.4]. The extension of this construction to the case of nonzero observable fields $\sigma _a,\sigma _b$ follows a similar route as in the 4-dimensional nearest neighbour case in [5, 32], as we now explain. The coordinates for the RG map are discussed in Sect. 3.1, the domain of the RG map is discussed in Sect. 3.2, and the main estimates for the RG map are given in Theorem 3.3. These estimates, combined with a new estimate derived from a cluster expansion, are used in Sects. 4–5 to control the flow generated by the RG map.

3.1 RG Coordinates

The RG map will be defined so as to express the sequence $Z_j$ defined by (2.11) as

$$\begin{aligned} Z_j = e^{\zeta _j} (I_j \circ K_j)({\varLambda }), \end{aligned}$$

(3.1)

for a real sequence $\zeta _j$ and sequences of maps $I_j : \mathcal {P}_j \rightarrow \mathcal {N}$ and $K_j : \mathcal {P}_j \rightarrow \mathcal {N}$. The perturbative coordinate $I_j$ is an explicit function of $V_j \in \mathcal {V}$, and

$$\begin{aligned} \zeta _j = -u_j |{\varLambda }| + \tfrac{1}{2} (q_{a,j} + q_{b,j}) \sigma _a\sigma _b. \end{aligned}$$

(3.2)

The nonperturbative coordinate $K_j$ is discussed in detail below. By (2.3), (3.1) holds at scale $j = 0$ with $u_0 = q_{x,0} = 0$, $I_0 = e^{-V_0}$, and . We sometimes write an element of $\mathcal {U}$ as $U = (\zeta , V)$ with $V \in \mathcal {V}$, where $\zeta$ encodes $u, q_a, q_b$.

We express the map $Z_j \mapsto Z_{j+1}$ of (2.11) via a map $(V_j, K_j) \mapsto (\delta \zeta _{j+1}, V_{j+1}, K_{j+1})$, the renormalisation group (RG) map, in such a manner that

$$\begin{aligned} \mathbb {E}_{j+1}\theta (I_j \circ K_j)({\varLambda }) = e^{\delta \zeta _{j+1}} (I_{j+1} \circ K_{j+1})({\varLambda }) \end{aligned}$$

(3.3)

with $I_{j+1} = I_{j+1}(V_{j+1})$ and $\delta \zeta _{j+1} = \zeta _{j+1} - \zeta _j$. This ensures that $Z_{j+1}$ has the form (3.1) with $\zeta _{j+1} = \zeta _j + \delta \zeta _{j+1}$.

3.1.1 Perturbative Coordinate

The form of the perturbative coordinate $I_j$ is as follows. Given a ${\varLambda }\times {\varLambda }$ matrix w, we define the operator $\mathcal {L}_w = \frac{1}{2} \sum _{u,v \in {\varLambda }} w_{u,v} \sum _{i=1}^n \frac{\partial }{\partial \varphi _{u}^i} \frac{\partial }{\partial \varphi _{v}^i}$. Recall the projections defined in Sect. 2.3.1. Given $V',V'' \in \mathcal {V}$, we also define

$$\begin{aligned} F_{w}(V',V'')&= e^{\mathcal {L}_{w}} \big (e^{-\mathcal {L}_{w}}V'\big ) \big (e^{-\mathcal {L}_{w}}V'' \big ) - V'V'', \end{aligned}$$

(3.4)

$$\begin{aligned} F_{\pi ,w}(V',V'')&= F_{w}(V',\pi _\varnothing V'') + F_{w}(\pi _* V',V''). \end{aligned}$$

(3.5)

For $j \ge 0$, we write the partial sums of the covariance decomposition as

$$\begin{aligned} w_j = \sum _{i=1}^j C_i, \qquad w_0=0. \end{aligned}$$

(3.6)

As in [7, (3.21)], for $B \in \mathcal {B}_j$ we define

$$\begin{aligned} W_j(V,x) = \frac{1}{2} (1-\hbox {Loc}_x) F_{\pi ,w_j}(V_x,V({\varLambda })), \quad \quad W_j(V,B) = \sum _{x \in B} W_j(V,x). \end{aligned}$$

(3.7)

The polynomial $W_j(V, B)$ in the fields is thus an explicit quadratic function of V. In particular, $W_j^\varnothing$ is an even polynomial in the fields, and $W_j$ is quadratic in the coupling constants and is irrelevant in the RG sense. Finally, for $V\in \mathcal {V}$, we define $I_j = I_j(V, \cdot \,) : \mathcal {P}_j \rightarrow \mathcal {N}$ by

$$\begin{aligned} I_j(V, X) = e^{-V(X)} \prod _{B\in \mathcal {B}_j(X)} (1 + W_j(V, B)). \end{aligned}$$

(3.8)

As in (2.5), we write $D_{\sigma _a}=\frac{\partial }{\partial \sigma _a}|_{\sigma _a = \sigma _b = 0}$. We also write $\bar{D} = \sum _{x\in {\varLambda }} \frac{\partial }{\partial \varphi _x^1}|_{\varphi =0}$ and $\bar{D}^2 = \sum _{x,y\in {\varLambda }} \frac{\partial ^2}{\partial \varphi _x^1 \partial \varphi _y^1}|_{\varphi =0}$. We will later make use of the following corollary of Lemma 2.1.

Corollary 3.1

For $V \in \mathcal {V}$ and $x \in {\varLambda }$, and with $W_j=W_j(V,x)$,

$$\begin{aligned} \bar{D} W^\varnothing _j = \bar{D}^2 W^\varnothing _j = 0. \end{aligned}$$

(3.9)

Moreover, if $j < j_{ab}$, then

$$\begin{aligned} \bar{D} D_{\sigma _a} W_j = 0. \end{aligned}$$

(3.10)

Proof

The fact that $\bar{D} W^\varnothing _j = 0$ is immediate since $W_j^\varnothing$ is even in the fields. Also, since , it follows from Lemma 2.1 that $\bar{D}^2 W_j^\varnothing =0$. To see that $\bar{D} D_{\sigma _a} W_j = 0$, note that $D_{\sigma _a} W_j = \tfrac{1}{2} (1 - \mathrm{loc}^a_x) F_{a}$, with $F_a \in \mathcal {N}^\varnothing$ the coefficient of $\sigma _a$ in $F_{\pi ,w_j}(V_x, V({\varLambda }))$. Thus, by definition (2.17) of the pairing, , which vanishes by Lemma 2.1 when $j < j_{ab}$. This completes the proof. $\square$

3.1.2 Nonperturbative Coordinate

We now define the space $\mathcal {K}_j$ of maps $K : \mathcal {P}_j \rightarrow \mathcal {N}$ which contains the nonperturbative RG coordinate. With $\mathcal {N}$ replaced by $\mathcal {N}^\varnothing$, such a space is defined in [31, Definition 6.2], and, as in [16], we extend it here to include observables. The symmetries (Euclidean covariance, gauge invariance, supersymmetry, and O(n)-invariance) used in Definition 3.2 are defined in [16, Sect. 1.6] and [4, Sect. 2.3]. For $n \ge 1$, as a replacement for the gauge invariance which holds for $n=0$ we also introduce sign invariance, which is invariance under the map $(\sigma ,\varphi ) \mapsto (-\sigma ,-\varphi )$. Note that $V_0$ of (2.3) is sign invariant. It can be verified that the property of sign invariance is preserved by the map $K \mapsto K_+$ of [16].

Definition 3.2

For $j<N$, let $\mathcal {C}\mathcal {K}_{j} = \mathcal {C}\mathcal {K}_{j} ({\varLambda })$ denote the real vector space of functions $K : \mathcal {C}_j \rightarrow \mathcal {N}$ with the following properties:

• Field Locality: For all $X\in \mathcal {P}_j({\varLambda })$, $K (X) \in \mathcal {N}(X^{\Box })$. Also, (i) $\pi _{a} K (X)=0$ unless $a\in X$, (ii) $\pi _{b} K (X)=0$ unless $b\in X$, and (iii) $\pi _{ab} K (X)=0$ unless $a\in X$ and $b\in X^\Box$ or vice versa, and $\pi _{ab} K(X)=0$ if $X\in \mathcal{S}_j$ and $j<j_{ab}$.

• Symmetry: (i) $\pi _\varnothing K$ is Euclidean covariant, (ii) if $n = 0$, $\pi _\varnothing K$ is supersymmetric and K is gauge invariant and has no constant part; if $n \ge 1$, $\pi _\varnothing K$ is O(n)-invariant and K is sign invariant.

Let $\mathcal {K}_{j} = \mathcal {K}_{j} ({\varLambda })$ be the real vector space of functions $K : \mathcal {P}_j \rightarrow \mathcal {N}$ which have the above field locality and symmetry properties, and, in addition:

• Component Factorisation: for all polymers X, $K (X) = \prod _{Y \in \mathrm{Comp}( X)}K (Y)$.

The nonperturbative coordinate $K_j$ appearing in (3.1) is an element of $\mathcal {K}_j$. An element of $\mathcal {K}_{j}$ determines an element of $\mathcal {C}\mathcal {K}_{j}$ by restriction to $X\in \mathcal {C}_j$. Also, an element of $\mathcal {C}\mathcal {K}_{j}$ determines an element of $\mathcal {K}_{j}$ by the factorisation property. The same symbol is used for both elements related by this correspondence. Since the empty set is not a connected set, becomes $0 \in \mathcal {C}\mathcal {K}_{j}$ under this correspondence.

3.2 Norms and RG Domain

We now specify the domain of the RG map, which requires specification of norms on the spaces $\mathcal {V}$ and $\mathcal {C}\mathcal {K}$. Without the observables fields, the norms are discussed in [31, Sect. 6.2]. For the nearest-neighbour 4-dimensional case, the adaptation of the norms to include observables is discussed, e.g., in [32, Sect. 5.1]. For our current long-range setting, we need only adjust some norm parameters, compared to [32, Sect. 5.1].

As in [31, (5.49)], the small number $\bar{s}$ in Theorem 1.1 is given by

$$\begin{aligned} \bar{s}=\frac{1}{a} (1-L^{-\varepsilon }) = O(\varepsilon ), \end{aligned}$$

(3.11)

with the constant a specified in [31, Lemma 5.3] (not to be confused with the point $a\in \mathbb Z^{d}$ used for the two-point function). Recall the mass scale $j_m$ defined in (2.21). Following [31, (3.19)–(3.20), (6.24)], we fix

$$\begin{aligned} \alpha ' \in (0, \tfrac{1}{2}\alpha ), \end{aligned}$$

(3.12)

and define the bulk parameters

$$\begin{aligned} \ell _j&= \ell _0 L^{-\frac{1}{2} (d-\alpha )j} L^{- \frac{1}{2} (\alpha +\alpha ') (j-j_m)_+} = {\left\{ \begin{array}{ll} \ell _0 L^{-\frac{1}{2} (d-\alpha )j} &{} (j \le j_m) \\ \ell _0 L^{-\frac{1}{2} (d-\alpha )j_m}L^{-\frac{1}{2} (d+\alpha ')(j-j_m)} &{} (j > j_m), \end{array}\right. } \end{aligned}$$

(3.13)

$$\begin{aligned} h_j&= \frac{1}{\bar{s}^{1/4}}k_0 L^{-\frac{1}{2} (d-\alpha )j} = \frac{1}{\bar{s}^{1/4}}\frac{k_0 }{\ell _0} \ell _j \quad \quad (j \le j_m) . \end{aligned}$$

(3.14)

Here $\ell _0$ can be chosen large (depending on L) and $k_0$ is a fixed (small) constant. We use $\mathfrak {h}_j$ to refer to either of the bulk parameters $\ell _j,h_j$.

Now that observables are present, the pair of parameters $\mathfrak {h}_j$ is supplemented by the pair

$$\begin{aligned} \mathfrak {h}_{\sigma ,j} = \ell _{j \wedge j_{ab}}^{-1} 2^{(j - j_{ab})_+} \times {\left\{ \begin{array}{ll}\bar{s}&{} (\mathfrak {h}= \ell ) \\ \bar{s}^{1/4} &{} (\mathfrak {h}= h). \end{array}\right. } \end{aligned}$$

(3.15)

We only use $h_{\sigma ,j}$ for $j \le j_m$. Recall that we assume that the coalescence scale $j_{ab}$ is smaller than the mass scale $j_m$, since the limit $m^2 \downarrow 0$ will be taken before considering arbitrarily large $|a-b|$.

For $U \in \mathcal {U}\simeq \mathbb R^7$, we define the scale-dependent norm

$$\begin{aligned} \Vert U\Vert _\mathcal {U}= \max \Big \{ |g| L^{\varepsilon (j \wedge j_m)}, |\nu | L^{\alpha (j \wedge j_m)}, |u| L^{j d}, \ell _j \ell _{\sigma ,j} (|\lambda _a| \vee |\lambda _b|), \ell _{\sigma ,j}^2 (|q_a| \vee |q_b|) \Big \} . \end{aligned}$$

(3.16)

We denote the restriction of $\Vert \cdot \Vert _\mathcal {U}$ to $\mathcal {V}$ by the same symbol. Given $C_\mathcal {D}>0$, we define the domain

$$\begin{aligned} \mathcal {D}_j = \{ V \in \mathcal {V}: \Vert V\Vert _\mathcal {U}\le C_\mathcal {D}\bar{s}, \; g > C_\mathcal {D}^{-1} \bar{s}L^{-\varepsilon (j \wedge j_m)} \} \subset \mathcal {V}. \end{aligned}$$

(3.17)

Note that $\mathcal {D}_j$ is a domain in $\mathcal {V}$, and as such, does not involve the coupling constants u or q.

A sequence $\mathcal {W}^\varnothing _j$ of Banach spaces is defined in terms of the $T_\varphi (\mathfrak {h}_j)$ seminorms in [31] (they are denoted $\mathcal {W}_j$ there). We extend $\mathcal {W}^\varnothing _j$ to a space $\mathcal {W}_j \subset \mathcal {C}\mathcal {K}_j$ whose definition is the same with the exception that the $T_\varphi$ seminorms are defined on the extended space $\mathcal {N}$. As in [31, Remark 6.3], we define a sequence

$$\begin{aligned} {\vartheta }_j = L^{-\tfrac{1}{4} \alpha (j - j_m)_+}. \end{aligned}$$

(3.18)

Given a parameter $t> 0$, the domain of the RG map is defined by

$$\begin{aligned} \mathbb {D}_j = \mathcal {D}_j \times B_{\mathcal {W}_j}(t\vartheta _j^3 \bar{s}^3), \end{aligned}$$

(3.19)

where $B_{\mathcal {W}_j}(r)$ is the open ball of radius r in the Banach space $\mathcal {W}_j$.

3.3 Estimates on RG Map

We now specify the RG map $(V_j, K_j) \mapsto (U_{j+1},K_{j+1})= (\delta \zeta _{j+1}, V_{j+1}, K_{j+1})$ and state our bounds on it. To shorten notation, we condense indices and write, e.g., (V, K) for $(V_j, K_j)$ and $(U_+,K_+)$ for $(U_{j+1},K_{j+1})$. The definition of the maps $U_+,K_+$ is described in a general setup in [7, 16], and is adapted to the long-range model with $\sigma _a=\sigma _b=0$ in [31]. The same definitions extend to include observables.

In particular, the map $(V,K) \mapsto U_+=(\delta \zeta _+, V_+) = {{\mathrm{PT}}}(V) + R_+(V,K)$ is explicit and consists of a perturbative part ${{\mathrm{PT}}}$, incorporating second-order perturbation theory, and a nonperturbative, third-order error $R_+$. The explicit map ${{\mathrm{PT}}}$ is the one defined in [7] for $n=0$, extended in [4] to $n \ge 1$, and used in [32] for general $n \ge 0$. Let $\lambda$ denote $\lambda _a$ or $\lambda _b$, and let q denote $q_a$ or $q_b$. We denote the $\lambda , q$ components of the map ${{\mathrm{PT}}}$ by $\lambda _\mathrm{pt}, \delta q_\mathrm{pt}$. For $j<N$, and with $w_j$ given by (3.6), let

$$\begin{aligned} w_j^{(1)} = \sum _{y \in \mathbb Z^{d}}w_{j;x,y}. \end{aligned}$$

(3.20)

By [31, (5.10) and Lemma 5.2],

$$\begin{aligned} w_j^{(1)} = O(L^{\alpha j}). \end{aligned}$$

(3.21)

Let

$$\begin{aligned} \delta [\nu w^{(1)}] = (\nu + (n+2) g C_{+;0,0})w_{+}^{(1)} - \nu w^{(1)}. \end{aligned}$$

(3.22)

Recall the definition of the coalescence scale $j_{ab}$ in (2.24). Then, as in [32, Proposition 3.2], for general $n \ge 0$ the observable part of the map ${{\mathrm{PT}}}$ is the map $V \mapsto (\lambda _\mathrm{pt}, \delta q_\mathrm{pt})$ given by

$$\begin{aligned} \lambda _{\mathrm {pt}}&= {\left\{ \begin{array}{ll} (1 - \delta [\nu w^{(1)}])\lambda &{} (j+1 < j_{ab}) \\ \lambda &{} (j+1 \ge j_{ab}), \end{array}\right. } \end{aligned}$$

(3.23)

$$\begin{aligned} \delta q_{\mathrm {pt}}&= \lambda _a\lambda _b \,C_{j+1;a,b} . \end{aligned}$$

(3.24)

Note that $\lambda _\mathrm{pt}= \lambda$ for all scales $j \ge j_{ab}-1$, i.e., the flow of $\lambda$ stops evolving after scale $j_{ab}-1$. Conversely, since $C_{j+1;a,b}=0$ for $j+1\le j_{ab}$, nonzero $\delta q_{\mathrm{pt}}$ can occur only at scales $j \ge j_{ab}$. The map $(V,K) \mapsto U_+$ is now defined by

$$\begin{aligned} U_+ = {{\mathrm{PT}}}(\hat{V}) \quad \text {with} \quad \hat{V} = V - \!\!\!\! \sum _{Y \in \mathcal{S} ({\varLambda }) : Y \supset B} \!\!\!\! \mathrm{Loc} _{Y,B} I^{-Y} K (Y) . \end{aligned}$$

(3.25)

The localisation operator $\mathrm{Loc} _{Y,B}$ is defined in [14, Definition 1.17]. The higher-order correction $R_+ : \mathcal {V}\rightarrow \mathcal {U}$ to the perturbative calculation is then defined by

$$\begin{aligned} R_+(V, K) = {{\mathrm{PT}}}(\hat{V}) - {{\mathrm{PT}}}(V), \end{aligned}$$

(3.26)

so that $U_+= {{\mathrm{PT}}}(V)+R_+(V,K)$. We do not need the explicit form of $R_+$ and only use the bounds of Theorem 3.3 below.

The map $(V, K)\mapsto K_+$ is also given explicitly in [16], but it is complicated to write down. Like $R_+$, this nonperturbative part of the RG map is of order $O(\bar{s}^3)$. It is part of the statement of Theorem 3.3 below that the formula for $K_+$ constructed in [16] is well-defined on the domain specified in Theorem 3.3. We do not need to know more here about $K_+$ than the estimates provided by Theorem 3.3.

The RG map depends on the mass $m^2$ through its dependence on the covariance $C_{+}$. We require continuity in the mass in the limit $m^2\downarrow 0$, which can only be taken after the infinite-volume limit $N\rightarrow \infty$. Given small $\delta > 0$, we define the mass domain for the RG map by

$$\begin{aligned} \mathbb {I}_j = {\left\{ \begin{array}{ll} {[}0,\delta ] &{} (j<N) \\ {[}\delta L^{-\alpha (N-1)},\delta ] &{} (j=N). \end{array}\right. } \end{aligned}$$

(3.27)

The special attention to $j=N$ is due to the fact that the final covariance $C_{N,N}$ is only defined for $m^2>0$, and it obeys good estimates for $m^2\in \mathbb {I}_N$.

The following extends [31, Theorem 6.4] to allow for the presence of observables. Its estimates appear identical to [31, Theorem 6.4], but it is in fact an extension since the domain and range of the RG map now include observables in $(V,K)\in \mathbb {D}$, as well as in $R_+$ and $K_+$. Note that the map $R_+$, which acts on (V, K) with $V \in \mathcal {V}$, produces a polynomial in $\mathcal {U}$ which in particular contains the nonperturbative contributions to $\delta \zeta$. The bound (3.29) on $R_+$ controls these nonperturbative contributions to $\delta \zeta$. Note that the estimates (3.29) hold for $m^2 \in \mathbb {I}_+$, but the continuity is in the smaller interval $m^2 \in [0,L^{-\alpha j}]$. A restriction like this on the continuity interval is essential, because larger $m^2$ will put j above the mass scale, at which point the spaces themselves become dependent on $m^2$ through their dependence on $\ell _j$ and a continuity statement becomes meaningless.

Theorem 3.3

Let $d =1,2,3$; $n \ge 0$; $\alpha = \frac{1}{2} (d+\varepsilon )$ and $j<N$. Let $C_\mathcal {D}$ and L be sufficiently large, and let $\varepsilon$ be sufficiently small. There exist $c>0$, $C_\mathrm{RG}>0$, $\delta >0$, such that, with the domain $\mathbb {D}$ defined using $t=4 C_\mathrm{RG}$, the maps

$$\begin{aligned} R_+:\mathbb {D}\times \mathbb {I}_+ \rightarrow \mathcal {U}, \quad K_+:\mathbb {D}\times \mathbb {I}_+ \rightarrow \mathcal {W}_{+} \end{aligned}$$

(3.28)

are analytic in (V, K), provide a solution to (3.3), and satisfy the estimates

$$\begin{aligned} \Vert R_+\Vert _{\mathcal {U}_+}&\le c {\vartheta }_+ \bar{s}^{3} , \quad \quad \Vert K_+\Vert _{\mathcal {W}_+} \le C_\mathrm{RG}{\vartheta }_+^3 \bar{s}^{3}. \end{aligned}$$

(3.29)

The coordinate in $R_+$ corresponding to $\delta q_{a,j},\delta q_{b,j}$ is identically zero for $j\le j_{ab}$, and the coordinate corresponding to $\lambda _{a,j},\lambda _{b,j}$ is identically zero for $j \ge j_{ab}$. In addition, $R_+,K_+$ are jointly continuous in $m^2 \in [0,L^{-\alpha j}], V,K$.

Proof

The theorem is a consequence of the main result of [16], which focusses on the 4-dimensional nearest-neighbour case. For the long-range model, the appropriate modifications for the bulk part of the RG map are discussed in [31], and we assume familiarity with both the methodology and the modifications. In order to include observables, only minor further modifications are required, compared to [15, 16].

One requirement is to verify that, for $V\in \mathcal {D}$, the basic small parameters $\varepsilon _V$ and $\bar{\varepsilon }$ obey appropriate estimates when observables are present, as in [15, Sects. 3.2–3.3]. We verify this here; this verification validates our choice (3.15) for the norm parameters. (In fact, somewhat larger domains are used in [15, Sects. 3.2–3.3]; the main ideas are present for $V\in \mathcal {D}$, which we consider here, and the extension to the larger domains is a matter of bookkeeping.) A second requirement is to verify that the “crucial contraction” is maintained in the presence of observables, and we also verify this here.

Bound on $\varepsilon _V$. Let $V\in \mathcal {D}$. For $\varepsilon _V$, it suffices to observe that for $|\lambda _a| \le C_\mathcal {D}\bar{s}\ell ^{-1}\ell _\sigma ^{-1}$,

$$\begin{aligned} \Vert \lambda _a \sigma _a \varphi _a^1\Vert _{T_0(\mathfrak {h})}&= |\lambda _a| \mathfrak {h}_{\sigma }\mathfrak {h}\le C_\mathcal {D}\bar{s}\frac{\mathfrak {h}_{\sigma }}{\ell _\sigma } \frac{\mathfrak {h}}{ \ell } = {\left\{ \begin{array}{ll} C_\mathcal {D}\bar{s}&{} (\mathfrak {h}=\ell ) \\ C_\mathcal {D}k_0 \ell _0^{-1} &{} (\mathfrak {h}=h), \end{array}\right. } \end{aligned}$$

(3.30)

which implies stability on the domain $\mathcal {D}$ of (3.17), and complements the arguments of [16, 31].

Bound on $\bar{\varepsilon }$. We must also verify the analogue of [15, Lemma 3.4]. To state the desired estimate, as in [31, (6.56)] we define the norm parameter

$$\begin{aligned} \hat{\ell }_j = \hat{\ell }_0 \ell _j L^{-\frac{1}{2} (\alpha -\alpha ')(j-j_m)_+}, \end{aligned}$$

(3.31)

and as in [31, (6.25)] we define the small parameter

$$\begin{aligned} \bar{\varepsilon }&= \bar{\varepsilon }_j(\mathfrak {h}) = {\left\{ \begin{array}{ll} \bar{s}{\vartheta }_j &{} (\mathfrak {h}=\ell ) \\ \bar{s}^{1/4} &{} (\mathfrak {h}=h,\; j \le j_m) . \end{array}\right. } \end{aligned}$$

(3.32)

We write $U_\mathrm{pt}= {{\mathrm{PT}}}(V)$ and let $\delta V = \theta V - U_\mathrm{pt}$. Our goal then is to show that, for $V \in \mathcal {D}$,

$$\begin{aligned} \max _{B\in \mathcal {B}} \Vert \delta V(B)\Vert _{T_0(\mathfrak {h}\sqcup \hat{\ell })} \le C_{\delta V} \bar{\varepsilon }\end{aligned}$$

(3.33)

where $C_{\delta V}$ is an L-dependent constant.

It is argued in [31, Sect .6.4.4] that (3.33) holds with $\delta V$ replaced by $\delta V^\varnothing = \pi _\varnothing \delta V$. Thus, it suffices to establish (3.33) with $\delta V$ replaced by $\delta V^* = \pi _* \delta V$. This can be done by writing

$$\begin{aligned} \Vert \delta V^*(B)\Vert _{T_0(\mathfrak {h}\sqcup \hat{\ell })}\le \Vert \theta V^*(B) - V^*(B)\Vert _{T_0(\mathfrak {h}\sqcup \hat{\ell })} + \Vert V^*(B) - U_\mathrm{pt}^*(B)\Vert _{T_0(\mathfrak {h}\sqcup \hat{\ell })} \end{aligned}$$

(3.34)

and applying the triangle inequality to estimate each of the two terms on the right-hand side.

For instance, if $a \in B$, then the $\sigma _a$ term of $\theta V^*(B) - V^*(B)$ is $\lambda _a \sigma _a \zeta ^1_a$. By definition of the norm, by (3.31), (3.15), (3.32), (3.18), and by the fact that $\alpha ' < \tfrac{1}{2} \alpha$,

$$\begin{aligned} \Vert \lambda _a\sigma _a\zeta ^1_a\Vert _{T_0(\mathfrak {h}\sqcup \hat{\ell })} = |\lambda _a|\mathfrak {h}_{\sigma }\hat{\ell }\le C_\mathcal {D}\bar{s}\hat{\ell }_0 L^{-\frac{1}{2} (\alpha - \alpha ') (j - j_m)_+} \frac{\mathfrak {h}_\sigma }{\ell _\sigma } \le C_\mathcal {D}\hat{\ell }_0 \bar{\varepsilon }(\mathfrak {h}). \end{aligned}$$

(3.35)

The $\sigma _a$ term of $V^*(B) - U_\mathrm{pt}^*(B)$ is zero above the coalescence scale, whereas if $j+1 < j_{ab}$ then it is $\delta [\nu w^{(1)}] \lambda _a \sigma _a \varphi ^1_a$, by (3.23). Thus, by (3.30), it is sufficient to show that

$$\begin{aligned} |\delta [\nu w^{(1)}]| \le \bar{s}^{1/4} {\vartheta }. \end{aligned}$$

(3.36)

By its definition in (3.22),

$$\begin{aligned} \delta [\nu w^{(1)}] = \nu \sum _x C_{+:0,x} + (n + 2) g C_{+;00} w^{(1)}_+. \end{aligned}$$

(3.37)

By (3.17) and (2.23), and the finite-range property (2.7), the first term is bounded by

$$\begin{aligned} O(\bar{s}) L^{-\alpha (j \wedge j_m)} L^{jd} L^{-(d - \alpha ) j - 2 \alpha (j - j_m)_+} = O(\bar{s}) L^{-\alpha (j - j_m)_+} , \end{aligned}$$

(3.38)

and the second term is bounded by

$$\begin{aligned} O(\bar{s}) L^{-\varepsilon (j \wedge j_m)} L^{-(d - \alpha ) j - 2 \alpha (j - j_m)_+} L^{\alpha j} = O(\bar{s}) L^{- d (j - j_m)_+}. \end{aligned}$$

(3.39)

These bounds do better than what is required by (3.36).

For the $\sigma _a\sigma _b$ term, we can take $j \ge j_{ab}$. The $\sigma _a \sigma _b$ term of $\theta V^* - V^*$ is always 0 and the coefficient of $\sigma _a\sigma _b$ in $V^*(B) - U_\mathrm{pt}^*(B)$ is at most $|C_{+;a,b} \lambda _a\lambda _b|$. By (3.17), (2.23), and (3.13) and the fact that $\alpha '<\frac{1}{2} \alpha$,

$$\begin{aligned} \Vert C_{+;a,b} \lambda _a\lambda _b \sigma _a\sigma _b\Vert _{T_0(\mathfrak {h}\sqcup \hat{\ell })} \le O(\bar{s}^2) \frac{|C_{+;a,b}|}{\ell ^2} \frac{\mathfrak {h}_\sigma ^2}{\ell _\sigma ^2} \le O(\bar{s}^2) {\vartheta }^2 \frac{\mathfrak {h}_\sigma ^2}{\ell _\sigma ^2}. \end{aligned}$$

(3.40)

When $\mathfrak {h}= h$ (hence $j < j_m$) this is $O(\bar{\varepsilon }(h)^2)$, and when $\mathfrak {h}= \ell$ it is $O(\bar{\varepsilon }(\ell )^2)$. This is better than what is required for (3.33).

Crucial contraction The adaptation of the crucial contraction to the long-range model is provided for the bulk in [31, Sect. 6.4.5–6.4.6]. We now extend the adaptation to include observables.

Below the mass scale, the least irrelevant of the sign invariant monomials involving the observable fields each have two additional spin fields compared to their marginal counterparts $\sigma _a \varphi _a^1$ and $\sigma _a\sigma _b$ (the latter occurs only above the coalescence scale), so have dimension which is larger by $2[\varphi ]=d-\alpha$. Compared to [31, (6.64)], this gives rise to $\gamma = L^{-(d-\alpha )}$, and there is no factor $L^{d}$ for observables, so the gain here is proportional to $L^{-(d-\alpha )}$. The worst $\gamma$ occurs for $d=1$, where we have $\gamma = L^{-\frac{d}{2} + \frac{\varepsilon }{2}} = L^{-\frac{1}{2} + \frac{\varepsilon }{2}}$. This is consistent with the values of $L^d\gamma$ reported for the bulk in [31, (6.64)].

Above the mass scale, we extend the discussion in [31, Sect. 6.4.6], as follows. For the perturbative contribution to K, we have already verified that we can continue to use the $\bar{\varepsilon }$ given by (3.32) when observables are present, and there is therefore no change to [31] concerning this issue. It remains to consider the crucial contraction.

We recall and invoke our assumption that $j_{ab}<j_m$. Now $d_+^a=0$, so the least irrelevant monomial in $\mathcal {N}^a$ is $\sigma _a \varphi$. This scales as

$$\begin{aligned} \ell _{\sigma ,j}\ell _j&= \frac{\ell _j}{\ell _{j_{ab}}}2^{j-j_{ab}}\bar{s}= \frac{L^{-\frac{1}{2} (d-\alpha )j}L^{-(\alpha +\alpha ')(j-j_m)}}{L^{-\frac{1}{2} (d-\alpha )j_{ab}}}2^{j-j_{ab}} \bar{s}\nonumber \\&\le L^{-\frac{1}{2} (d+ \alpha ')(j-j_m)} 2^{j-j_m}\bar{s}. \end{aligned}$$

(3.41)

A change from scale j to scale $j+1$ in the above right-hand side gives rise to a factor $2L^{-\frac{1}{2} (d+\alpha ')}$. As in [31, (6.68)–(6.69)], the essential condition here is that the product of this factor with ${\vartheta }^{-3} = L^{\frac{3}{4} \alpha }$ should be bounded above by an inverse power of L. This condition is indeed satisfied, since

$$\begin{aligned} \tfrac{1}{2}(d+\alpha ') - \tfrac{3}{4} \alpha = \tfrac{1}{2}\left( d+\alpha ' - \tfrac{3}{4}(d +\varepsilon ) \right) = \tfrac{1}{2}\left( \tfrac{1}{4} d+\alpha ' - \tfrac{3}{4}\varepsilon ) \right) > \tfrac{1}{8} d. \end{aligned}$$

(3.42)

Similarly, the least irrelevant monomial in $\mathcal {N}^{ab}$ that is sign invariant is of the form $\sigma _a \sigma _b \varphi \varphi$, and has scaling dimension twice that considered in the previous paragraph, so twice as good. Thus the crucial contraction is not harmed by the presence of observables.

Estimate for $R_+$ above the mass scale Finally, we consider the extension of [31, Sect. 6.4.7] to include observables. The observable terms have the same $T_0$ and $\mathcal {U}$ norms: $\Vert \sigma _a\varphi _a^1\Vert _{T_0} = \ell _\sigma \ell = \Vert \sigma _a\varphi _a^1\Vert _{\mathcal {U}}$ and $\Vert \sigma _a\sigma _b\Vert _{T_0} = \ell _\sigma ^2 = \Vert \sigma _a\sigma _b\Vert _{\mathcal {U}}$. This leads to an extension to [31, Lemma 6.6], as follows. Let

(3.43)

(3.44)

The estimates of [31, Lemma 6.6] now become

$$\begin{aligned} \Vert F_1\Vert _{\mathcal {U}} \le c_L L^{\alpha '(j-j_m)_+}\Vert F_1(B)\Vert _{T_0}, \quad \quad \Vert F_2(B)\Vert _{T_0} \le c \Vert F_2\Vert _{\mathcal {U}}, \end{aligned}$$

(3.45)

i.e., the bound remains the same for $F_1$ but loses a helpful factor $L^{-\alpha '(j-j_m)_+}$ for $F_2$. The bound on $F_1$ then implies, as in [31, (6.74)], that

$$\begin{aligned} \Vert R_+\Vert _\mathcal {U}\le O(L^{\alpha '(j-j_m)_+})\Vert R_+(B)\Vert _{T_0}, \end{aligned}$$

(3.46)

and the bound (3.29) follows from this as in [31, Sect. 6.4.7].

The introduction of observables does lead to a change in the bounds on F, W, P in [31, Lemma 6.7], due to the weakened estimate for $F_2$ in (3.45). The change is to replace the factors $L^{-(\alpha +\alpha ')(j-j_m)_+}$ and $(c/L)^{-(\alpha +\alpha ')(j-j_m)_+}$ in the three upper bounds of [31, Lemma 6.7] by the worse factor ${\vartheta }^2=L^{-\frac{1}{2} \alpha (j-j_m)_+}$. Since we seek an upper bound which includes the factor ${\vartheta }^2$ in $\bar{\varepsilon }^2$, the weakened bounds remain more than good enough.

For general reasons, $\pi _{ab}W=0$ [15, Proposition 4.10], so there can be no such term in W. Thus, in the proof of [31, Lemma 6.7], only one factor $L^{-\alpha '(j-j_m)_+}$ can be lost by application of (3.45), not two. Also, by direct calculation, the relevant contribution to F is $F_C(\lambda _a\sigma _a\varphi _a^1,\lambda _b\sigma _b\varphi _b^1) = \tfrac{1}{2}\lambda _a\lambda _bC_{a,b}\sigma _a\sigma _b$, whose $T_0(\ell )$ norm is given as in the first inequality of (3.40) to be at most $L^{-(\alpha -\alpha ')(j-j_m)}$, which is better than the required ${\vartheta }^2$. The bound on P follows from the bounds on F, W as in [15, Proposition 4.1]. $\square$

In the absence of observables, Theorem 3.3 is used in [31] to construct a global RG flow $(g_j,\nu _j,K_j^\varnothing )$ that remains in the RG domain for all j. This requires tuning the initial $\nu$ to a mass-dependent critical value $\nu _0^c(m^2)$; this value converges to the critical point $\nu _c(g;n)$ as $m^2 \downarrow 0$ (see [31, (8.93)–(8.94)]). Throughout the remainder of the present paper, we always take $(g_j,\nu _j)$ to be this global flow of coupling constants. For general reasons this flow is the same in the presence of observables as in their absence: see [16, (1.68)–(1.69)]. The main task for the proof of Theorem 1.1 is to apply the estimates of Theorem 3.3 to control, in addition, the flow of the observable coupling constants $\lambda$ and $\delta q$, and the observable part of the coordinate K. The flow of $\delta q$ and K is analysed as in the 4-dimensional nearest-neighbour case [5, 32].

The flow of $\lambda$ is marginal, for the same reasons as in the 4-dimensional case. In [5, 32], the perturbative approximation (3.23) to the recursion for $\lambda$ is solved along the lines of the rough computation

$$\begin{aligned} \lambda _j&= \prod _{k=1}^{j-1} (1-\delta [\nu w^{(1)}]) = \exp \left[ \sum _{k=1}^{j-1} \log ( 1-\delta [\nu w^{(1)}]) \right] \nonumber \\&\approx \exp \left[ -\sum _{k=1}^{j-1} \delta [\nu w^{(1)}] \right] = \exp \left[ -\nu _{j} w_{j}^{(1)} \right] \approx 1 - \nu _{j} w_{j}^{(1)} . \end{aligned}$$

(3.47)

In [5, 32], the errors introduced by the map $R_+$ into (3.23) were summable over all scales because of the decay of the marginal coupling constant $g_j$ with the scale (Gaussian fixed point), and the above computation survives the introduction of these errors.

For the long-range model, the fixed point is non-Gaussian, and the corrections due to $R_+$ are not summable. Instead of trying to follow the route laid out in [5, 32], we derive an exact relation between $\lambda _{a,j}$ and the known bulk coupling constants, similar to (3.47), which gives better control of its flow than the recursion. This is done in Sect. 4.

4 Flow of $\lambda$

According to (3.23) and Theorem 3.3, the flow of $\lambda _{a,j}$ under the RG map is nontrivial only until scale $j_{ab}-1$, and stops beyond this scale. Conversely, $q_{a,j}=0$ for $j < j_{ab}$, and the flow of $q_{a,j}$ is nontrivial only for scales $j \ge j_{ab}$. Our goal now is to determine the form of the flow until scale $j_{ab}$. Since we later take the limit $m^2\downarrow 0$ before studying large $j_{ab}\sim \log _L \vert a-b\vert$, we can and do assume that $j_{ab}< j_m$. We will prove the following proposition.

Proposition 4.1

Let $n \ge 0$, let L be sufficiently large, let ${\varLambda }_N$ be the torus of period $L^N$, and let $\varepsilon$ be sufficiently small. Let $g \in [\frac{63}{64}\bar{s},\frac{65}{64}\bar{s}]$, and let $m^2 \in [L^{-\alpha (N-1)}, \delta ]$ with $\delta >0$ sufficiently small. Let $j_{ab}< j_m < N$. Let $g_0=g$, let $\nu _0$ be the critical value $\nu _0^c(m^2)$ constructed in [31], and let $\lambda _{a,0}=\lambda _{b,0}=1$. Then the RG map can be iterated to scale $j_{ab}$, i.e., it produces a sequence $(V_j,K_j)\in \mathbb {D}_j$ with initial condition $(V_0,0)$, such that (3.1) holds for all $j \le j_{ab}$ with $I_j=I_j(V_j)$ and $\zeta _j = \sum _{k=1}^j \delta \zeta _j$. Moreover, $q_{x,j}=0$, and for the component $\lambda _{x,j}$ of this flow we have the stronger statement

$$\begin{aligned} \lambda _{x,j} = 1 -\nu _j w_j^{(1)} +O(\bar{s}^2) \quad \quad (j < j_{ab}, \; x=a,b). \end{aligned}$$

(4.1)

The proof of Proposition 4.1 is given in Sect. 4.1 below. Its statement holds trivially at $j=0$, and will be established inductively for higher scales. The induction for the bulk quantities $U_j^\varnothing ,K_j^\varnothing$ is the result of [31], and is unaffected by the presence of observables.

The main additional ingredient for the induction of the observable parts is to establish the flow (4.1) of $\lambda _{a,j}$. To achieve this, in Lemma 4.2 we use integration by parts to obtain a relation between $\lambda _{a,j}$, quantities of the bulk flow, and the observable parts of the coordinate $K_j$. This is achieved by taking suitable derivatives of the identity $Z_j=\mathbb {E}_{w_j} \theta Z_0$. The contribution due to $K_j$ is bounded uniformly in the volume using a cluster expansion, in Sect. 4.2.

The formula (4.1) for $\lambda _{x,j}$ has a natural counterpart for the nearest-neighbour 4-dimensional case, with error term $O(g_j^2)$ instead of $O(\bar{s}^2)$. In that context, $-\nu _j w_j^{(1)} +O(g_j^2) \rightarrow 0$ as $j \rightarrow \infty$. This provides insight into the fact that $\lim _{j\rightarrow \infty } \lambda _{x,j}=1$ in [5, Lemma 4.6] and [32, Corollary 6.4]. For the long-range model considered in Proposition 4.1, the non-Gaussian fixed point leads to a limit which is not equal to 1.

4.1 Integration by Parts

For notational convenience we restrict attention to $n \ge 1$; small modifications apply for $n=0$. Recall that $\bar{D}$ and $D_{\sigma _a}$ are defined above Corollary 3.1, and that $Z_j=\mathbb {E}_{w_j} \theta Z_0$. Let

$$\begin{aligned} z_j=z_j({\varLambda }) = e^{-\zeta _j}\frac{Z_j({\varLambda })}{I_j({\varLambda })}, \quad \quad \mathcal {L}_j=\mathcal {L}_j({\varLambda }) = \log z_j({\varLambda }). \end{aligned}$$

(4.2)

Then we have

$$\begin{aligned} Z_j = e^{\zeta _j}I_j({\varLambda }) z_j({\varLambda }) = e^{\zeta _j}I_j({\varLambda }) e^{\mathcal {L}_j({\varLambda })}. \end{aligned}$$

(4.3)

The existence of the logarithm $\mathcal {L}_j$ is discussed in Sect. 4.2, where it is constructed as an element of a Banach space $T_0(\ell _j)$ which only examines derivatives at zero field, using a cluster expansion. Bounds on $\mathcal {L}_j$ and its derivatives at zero field are proved in Proposition 4.4 below.

Lemma 4.2

The functions $I_j$ and $\mathcal {L}_j$ are related by the identity

$$\begin{aligned} \bar{D} D_{\sigma _a} I_j({\varLambda }) + \bar{D} D_{\sigma _a} \mathcal {L}_j({\varLambda }) = 1 + \frac{1}{|{\varLambda }|} w_j^{(1)} \left[ \bar{D}^2 I_j^\varnothing ({\varLambda }) + \bar{D}^2 \mathcal {L}^\varnothing _j({\varLambda }) \right] . \end{aligned}$$

(4.4)

Proof

By definition, followed by Gaussian integration by parts,

$$\begin{aligned} D_{\sigma _a} \mathbb {E}_{w_j} \theta Z_0&= \mathbb {E}_{w_j} (\varphi _a^1+ \zeta _a^1) Z_0^\varnothing (\varphi +\zeta ) \nonumber \\&= \varphi _a^1 \mathbb {E}_{w_j} Z_0^\varnothing (\varphi +\zeta ) + \sum _{y \in {\varLambda }} w_{j;a,y}\mathbb {E}_{w_j} \frac{\partial }{\partial \zeta _y^1} Z_0^\varnothing (\varphi +\zeta ). \end{aligned}$$

(4.5)

On the right-hand side, $\frac{\partial }{\partial \zeta _y^1}$ can be replaced by $\frac{\partial }{\partial \varphi _y^1}$, and the latter commutes with the expectation. Then application of $\bar{D} = \sum _{x\in {\varLambda }}\frac{\partial }{\partial \varphi _x^1}|_{\varphi =0}$ gives

$$\begin{aligned} \bar{D} D_{\sigma _a} \mathbb {E}_{w_j} \theta Z_0&= \mathbb {E}_{w_j} Z_0^\varnothing + \sum _{y \in {\varLambda }} w_{j;a,y} \sum _{x \in {\varLambda }} \frac{\partial ^2}{\partial \varphi _y^1\partial \varphi _x^1} \Big |_{\varphi =0} \mathbb {E}_{w_j} Z_0^\varnothing (\varphi +\zeta ), \end{aligned}$$

(4.6)

which by translation invariance and by definition of $Z_j$ is the same as

$$\begin{aligned} \bar{D} D_{\sigma _a} Z_j&= Z_j^\varnothing |_{\varphi =0} + w_j^{(1)} \frac{1}{|{\varLambda }|} \bar{D}^2 Z_j^\varnothing . \end{aligned}$$

(4.7)

Now we divide both sides of (4.7) by $Z_j^\varnothing |_{\varphi =0}$ and use (4.3). Since $I_j|_{\varphi =0}=1$, and since $\bar{D} Z^\varnothing _j=\bar{D} I_j|_{\sigma _a=\sigma _b=0} = D_{\sigma _a} I_j\vert _{\varphi =0} = 0$ by symmetry, the result is (4.4). $\square$

Note that the right-hand side of (4.4) involves only bulk quantities, while the left-hand side depends on $\lambda _{a,j}$ through $I_j({\varLambda })$ and $D_{\sigma _a} \mathcal {L}_j({\varLambda })$, and also on the observable part of the irrelevant coordinate $K_j$ (through $D_{\sigma _a}\mathcal {L}_j({\varLambda })$). For the explicit terms, we have the following identities.

Lemma 4.3

For $j \le N$ and $V \in \mathcal {V}$,

$$\begin{aligned} \bar{D}^2 I_j^\varnothing ({\varLambda }) = - \nu _j |{\varLambda }|, \end{aligned}$$

(4.8)

and if $j < j_{ab}$ then

$$\begin{aligned} \bar{D} D_{\sigma _a} I_j({\varLambda }) = \lambda _{a,j}. \end{aligned}$$

(4.9)

Proof

We differentiate the formula $I_{j}({\varLambda }) = e^{-V_j({\varLambda })}\prod _{B\in \mathcal {B}_j({\varLambda })} (1+ W_j(V_j,B))$, which is (3.8). We apply the product rule, Corollary 3.1, and the facts that $\bar{D} V_j^\varnothing =0$ and $I_j\vert _{\varphi =0}=1$, to obtain

$$\begin{aligned} \bar{D}^2 I_{j}^\varnothing ({\varLambda })&= -\nu _j\vert {\varLambda }\vert + \sum _{B\in \mathcal {B}_j({\varLambda })}\bar{D}^2 W_{j}^\varnothing (V_j,B) = -\nu _j \vert {\varLambda }\vert . \end{aligned}$$

(4.10)

Similarly, for $j < j_{ab}$, we also use $D_{\sigma _a} V_j({\varLambda }) = -\lambda _{a,j} \varphi ^1_a$ to obtain

$$\begin{aligned} \bar{D} D_{\sigma _a} I_{j}({\varLambda }) = \lambda _{a,j} + \sum _{B\in \mathcal {B}_j({\varLambda })}\bar{D} D_{\sigma _a} W_{j}(V_j,B) = \lambda _{a,j} , \end{aligned}$$

(4.11)

and the proof is complete. $\square$

We now state our bounds for the terms in (4.4) involving $\mathcal {L}_j$. The hypothesis $(V_j,K_j)\in \mathbb {D}_j$ of Proposition 4.4 will be established inductively.

Proposition 4.4

Let $j \le j_{ab}$, let $\mathcal {L}_j$ be defined as in (4.2), and assume that $Z_j = e^{\zeta _j}(I_j\circ K_j)$ with $I_j=I_j(V_j)$ and $(V_j,K_j) \in \mathbb {D}_j$. Then there is a constant $c_1>0$ such that

$$\begin{aligned} |\bar{D} D_{\sigma _a} \mathcal {L}_j({\varLambda })|&\le c_1 \bar{s}^2, \quad \quad |\bar{D}^2 \mathcal {L}_j^\varnothing ({\varLambda }) | \le c_1 \vert {\varLambda }\vert L^{-\alpha j}\bar{s}^3. \end{aligned}$$

(4.12)

We defer the proof of Proposition 4.4 to Sect. 4.2.

Proof

of Proposition 4.1 The proof is by induction on j. The statement of Proposition 4.1 for $j=0$ is trivial. Without loss of generality, we consider the case $x=a$. We assume that we have (3.1) for $Z_k$ with $(V_k,K_k)$ constructed inductively using the RG map for $k \le j$, and we make the constant in the hypothesis (4.1) explicit by assuming that, with $c_1$ from (4.12),

$$\begin{aligned} |\lambda _{a,j}-1+\nu _jw_j^{(1)}| \le 2c_1 \bar{s}^2. \end{aligned}$$

(4.13)

Then we have (3.1) with a pair of RG coordinates $(V_j,K_j)\in \mathbb {D}_j$, satisfying in addition (4.13). Theorem 3.3 guarantees the existence of RG coordinates $(U_{j+1},K_{j+1})=(\delta \zeta _{j+1}=\delta u_{j+1},V_{j+1},K_{j+1})$ at scale $j+1$ such that $Z_{j+1}$ obeys (3.1), with $U_{j+1} = {{\mathrm{PT}}}_{j}(V_j) + R_{j+1}(V_j,K_j)$, and bounds on $R_{j+1}(V_j,K_j)$ and $K_{j+1}$ as in (3.29).

It has been proved in [31] that the bulk part of $V_{j+1}$ lies in $\mathbb {D}_{j+1}$. The second bound in (3.29) is sufficient to guarantee that $K_{j+1}$ also lies in $\mathbb {D}_{j+1}$. Therefore, to complete the proof that $(V_{j+1},K_{j+1})\in \mathbb {D}_{j+1}$, we only need to show that $\vert \lambda _{j+1}\vert < C_{\mathcal {D}}$, where $C_{\mathcal {D}} >1$ is the constant in (3.17). By (3.23), and by the first bound of (3.29) together with the definition of the norm in (3.16), we have $\lambda _{j+1} = (1+O(\bar{s})) \lambda _j + O(\bar{s}^2)$. It now follows immediately from (4.13) that $0< \lambda _{j+1} = 1+O(\bar{s})<C_{\mathcal {D}}$, since $\bar{s}$ can be chosen small enough. This proves that $(V_{j+1},K_{j+1})\in \mathbb {D}_{j+1}$.

To complete the induction, we must prove (4.13) with j replaced by $j+1$. Since (4.1) is only required for scales below the coalescence scale, we may assume here that $j+1<j_{ab}$. The bounds of Proposition 4.4 at scale $j+1$ can be applied, since the hypothesis $(V_{j+1},K_{j+1})\in \mathbb {D}_{j+1}$ has now been verified. Also, the hypothesis $j+1<j_{ab}$ of Lemma 4.3 is satisfied. We use Lemma 4.3 in conjunction with (4.4), and apply Proposition 4.4. This gives

$$\begin{aligned} \big \vert \lambda _{a,j+1} -1 + \nu _{j+1}w_{j+1}^{(1)}\big \vert \le c_1\bar{s}^2 + c_1 w_{j+1}^{(1)}L^{-\alpha (j+1)} \bar{s}^3 \le 2c_1 \bar{s}^2, \end{aligned}$$

(4.14)

by (3.21) and by taking $\bar{s}$ sufficiently small. This advances (4.13) to scale $j+1$, and completes the proof. $\square$

4.2 Cluster Expansion

In this section, we use a cluster expansion to construct a formula for $\mathcal {L}_j = \log z_j$ and prove Proposition 4.4. Let $p(X) = K_j(X)/I_j(X)$. By (3.1), (4.2), and by definition of the circle product,

$$\begin{aligned} z_j = I_j({\varLambda })^{-1} (I_j\circ K_j)({\varLambda }) = \sum _{X\in \mathcal {P}_j({\varLambda })} p(X), \end{aligned}$$

(4.15)

where the term in the sum with $X=\varnothing$ is interpreted as 1. In the sum, we decompose $X\in \mathcal {P}_j$ into its connected components $X_1,\ldots ,X_n \in \mathcal {C}_j$, which may be labelled in n! different ways. For $X,X'\in \mathcal {C}_j$, we set $g(X,X')=-1$ if X and $X'$ touch, and otherwise set $g(X,X')=0$. Using the component factorisation property of $K_j$, we obtain

$$\begin{aligned} z_j = \sum _{n= 0}^\infty \frac{1}{n!} \sum _{X_1,\ldots ,X_n\in \mathcal {C}_j} p(X_1)\cdots p(X_n) \prod _{1 \le i<j \le n}(1+g(X_i,X_j)), \end{aligned}$$

(4.16)

where the $n=0$ term is again interpreted as 1. This has the form of the partition function of a polymer system, as defined, e.g., in [34, (1)]. It is a standard result, e.g., [19, 34], that $\log z_j$ can be written as a cluster expansion and accurately bounded, provided the polymer activities p(X) obey suitable estimates. In the following proof, we discuss this in detail and invoke a convergence criterion from [34]; see also [21, 30] for pedagogical introductions to the cluster expansion. The verification of the criterion from [34] is an almost immediate consequence of the norm estimates in the definition of the domain $\mathbb {D}_j$.

Since we are interested only in the derivatives of $\mathcal {L}_j$ at zero external and observable fields, we do not construct $\mathcal {L}_j$ as a function of these fields (even though this would also be possible for suitably small fields), but rather as a Taylor polynomial (jet) of order $p_{\mathcal {N}}$ in the fields around zero. In other words, we work on the quotient of $\mathcal {N}$ by the ideal of elements of $F\in \mathcal {N}$ with $\Vert F\Vert _{T_0(\ell _j)} = 0$. On this quotient, the $T_0(\ell _j)$ seminorm becomes a norm, and the quotient becomes a finite-dimensional Banach algebra. This is discussed in detail in [16, Sect. 1.7.3, Appendix A]. We adopt the point of view in the following that we work in this normed space, and write simply $\Vert \cdot \Vert$ for $\Vert \cdot \Vert _{T_0(\ell _j)}$. Although the results of [34] are stated for complex-valued p(X), the proofs hold verbatim for values in any Banach algebra. The completeness of the Banach algebra is important for the existence of $\mathcal {L}_j = \log z_j$, which is defined in terms of an infinite sum.

The estimates we use, for $(V_j,K_j) \in \mathbb {D}_j$ and $X \in \mathcal {C}_j$, are:

$$\begin{aligned} \Vert 1/I_j(X)\Vert \le 2^{|X|}, \quad \quad \Vert K_j(X)\Vert \le M \bar{s}^{3+ a(|X|-2^d)_+}, \end{aligned}$$

(4.17)

where $a>0$ is small; here, $|X| = |X|_j$ denotes the number of j-blocks in X. The bound on $I_j^{-X}$ is a small adaptation of [15, Proposition 2.2] to our long-range setting, and the bound on $K_j$ follows from the definition of the $\mathcal {W}_j$ norm and (3.19). Absorbing the factor $2^{|X|}$ by replacing M by $M'>M$, replacing a by $a'\in (0,a)$, and using the fact that $\bar{s}$ is sufficiently small, we conclude that the polymer activity obeys the bound

$$\begin{aligned} \Vert p(X)\Vert \le M' \bar{s}^{3+ a'(|X|-2^d)_+}. \end{aligned}$$

(4.18)

The following lemma uses this bound and will be employed to verify the hypothesis of [34, Theorem 1].

Lemma 4.5

If $B\in \mathcal {B}_j$, then for $\bar{s}$ sufficiently small (depending only on d)

$$\begin{aligned} \sum _{Y\in \mathcal {C}_j} |g(B, Y)| \Vert p(Y)\Vert e^{|Y|} \le O(\bar{s}^3), \end{aligned}$$

(4.19)

where the constant depends on d.

Proof

The number of connected polymers $Y \in \mathcal {C}_j$ that touch a block Y and have size $|Y|=n$ is at most $A^n$ for some d-dependent constant A. Thus,

$$\begin{aligned} \sum _{Y\in \mathcal {C}_j} |g(B, Y)| \Vert p(Y)\Vert e^{|Y|} \le M' \bar{s}^3 \sum _{n=1}^{|{\varLambda }|} (Ae)^n \bar{s}^{a' (n-2^d)_+}. \end{aligned}$$

(4.20)

We split the sum on the right-hand side into sums with $n \le 2^d$ and $n > 2^d$. The first of these is a constant that depends only on d. Taking $\bar{s}$ small so that $A e \bar{s}^{a'} < 1$, the second sum is bounded as

$$\begin{aligned} \bar{s}^{-a' 2^d} \sum _{n=2^d+1}^{|{\varLambda }|} (A e \bar{s}^{a'})^n = O(\bar{s}^{a'}) \end{aligned}$$

(4.21)

with a d-dependent constant, which suffices. $\square$

Proof

of Proposition 4.4 By Lemma 4.5, if $\bar{s}$ is sufficiently small, then

$$\begin{aligned} \sum _{Y\in \mathcal {C}_j} |g(X, Y)| \Vert p(Y)\Vert e^{|Y|} \le \sum _{B\in \mathcal {B}_j(X)} \sum _{Y \in \mathcal {C}_j} |g(B, Y)| \Vert p(Y)\Vert e^{|Y|} \le |X|, \end{aligned}$$

(4.22)

which verifies the hypothesis [34, (3)] with $a(A)=|A|$. Also by Lemma 4.5,

$$\begin{aligned} \sum _{Y \in \mathcal {C}_j}\Vert p(Y)\Vert e^{|Y|} \le \sum _{B\in \mathcal {B}_j} \sum _{Y \in \mathcal {C}_j} |g(B, Y)| \Vert p(Y)\Vert e^{|Y|} \le O( L^{(N-j)d} \bar{s}^3) < \infty , \end{aligned}$$

(4.23)

which verifies the other hypothesis of [34, Theorem 1].

Let $u(X_1,\ldots ,X_n)$ denote the Ursell function, defined in [34, (2)] (with g written as $\zeta$). We conclude from [34, Theorem 1] that $\mathcal {L}_j$ is given by the absolutely convergent sum

$$\begin{aligned} \mathcal {L}_j = \sum _{n=1}^\infty \sum _{X_1,\ldots ,X_n\in \mathcal {C}_j({\varLambda })} p(X_1)\cdots p(X_n)u(X_1,\ldots ,X_n) , \end{aligned}$$

(4.24)

and that for all $X_1\in \mathcal {C}_j$ we have

$$\begin{aligned} \sum _{n=1}^\infty n \sum _{X_2,\ldots ,X_n\in \mathcal {C}_j} \Vert p(X_2)\Vert \cdots \Vert p(X_n)\Vert |u(X_1,\ldots , X_n)| \le e^{|X_1|}, \end{aligned}$$

(4.25)

with the $n=1$ term in (4.25) interpreted as 1.

By (4.24)–(4.25) (the factor n in (4.25) is not needed) and (4.23),

$$\begin{aligned} \Vert \mathcal {L}_j^\varnothing \Vert \le \sum _{X_1\in \mathcal {C}_j} \Vert p(X_1)\Vert e^{|X_1|} = O(L^{(N-j)d} \bar{s}^3). \end{aligned}$$

(4.26)

Similarly, for $\Vert D_{\sigma _a} \mathcal {L}_j \Vert$, we use the product rule for differentiation, this time using the factor n (due to the product rule) in (4.25). With the definition of the $T_0$ seminorm in (2.19) and of $\ell _{\sigma ,j}$ in (3.15) for $j \le j_{ab}$, we obtain

$$\begin{aligned} \Vert D_{\sigma _a} \mathcal {L}_j \Vert \le \sum _{X\in \mathcal {C}_j : X \ni a} \Vert D_{\sigma _a} p(X)\Vert e^{|X|} = \sum _{X\in \mathcal {C}_j : X \ni a} \ell _{\sigma ,j}^{-1} \Vert p(X)\Vert e^{|X|} = O(\ell _j \bar{s}^2). \end{aligned}$$

(4.27)

For $F \in \mathcal {N}$, and , with the test functions of Lemma 2.1. These two test functions have $\Phi _j$-norms (as defined, e.g., in [31, (6.8)])

(4.28)

Therefore, for $m=1,2$,

$$\begin{aligned} |\bar{D}^m F| \le \Vert F \Vert \ell _j^{-m} . \end{aligned}$$

(4.29)

In particular, since $L^{Nd}=|{\varLambda }|$,

$$\begin{aligned} |\bar{D} D_{\sigma _a} \mathcal {L}_j|&\le \Vert D_{\sigma _a} \mathcal {L}_j \Vert \ell _j^{-1} = O(\bar{s}^2), \end{aligned}$$

(4.30)

$$\begin{aligned} |\bar{D}^2 \mathcal {L}_j^\varnothing |&\le \Vert \mathcal {L}_j^\varnothing \Vert \ell _j^{-2} = O(|{\varLambda }| L^{-\alpha j} \bar{s}^3), \end{aligned}$$

(4.31)

and the proof is complete. $\square$

5 Full RG Flow and Proof of Theorem 1.1

In Proposition 4.1, the RG flow $(\zeta _j,V_j,K_j)$ is constructed for scales $j\le j_{ab}$. The sequence $\zeta _j$ of (3.2) contains in particular the coupling constants $q_{a,j},q_{b,j}$; recall that $q_{x,j}=0$ for $j \le j_{ab}$. In Sect. 5.1, we apply Theorem 3.3 inductively to continue the RG flow $(\zeta _j,V_j,K_j)$ to scales $j_{ab} < j \le N$. Using the extended flow, we prove Theorem 1.1 in Sect. 5.2. The analysis proceeds as in [5, 32].

Once the RG flow has been extended to all scales, the combination of (2.12) and (3.1) gives, at the final scale $j=N$, the representation

$$\begin{aligned} \mathbb {E}_C e^{- V_{0} ({\varLambda })} = Z_N \big |_{\varphi =0} = e^{\zeta _N}(I_N({\varLambda }) + K_N({\varLambda }) )\big |_{\varphi =0}. \end{aligned}$$

(5.1)

From this, we apply (2.5) to to calculate the two-point function as

$$\begin{aligned} G_{a,b,N}(g,\nu ;n)&= D_{\sigma _a\sigma _b}^2 \log \mathbb {E}_C e^{- V_{0} ({\varLambda })} = \frac{1}{2}(q_{a,N} + q_{b,N}) + A_N, \end{aligned}$$

(5.2)

with

$$\begin{aligned} A_N&= \frac{D^2_{\sigma _a\sigma _b}K_{N}}{1 + K_{N}^\varnothing }\Big |_{\varphi =0} - \frac{\left( D_{\sigma _a}K_{N}\right) \left( D_{\sigma _b}K_{N}\right) }{(1 + K_{N}^\varnothing )^2}\Big |_{\varphi =0} . \end{aligned}$$

(5.3)

5.1 Flow of q

The next proposition states that the RG flow exists for scales $j_{ab}\le j\le N$, and in particular analyses the flow of q and establishes control on the terms of the right-hand side of (5.2), which is needed to prove Theorem 1.1.

Proposition 5.1

Let $n \ge 0$, let L be sufficiently large, let ${\varLambda }_N$ be the torus of period $L^N$, and let $\varepsilon$ be sufficiently small. Let $g \in [\frac{63}{64}\bar{s},\frac{65}{64}\bar{s}]$, and let $m^2 \in [L^{-\alpha (N-1)}, \delta ]$ with $\delta >0$ sufficiently small. Suppose that $j_{ab}<j_m$. Starting with $(V_{j_{ab}},K_{j_{ab}})$ produced by Proposition 4.1, the RG map can be iterated to scale N, i.e., it produces a sequence $(V_j,K_j)\in \mathbb {D}_j$ such that (3.1) holds for all $j \le N$ with $I_j=I_j(V_j)$ and $\zeta _j = \sum _{k=1}^j \delta \zeta _j$. The $q_{x,j}$ component of $\zeta _j$ is given by

$$\begin{aligned} q_{x,j} = \lambda _{a, j_{ab}} \lambda _{b, j_{ab}} w_{j; a,b} + \sum _{i = j_{ab}}^{j - 1} r_{x,i} \quad \quad (x = a, b) \end{aligned}$$

(5.4)

with

$$\begin{aligned} |r_{x,i}| \le O(\bar{s}) \frac{1}{|a-b|^{d-\alpha }} 4^{-(i-j_{ab})_+}. \end{aligned}$$

(5.5)

Moreover,

$$\begin{aligned} \lim _{N \rightarrow \infty } A_N=0. \end{aligned}$$

(5.6)

Proof

For $j=j_{ab}$, we have $(V_{j_{ab}},K_{j_{ab}})\in \mathbb {D}_{j_{ab}}$ by Proposition 4.1. Also, (5.4)–(5.5) hold trivially, since $r_{x,j_{ab}}=0$ by Theorem 3.3 and hence $q_{x,j_{ab}}=\lambda _{a, j_{ab}} \lambda _{b, j_{ab}} w_{j; a,b}$ by (3.24).

We fix $j \ge j_{ab}$ and assume inductively that (3.1) holds with a pair of RG coordinates $(V_j,K_j)\in \mathbb {D}_j$ and that (5.4)–(5.5) hold. As in the proof of Proposition 4.1, Theorem 3.3 guarantees the existence of RG coordinates $(V_{j+1},K_{j+1})$ at scale $j+1$, with $V_{j+1} = {{\mathrm{PT}}}_{j+1}(V_j) + R_{j+1}(V_j,K_j)$, and bounds on $R_{j+1}(V_j,K_j)$ and $K_{j+1}$ as in (3.29).

As before, it has been proved in [31] that the bulk part of $V_{j+1}$ lies in $\mathbb {D}_{j+1}$. The coordinate $\lambda _{a,j}=\lambda _{a,j_{ab}}$ remains constant for $j> j_{ab}$, and thus still lies in $\mathbb {D}_{j+1}$. As before, the second bound in (3.29) is sufficient to guarantee that $K_{j+1}$ also lies in $\mathbb {D}_{j+1}$. This shows that $(V_{j+1},K_{j+1})\in \mathbb {D}_{j+1}$.

We now show that $q_{a,j+1}$ satisfies (5.4) at scale $j+1$ and that (5.5) holds. Using (5.4) and (3.24), and denoting by $r_{a,j}$ the component of $R_{j+1}(U_j,K_j)$ corresponding to the component $q_a$, we see that

$$\begin{aligned} q_{a,j+1} = q_{a,j} + \lambda _{a,j_{ab}}\lambda _{b,j_{ab}} C_{j+1;a,b} + r_{a,j} = \lambda _{a, j_{ab}} \lambda _{b, j_{ab}} w_{j+1; a,b} + \sum _{i = j_{ab}}^{j} r_{x,i}, \end{aligned}$$

verifying (5.4) at scale $j+1$. By definition of the norm in (3.16) and by our assumption that $j_{ab}<j_m$, Theorem 3.3 gives the bound

$$\begin{aligned} r_{x,j} \le \ell _{\sigma ,j+1}^{-2} \Vert R_{j+1}\Vert \le 1_{j \ge j_{ab}} \ell _{\sigma ,j+1}^{-2} O(\bar{s}^3)&= 1_{j \ge j_{ab}} L^{- j_{ab}(d-\alpha )} 4^{-(j-j_{ab})_+}O(\bar{s}) , \end{aligned}$$

(5.7)

which proves (5.5) since $L^{-j_{ab}(d-\alpha )}=O(|a-b|^{-(d-\alpha )})$ by (2.25).

Finally, we write $D^k_\sigma$ to mean no derivative for $k=0$, the derivative with respect to $\sigma _a$ or $\sigma _b$ for $k=1$, and the second derivative with respect to $\sigma _a,\sigma _b$ for $k=2$. Since $(V_N,K_N)\in \mathbb {D}_N$, it follows from (3.29), with the fact that the $\mathcal {W}_N$ norm bounds the $T_0(\ell _N)$ norm, that

$$\begin{aligned} |D_{\sigma }^l K_{N}({\varLambda })|_{\varphi =0}|&\le \ell _{\sigma ,N}^{-l}C_\mathrm{RG}\vartheta _N^3 \bar{s}^3 \le O(\bar{s}^{3-l}) \vartheta _N^3 \left( 2^{-(N-j_{ab})_+} L^{-\frac{1}{2} j_{ab}(d-\alpha )} \right) ^l . \end{aligned}$$

(5.8)

Since $\vartheta _N \rightarrow 0$ as $N \rightarrow \infty$, this implies (5.6) and completes the proof. $\square$

5.2 Proof of Theorem 1.1

With (5.2) and Proposition 5.1, it is now straightforward to complete the proof of our main result. In the proof, we write $C_{a,b}$ for the massless free two-point function $((-{\varDelta })^{\alpha /2})^{-1}_{a,b}$ on $\mathbb Z^{d}$. According to (1.6), $C_{a,b} \asymp |a-b|^{-(d-\alpha )}$. The proof uses the following lemma.

Lemma 5.2

Let $\nu _0 = \nu _0^c(m^2)$. Then for any $j < N$ the map $m^2 \mapsto (V_j, K_j)$ is continuous for $m^2 \in [0, L^{-\alpha j}]$. Moreover, the sequence $V_j$ is independent of N. In particular, for any $j < \infty$, the maps $V_j, K_j, R_{+,j}$ depend continuously on $m^2$ at $m^2 = 0$.

Proof

We show by induction that $(V_j, K_j)$ depends continuously on $m^2 \in [0, L^{-\alpha j}]$. The case $j = 0$ follows from [31, (7.14)] and [31, Corollary 7.5]. Now suppose the inductive hypothesis holds for some $j \ge 0$. Then the case $j + 1$ follows from (3.23), [31, Lemma 5.2], and Theorem 3.3. The fact that $V_j$ is independent of N is [16, Proposition 1.18].$\square$

Proof

of Theorem 1.1 We first take the limit $N \rightarrow \infty$, then take the limit $m^2 \downarrow 0$, and finally consider large $|a-b|$. By (5.4) with $j=N$,

$$\begin{aligned} q_{x,N} = \lambda _{a, j_{ab}} \lambda _{b, j_{ab}} w_{N; a,b} + \sum _{i = j_{ab}}^{N - 1} r_{x,i} . \end{aligned}$$

(5.9)

By Proposition 5.1, the remainder term is bounded uniformly in N and in $m^2 \in [L^{-\alpha (N-1)},\delta ]$ by

$$\begin{aligned} \Big | \sum _{i = j_{ab}}^{N - 1} r_{x,i} \Big | \le O(\bar{s}) \frac{1}{|a-b|^{d-\alpha }} \le O(\bar{s}) C_{a,b}. \end{aligned}$$

(5.10)

By dominated convergence, and by the continuity of $r_{x,i}$ (a component of $R_{i+1}$) at $m^2=0$ guaranteed by Lemma 5.2, $\lim _{m^2 \downarrow 0}\lim _{N \rightarrow \infty }\sum _{i = j_{ab}}^{N - 1} r_{x,i}$ exists and is bounded by $O(\bar{s}) C_{a,b}$. For the main term, since $\lambda _{j_{ab}}=1+O(\bar{s})$ by Proposition 4.1, it follows from the definition of $w_N$ in (3.6) (together with the fact that that the covariance appearing in (3.24) is always the infinite-volume one), that

$$\begin{aligned} \lim _{m^2 \downarrow 0}\lim _{N\rightarrow \infty } \lambda _{j_{ab}}^2 w_{N,a,b} = (1+O(\bar{s})) C_{a,b} . \end{aligned}$$

(5.11)

The existence of the above limit as $m^2 \downarrow 0$ is a consequence of the fact that $w_{\infty ,a,b} = ((-{\varLambda })^{\alpha /2}+m^2)^{-1}_{ab} \rightarrow C_{a,b}$, together with the mass continuity of $\lambda _{j_{ab}}$, which follows from Lemma 5.2. We apply (5.6) in (5.2), and find that the critical two-point function obeys

$$\begin{aligned} G_{a,b} = \lim _{N \rightarrow \infty } G_{a,b,N} = \frac{1}{2}(q_{a,\infty } + q_{b,\infty }) = (1 +O(\bar{s})) C_{a,b} +O(\bar{s}) C_{a,b} = (1 +O(\bar{s})) C_{a,b} . \end{aligned}$$

(5.12)

This completes the proof. $\square$