Continuous Cluster Expansion for Field Theories

TL;DR

This paper introduces a novel, invariant cluster expansion method and applies it to construct connected Schwinger functions for regularized b4^4b4 field theory in a continuous manner.

Contribution

It presents a new invariant cluster expansion technique and demonstrates its application to continuous construction of Schwinger functions in b4^4b4 theory.

Findings

Invariant cluster expansion without breaking symmetries

Construction of connected Schwinger functions in a continuous setting

Application to regularized b4^4b4 field theory

Abstract

A new version of the cluster expansion is proposed without breaking the translation and rotation invariance. As an application of this technique, we construct the connected Schwinger functions of the regularized $\phi^4$ theory in a continuous way.

Equations

$$Z_{\Lambda }[J]=\int d\mu (\varphi )e^{-\lambda {\int }_{\Lambda }dx{\varphi }_x^4+{\int }_{\Lambda }dx{J}_x{\varphi }_x},J\in \mathcal{S}({\mathbb{R}}^d).$$

$$C_{x,y}=\frac{1}{(4\pi {)}^{d/2}}{\int }_{\frac{1}{2}}^1\frac{d\alpha }{{\alpha }^{d/2}}{e}^{-\alpha -|x-y{|}^2/4\alpha },$$

$$((-\Delta +1{)}^{-1}{)}_{x,y}=\frac{1}{(4\pi {)}^{d/2}}{\int }_0^{\infty }\frac{d\alpha }{{\alpha }^{d/2}}{e}^{-\alpha -|x-y{|}^2/4\alpha }.$$

$$\displaystyle S^{c}_{r,\Lambda;w_{1},\dots,w_{r}}=\tfrac{\delta^{r}}{\delta J_{w_{1}}\!\cdots\delta J_{w_{r}}}\ln Z_{\Lambda}\big{|}_{J=0},\;\;w_{1},\dots,w_{r}\!\in\!\mathbb{R}^{d}.$$

$$\displaystyle B_{x_{1},\dots,x_{n}}=\big{\{}y\!\in\!\mathbb{R}^{d}\!:\min\nolimits_{1\leq j\leq n}|y\!-\!x_{j}|\!\leq\!1\big{\}}$$

$$\displaystyle C_{\bm{t};\bm{x}}=t_{n}C_{\bm{t}^{\prime};\bm{x}^{\prime}}+(1\!-\!t_{n})\big{(}\chi_{B_{\bm{x}}}C_{\bm{t}^{\prime};\bm{x}^{\prime}}\chi_{B_{\bm{x}}}\!+\!\chi_{(B_{\bm{x}})^{\mathsf{c}}}C\chi_{(B_{\bm{x}})^{\mathsf{c}}}\big{)}$$

$$0\le (C_{\mathbf{t};\mathbf{x}}{)}_{x,y}\le c_1{e}^{-2|x-y|}$$

$$\displaystyle\frac{\partial C_{\bm{t};\bm{x}}}{\partial t_{n}}=\sum_{k=1}^{n}\Big{(}\prod_{l=k}^{n-1}t_{l}\Big{)}\Big{(}\chi_{B^{\prime}_{x_{1},\dots,x_{k}}}C\chi_{(B_{\bm{x}})^{\mathsf{c}}}+\chi_{(B_{\bm{x}})^{\mathsf{c}}}C\chi_{B^{\prime}_{x_{1},\dots,x_{k}}}\Big{)}$$

$$\frac{\partial }{\partial t_n}⟨F{⟩}_{\mathbf{t};\mathbf{x}}^{\varphi }$$

$$\displaystyle=\!\int_{(B_{\bm{x}})^{\mathsf{c}}}\!\!\!dx\,\Big{\langle}\Big{(}\sum_{k=1}^{n}\Big{(}\prod_{l=k}^{n-1}t_{l}\Big{)}\Delta_{x;x_{1},\dots,x_{k}}^{\phi}\Big{)}F\Big{\rangle}_{\bm{t};\bm{x}}^{\phi},$$

$${\tilde{\mathbf{Z}}}_{\Lambda ;x_1,\dots ,x_n}$$

$$\displaystyle\quad\;\;e^{-\lambda\!\int_{\Lambda}\!dx\,\phi^{4}_{x}+\int_{\Lambda}\!dx\,J_{x}\phi_{x}}\Big{\rangle}_{(t_{j})_{j=1}^{n-1};(x_{j})_{j=1}^{n-1}}^{\phi}.$$

$$\displaystyle\sum_{\eta\in\mathcal{T}_{n}}\!\Big{(}\prod_{j=1}^{n-1}\!\int_{0}^{1}\!\!dt_{j}\Big{)}\Big{(}\prod_{j=2}^{n}\prod_{l=\eta(j)}^{j-2}t_{l}\Big{)}$$

$$\displaystyle\Big{\langle}\Big{(}\prod_{j=2}^{n}\Delta_{x_{j};x_{1},\dots,x_{\eta(j)}}^{\phi}\Big{)}e^{-\lambda\!\int_{\Lambda}\!dx\,\phi^{4}_{x}+\int_{\Lambda}\!dx\,J_{x}\phi_{x}}\Big{\rangle}_{(t_{j})_{j=1}^{n-1};(x_{j})_{j=1}^{n-1}}^{\phi}.$$

$$\displaystyle F[\phi]=\Big{(}\prod_{j=2}^{n}\sum_{k=1}^{j-1}\Big{(}\prod_{l=k}^{j-2}t_{l}\Big{)}\Delta_{x_{j};x_{1},\dots,x_{k}}^{\phi}\Big{)}e^{-\lambda\!\int_{\Lambda}\!dx\,\phi^{4}_{x}+\int_{\Lambda}\!dx\,J_{x}\phi_{x}}$$

$${\tilde{\mathbf{Z}}}_{\Lambda ;\mathbf{x}}={\mathbf{Z}}_{\Lambda ;\mathbf{x}}{Z}_{\Lambda \backslash B_{\mathbf{x}}}+{\int }_{(B_{\mathbf{x}}{)}^{\mathsf{c}}}dz{\tilde{\mathbf{Z}}}_{\Lambda ;\mathbf{x},z},$$

$$Z_{\Lambda \backslash B_{\mathbf{x}}}$$

$$\displaystyle=\sum_{m\geq 1}\Big{(}\prod_{j=1}^{m-1}\int_{(B_{z_{1},\dots,z_{j}})^{\mathsf{c}}}\!\!\!dz_{j+1}\Big{)}\bm{Z}_{\Lambda\backslash B_{\bm{x}};z_{1},\dots,z_{m}}Z_{\Lambda\backslash B_{\bm{x},z_{1},\dots,z_{m}}},$$

$$\displaystyle\frac{Z_{\Lambda\backslash B_{\bm{x},z_{1}}}}{Z_{\Lambda}}=\frac{Z_{\Lambda\backslash B_{\bm{x}}}}{Z_{\Lambda}}-\big{(}\bm{Z}_{\Lambda\backslash B_{\bm{x}};z_{1}}\!-\!1\big{)}\frac{Z_{\Lambda\backslash B_{\bm{x},z_{1}}}}{Z_{\Lambda}}\,-$$

$$\displaystyle\sum_{m\geq 2}\Big{(}\prod_{j=1}^{m-1}\int_{(B_{\bm{x},z_{1},\dots,z_{j}})^{\mathsf{c}}}\!\!\!dz_{j+1}\Big{)}\bm{Z}_{\Lambda\backslash B_{\bm{x}};z_{1},\dots,z_{m}}\frac{Z_{\Lambda\backslash B_{\bm{x},z_{1},\dots,z_{m}}}}{Z_{\Lambda}},$$

$$\displaystyle(\bm{A}_{\Lambda}\bm{f})_{\bm{x},z_{1}}=\bm{1}_{\bm{x}\not\in\mathcal{X}_{0}}\bm{f}_{\bm{x}}-\big{(}\bm{Z}_{\Lambda\backslash B_{\bm{x}};z_{1}}\!-\!1\big{)}\bm{f}_{\bm{x},z_{1}}\,-$$

$$\displaystyle\sum_{m\geq 2}\Big{(}\prod_{j=1}^{m-1}\int_{(B_{\bm{x},z_{1},\dots,z_{j}})^{\mathsf{c}}}\!\!\!dz_{j+1}\Big{)}\bm{Z}_{\Lambda\backslash B_{\bm{x}};z_{1},\dots,z_{m}}\bm{f}_{\bm{x},z_{1},\dots,z_{m}}.$$

$$\displaystyle\big{(}\bm{1}\!-\!\bm{A}_{0,\Lambda}\big{)}\bm{f}_{r,\Lambda}=\bm{e}\,\delta_{r,0}+\sum_{0<s\leq r}\bm{A}_{s,\Lambda}\bm{f}_{r-s,\Lambda}.$$

$$({\mathbf{A}}_{s,\Lambda }\mathbf{f}{)}_{w_1,\dots ,w_r;\mathbf{x},z_1}=-{\sum }_{\begin{array}{c}I,I^{\prime }\subset \{1,\dots ,r\}\\ |I|=s,I^{\prime }=\{1,\dots ,r\}\backslash I\end{array}}\sum _{m\ge 1}$$

$$\displaystyle\Big{(}\prod_{j=1}^{m-1}\int_{(B_{\bm{x},z_{1},\dots,z_{j}})^{\mathsf{c}}}\!\!\!dz_{j+1}\Big{)}\bm{Z}_{s,\Lambda\backslash B_{\bm{x}};(w_{i})_{i\in I};z_{1},\dots,z_{m}}\bm{f}_{(w_{i})_{i\in I^{\prime}};\bm{x},z_{1},\dots,z_{m}}.$$

$$\displaystyle\big{(}\bm{1}\!-\!\bm{A}_{0}\big{)}\bm{f}^{*}_{r}=\bm{e}\,\delta_{r,0}+\sum_{0<s\leq r}\bm{A}_{s}\bm{f}^{*}_{r-s}$$

$$\displaystyle(\bm{A}_{0}\bm{f})_{\bm{x},z_{1}}=\bm{1}_{\bm{x}\not\in\mathcal{X}_{0}}\bm{f}_{\bm{x}}-\big{(}\bm{Z}_{0,(B_{\bm{x}})^{\mathsf{c}};z_{1}}\!-\!1\big{)}\bm{f}_{\bm{x},z_{1}}-\sum_{m\geq 2}$$

$$\displaystyle\Big{(}\prod_{j=1}^{m-1}\int_{(B_{\bm{x},z_{1},\dots,z_{j}})^{\mathsf{c}}}\!\!\!dz_{j+1}\Big{)}\bm{Z}_{0,(B_{\bm{x}})^{\mathsf{c}};z_{1},\dots,z_{m}}\bm{f}_{\bm{x},z_{1},\dots,z_{m}}$$

$$({\mathbf{A}}_s\mathbf{f}{)}_{w_1,\dots ,w_r;\mathbf{x},z_1}=-{\sum }_{\begin{array}{c}I,I^{\prime }\subset \{1,\dots ,r\}\\ |I|=s,I^{\prime }=\{1,\dots ,r\}\backslash I\end{array}}\sum _{m\ge 1}$$

$$\displaystyle\Big{(}\prod_{j=1}^{m-1}\int_{(B_{\bm{x},z_{1},\dots,z_{j}})^{\mathsf{c}}}\!\!\!dz_{j+1}\Big{)}\bm{Z}_{s,(B_{\bm{x}})^{\mathsf{c}};(w_{i})_{i\in I};z_{1},\dots,z_{m}}\bm{f}_{(w_{i})_{i\in I^{\prime}};\bm{x},z_{1},\dots,z_{m}}.$$

$${\ell }_{x_1,\dots ,x_n}^{\prime }=\underset{T}{\min }\sum _{(i,j)\in T}|x_i-x_j|,$$

$${\ell }_{X_1,\dots ,X_n}^{\prime }=\underset{T}{\min }\sum _{(i,j)\in T}\underset{x\in X_i,x^{\prime }\in X_j}{\inf }|x-x^{\prime }|$$

$${\ell }_{x_1,\dots ,x_n}\le {\ell }_{x_1,\dots ,x_n,\dots ,x_{n+n^{\prime }}}\le {\ell }_{x_1,\dots ,x_n}+{\ell }_{x_n,\dots ,x_{n+n^{\prime }}},$$

$${\ell }_{x_1,\dots ,x_n;y_1,\dots ,y_m}\ge {\ell }_{x_1,\dots ,x_n;y_1,\dots ,y_m,\dots ,y_{m+m^{\prime }}}\ge {\ell }_{x_1,\dots ,x_n;y_1,\dots ,y_m}-{\ell }_{y_m,\dots ,y_{m+m^{\prime }}}.$$

$$\Vert \mathbf{f}{\Vert }_r=\frac{1}{r!}\underset{n\ge 1}{\sup }\underset{\mathbf{w}\in {\mathbb{R}}^{dr},\mathbf{x}\in {\mathcal{X}}_n}{\mathrm{esssup}}{2}^{1-n}{e}^{{\ell }_{\mathbf{w};\mathbf{x}}}|{\mathbf{f}}_{\mathbf{w};\mathbf{x}}|,$$

Full text

keywords Constructive Field Theory; Cluster Expansion; Translation and Rotation Invariance; Connected Schwinger Function

Mathematics Subject Classification 81T08

1. Introduction

As a constructive tool for field theories, the cluster expansion [19, 22] can be used to analyze the infinite volume limit in a rigorous manner. Contrary to the formal perturbative treatment, it is indeed convergent at least for weak coupling. Also its multiscale version, called phase cell expansion [3, 4], plays the central role in the constructive renormalization.

However, the traditional cluster expansion breaks the translation and rotation invariance explicitly, since it is based on a fixed discretization of space-time [7, 11, 18] or a wavelet decomposition of the fields [16]. Although some remedies [5, 9, 10] have been invented, none of them is fully satisfactory and a manifestly Euclidean invariant expansion is desired. Thanks to the Pauli’s principle, Fermionic field theories may be constructed continuously by some rearrangements and subtractions of the perturbative series [15, 17]. But continuous constructions of the Bosonic ones are more difficult, expect for some special cases such as $($Sine-Gordon$)_{2}$ with $\beta\!<\!4\pi$ [6, 12]. Recently, a new constructive technique for Bosonic field theories, called loop vertex expansion [20], has been well developed. As one of its successes, the new technique has been used to construct the regularized $\phi^{4}$ theory in a continuous way [21]. Unfortunately, when it is applied to the unregularized $\phi^{4}$ theory, the cluster expansion is still required to remove the volume cut-off [23].

In this paper, we provide a continuous version of the cluster expansion, which is based on dynamical discretizations of space-time instead of a fixed one and thus retains the translation and rotation invariance explicitly. As an example, we construct the connected Schwinger functions of the regularized $\phi^{4}$ theory in this way. Generalizations to other stable interactions of polynomial type are straightforward. Moreover, the phase cell expansion may be realized in a similar way and then it seems promising to construct unregularized Bosonic field theories continuously. The progress in this direction will appear in future publications.

2. Expansion

Let us denote by $\mathcal{B}$ the collection of all Borel subsets of $\mathbb{R}^{d}$ and by $\mathcal{B}^{\prime}$ the collection of all bounded ones in $\mathcal{B}$. We consider a regularized $\phi^{4}_{d}$ theory with the interacting part restricted in $\Lambda\!\in\!\mathcal{B}^{\prime}$. The generating functional of this theory is

[TABLE]

Here $d\mu$ is the Gaussian measure on the space of tempered distributions $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ with covariance

[TABLE]

which is a regularized version of the full covariance

[TABLE]

The connected Schwinger functions of the theory are

[TABLE]

(The $Z_{2}$ symmetry of the theory will not be used in the following, since the purpose of this paper is to provide a new expansion for general cases.) What we are really interested in is the limit theory as $\Lambda$ approaches $\mathbb{R}^{d}$, which can be seen as a simplified model for a single slice in the renormalization group of the full $\phi^{4}_{d}$ theory [8, 13, 14].

For $n\!\geq\!0$, we denote by $\mathcal{I}_{n}$ the set of all sequences $(t_{j})_{j=1}^{n}$ with $t_{1},\dots,t_{n}\!\in\![0,1]$ and by $\mathcal{X}_{n}$ the set of all sequences $(x_{j})_{j=1}^{n}$ with $x_{1},\dots,x_{n}\!\in\!\mathbb{R}^{d}$ satisfying $|x_{j}\!-\!x_{j^{\prime}}|\!>\!1$ for $j\!\neq\!j^{\prime}$. For $\bm{x}\!\in\!\mathcal{X}_{0}$, let $B_{\bm{x}}\!=\!B^{\prime}_{\bm{x}}\!=\!\varnothing$. For $(x_{j})_{j=1}^{n}\!\in\!\mathcal{X}_{n}$ with $n\!\geq\!1$, let

[TABLE]

and $B^{\prime}_{x_{1},\dots,x_{n}}=B_{x_{1},\dots,x_{n}}\backslash B_{x_{1},\dots,x_{n-1}}$. We recursively define $C_{\bm{t};\bm{x}}$ for $\bm{t}\!\in\!\mathcal{I}_{n}$ and $\bm{x}\!\in\!\mathcal{X}_{n}$ as follows. If $n\!=\!0$, $C_{\bm{t};\bm{x}}\!=\!C$. Otherwise,

[TABLE]

for $\bm{t}\!=\!(\bm{t}^{\prime},t_{n})$ and $\bm{x}\!=\!(\bm{x}^{\prime},x_{n})$, where $\chi_{A}$ denotes the characteristic function of subset $A$ of $\mathbb{R}^{d}$. By the convexity of the linear combination (2.6), the interpolated covariance $C_{\bm{t};\bm{x}}$ remains positive and has the bound

[TABLE]

from (2.2). Let $d\mu_{\bm{t};\bm{x}}$ be the Gaussian measure on $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ with covariance $C_{\bm{t};\bm{x}}$ and let $\langle F\rangle_{\bm{t};\bm{x}}^{\phi}$ be an abbreviation for $\int\!d\mu_{\bm{t};\bm{x}}(\phi)\,F[\phi]$. For $\bm{t}\!=\!(t_{j})_{j=1}^{n}\!\in\!\mathcal{I}_{n}$ and $\bm{x}\!=\!(x_{j})_{j=1}^{n}\!\in\!\mathcal{X}_{n}$ with $n\!\geq\!1$,

[TABLE]

and then, by the formula for infinitesimal change of covariance [7],

[TABLE]

where $\Delta_{x;\bm{x}}^{\phi}$ is written simply for $\int_{B^{\prime}_{\bm{x}}}\!dy\,\frac{\delta}{\delta\phi_{x}}C_{x,y}\frac{\delta}{\delta\phi_{y}}$.

We define, for $\Lambda\!\in\!\mathcal{B}^{\prime}$ and $(x_{j})_{j=1}^{n}\!\in\!\mathcal{X}_{n}$ with $n\!\geq\!1$,

[TABLE]

In particular, $\bm{\widetilde{Z}}_{\Lambda;x_{1}}\!=\!Z_{\Lambda}\!:=\!\int\!d\mu(\phi)\,e^{-\lambda\!\int_{\Lambda}\!dx\,\phi^{4}_{x}+\int_{\Lambda}\!dx\,J_{x}\phi_{x}}$, which is actually independent of $x_{1}$. For $n\!\geq\!2$, we call a map $\eta\!:\{2,3,...,n\}\!\to\!\{1,2,...,n\!-\!1\}$ with $\eta(j)\!<\!j$ an ordered tree with $n$ vertices and denote by $\mathcal{T}_{n}$ the set of all ordered trees with $n$ vertices. Then $\bm{\widetilde{Z}}_{\Lambda;x_{1},\dots,x_{n}}$ can be rewritten as

[TABLE]

Write simply $\bm{Z}_{\Lambda;\bm{x}}\!=\!\bm{\widetilde{Z}}_{\Lambda\cap B_{\bm{x}};\bm{x}}$ for $\Lambda\!\in\!\mathcal{B}$. By taking

[TABLE]

in (2) and performing the integrations over $t_{1},\dots,t_{n}$, we have

[TABLE]

where the first summand factorizes since both of the Gaussian measure at $t_{n}\!=\!0$ and the integrand factorize. Applying (2.13) successively, we obtain, for $z_{1}\!\not\in\!B_{\bm{x}}$,

[TABLE]

where the summands vanish for $m$ sufficiently large by the boundedness of $\Lambda$ and the ranges of integrations $(B_{z_{1},\dots,z_{j}})^{\mathsf{c}}$ can be replaced by $(B_{\bm{x},z_{1},\dots,z_{j}})^{\mathsf{c}}$. Then, dividing both sides of (2) by $Z_{\Lambda}$, we have the equation of Kirkwood-Salzburg type [1]

[TABLE]

which can be rewritten simply as $\bm{f}_{\Lambda}\!=\!\bm{e}\!+\!\bm{A}_{\Lambda}\bm{f}_{\Lambda}$. Here $\bm{e}$ and $\bm{f}_{\Lambda}$ are regarded as Borel measurable functions on $\bigcup_{n\geq 1}\mathcal{X}_{n}$ with $\bm{e}_{\bm{x}}\!\!=\!\bm{1}_{\bm{x}\in\mathcal{X}_{1}}$ and $\bm{f}_{\Lambda;\bm{x}}\!=\!\frac{Z_{\Lambda\backslash B_{\bm{x}}}}{Z_{\Lambda}}$, while $\bm{A}_{\Lambda}$ is regarded as a linear operator with

[TABLE]

(We say a function $\bm{f}$ on $\bigcup_{n\geq 1}\mathcal{X}_{n}$ is Borel measurable iff $\bm{f}|_{\mathcal{X}_{n}}$ is Borel measurable for each $n\!\geq\!1$.) Denoting \bm{Z}_{r,\Lambda;w_{1},\dots,w_{r}}\!=\!\tfrac{\delta^{r}}{\delta J_{w_{1}}\!\cdots\delta J_{w_{r}}}\bm{Z}_{\Lambda}\big{|}_{J=0} and \bm{f}_{r,\Lambda;w_{1},\dots,w_{r}}\!=\!\tfrac{\delta^{r}}{\delta J_{w_{1}}\!\cdots\delta J_{w_{r}}}\bm{f}_{\Lambda}\big{|}_{J=0}, we have

[TABLE]

Here $\bm{f}_{r,\Lambda}$ is regarded as a Borel measurable function on $\mathbb{R}^{d\times r}\times\bigcup_{n\geq 1}\mathcal{X}_{n}$, while $\bm{A}_{s,\Lambda}$ is regarded as a linear operator with $\bm{A}_{0,\Lambda}\!=\!\bm{A}_{\Lambda}[0]$ and, for $0\!<\!s\!\leq\!r$,

[TABLE]

Now letting $\bm{f}^{*}_{r}$ and $\bm{A}_{r}$ be the formal limits of $\bm{f}_{r,\Lambda}$ and $\bm{A}_{r,\Lambda}$ respectively as $\Lambda\!\to\!\mathbb{R}^{d}$, we have that

[TABLE]

with

[TABLE]

and, for $0\!<\!s\!\leq\!r$,

[TABLE]

To exhibit the exponential decay of the connected Schwinger functions, we review the following two tree-lengths for $x_{1},\dots,x_{n}\!\in\!\mathbb{R}^{d}$ [2]. One is

[TABLE]

where the minimum is taken over all trees $T$ connecting $x_{1},\dots,x_{n}$. The other is $\ell_{x_{1},\dots,x_{n}}\!=\!\inf\ell^{\prime}_{x_{1},\dots,x_{n},y_{1},\dots,y_{m}}$, where the infimum is taken over all finite subsets $\{y_{1},\dots,y_{m}\}$ of $\mathbb{R}^{d}$ (including $\varnothing$). In particular, $\ell^{\prime}_{x_{1}}\!=\!\ell_{x_{1}}\!=\!0$ and $\ell^{\prime}_{x_{1},x_{2}}\!=\!\ell_{x_{1},x_{2}}\!=\!|x_{1}\!-\!x_{2}|$. It is easy to see that $\ell^{\prime}_{x_{1},\dots,x_{n}}$ and $\ell_{x_{1},\dots,x_{n}}$ are symmetric in $x_{1},\dots,x_{n}$. Also we have the inequality $\tfrac{1}{2}\ell_{x_{1},\dots,x_{n}}\leq\ell^{\prime}_{x_{1},\dots,x_{n}}\leq\ell_{x_{1},\dots,x_{n}}$ [2], indicating the equivalence of $\ell^{\prime}_{x_{1},\dots,x_{n}}$ and $\ell_{x_{1},\dots,x_{n}}$ in some sense. We can also define these two tree-lengths for nonempty subsets $X_{1},\dots,X_{n}\!\subset\!\mathbb{R}^{d}$ as

[TABLE]

and $\ell_{X_{1},\dots,X_{n}}\!=\!\inf\ell^{\prime}_{X_{1},\dots,X_{n},\{y_{1}\},\dots,\{y_{m}\}}$ with the infimum also taken over all finite subsets $\{y_{1},\dots,y_{m}\}$ of $\mathbb{R}^{d}$. Writing simply $\ell_{x_{1},\dots,x_{n};y_{1},\dots,y_{m}}\!=\!\ell_{\{x_{1}\},\dots,\{x_{n}\},\{y_{1},\dots,y_{m}\}}$, we list the following useful inequalities for $\ell$ and leave the proofs of them to the reader:

[TABLE]

For $r\!\geq\!0$, let $\mathcal{F}_{r}$ be the Banach space of Borel measurable functions on $\mathbb{R}^{dr}\times\bigcup_{n\geq 1}\mathcal{X}_{n}$ with the norm

[TABLE]

where $\ell_{\bm{w};\bm{x}}$ is used to show the exponential decay of $\bm{f}_{\bm{w};\bm{x}}$ in $\bm{w}$. In particular,

[TABLE]

Also let $\|\bm{A}\|_{s\to r}=\sup_{\|\bm{f}\|_{s}=1}\|\bm{A}\bm{f}\|_{r}$ for linear operator $\bm{A}\!:\mathcal{F}_{s}\!\to\!\mathcal{F}_{r}$.

Theorem 1**.**

For $\lambda\!>\!0$ sufficiently small, we have $\|\bm{A}_{0}\|_{r\to r}\!\leq\!\tfrac{3}{4}$ and, for $0\!<\!s\!\leq\!r$, $\|\bm{A}_{s}\|_{r-s\to r}\!\leq\!c^{s}$.

By Theorem 1, (2.19) has a unique solution $(\bm{f}^{*}_{r})_{r\geq 0}$ with

[TABLE]

for $\lambda\!>\!0$ sufficiently small. Since $\|\bm{e}\|_{0}\!=\!1$, we have

[TABLE]

and then can obtain inductively $\|\bm{f}^{*}_{r}\|_{r}\!\leq\!4(5c)^{r}$.

By (2), we have

[TABLE]

where we have used the fact \tfrac{\delta}{\delta J_{w_{1}}}Z_{\Lambda\backslash B_{x_{1},\dots,x_{n}}}\big{|}_{x_{1}=w_{1}}\!\equiv\!0. Then, for $w_{1},\dots,w_{r}\!\in\!\mathbb{R}^{d}$,

[TABLE]

which can be rewritten as $S^{c}_{r,\Lambda}=\sum_{1\leq s\leq r}\bm{T}_{s,\Lambda}\bm{f}_{r-s,\Lambda}$ with

[TABLE]

The formal limit of $S^{c}_{r,\Lambda}$ as $\Lambda\!\to\!\mathbb{R}^{d}$ is $S^{c}_{r}=\sum_{1\leq s\leq r}\bm{T}_{s}\bm{f}^{*}_{r-s}$ with

[TABLE]

For $r\!\geq\!1$, let $\mathcal{F}^{\prime}_{r}$ be the Banach space of Borel measurable functions $f\!:\mathbb{R}^{d\times r}\!\to\!\mathbb{R}$ with the norm

[TABLE]

where $\ell_{\bm{w}}$ is used to show the exponential decay of $\bm{f}_{\bm{w}}$ in $\bm{w}$. Also let

[TABLE]

for linear operator $\bm{T}\!:\mathcal{F}_{s}\!\to\!\mathcal{F}^{\prime}_{r}$.

Theorem 2**.**

For $\lambda\!>\!0$ sufficiently small and $1\!\leq\!s\!\leq\!r$, $\|\bm{T}_{s}\|^{\prime}_{r-s\to r}\leq c^{s}$.

By Theorem 2, we have, for $\lambda\!>\!0$ sufficiently small,

[TABLE]

which is equivalent to

[TABLE]

3. Estimation

Let $\mathcal{H}_{n}$ be the set of all sequences $(h_{j})_{j=1}^{n}$ with $h_{j}\!\in\!\mathbb{N}$. For $(h_{j})_{j=1}^{n},(h^{\prime}_{j})_{j=1}^{n}\!\in\!\mathcal{H}_{n}$, we write $(h_{j})_{j=1}^{n}\!\leq\!(h^{\prime}_{j})_{j=1}^{n}$ iff $h_{j}\!\leq\!h^{\prime}_{j}$ for $1\!\leq\!j\!\leq\!n$ and $(h_{j})_{j=1}^{n}\!\pm\!(h^{\prime}_{j})_{j=1}^{n}\!:=\!(h_{j}\!\pm\!h^{\prime}_{j})_{j=1}^{n}$. We define, for $\bm{x}\!=\!(x_{j})_{j=1}^{n}\!\in\!\mathcal{X}_{n}$ and $\bm{h}\!=\!(h_{j})_{j=1}^{n}\!\in\!\mathcal{H}_{n}$,

[TABLE]

Then we have

[TABLE]

and in general

[TABLE]

In order to deal with some singular functional conveniently in the following, we regard them as signed measures on $\mathbb{R}^{d}$. (For signed measure $\nu$, we have the Jordan decomposition $\nu\!=\nu^{+}\!-\!\nu^{-}$, where $\nu^{+}$, $\nu^{-}$ are the positive and negative variations of $\nu$, and we have the total variation of $\nu$ as $|\nu|\!=\!\nu^{+}\!+\!\nu^{-}$. Furthermore, for signed measures $\nu_{1}$, $\nu_{2}$, we write $\nu_{1}\!\geq\!\nu_{2}$ iff $\nu_{1}\!-\!\nu_{2}$ is a positive measure. In particular, we have $|\delta_{z}(x)|\!=\!\delta_{z}(x)\!\geq\!0$.)

Then it is easy to show that $\sum_{\bm{h}\in\mathcal{H}_{n}}\!\|\phi_{z}\|_{\bm{x};\bm{h}}^{\phi}\!\leq\!1\!+\!|\phi_{z}|$ and, by (3.3),

[TABLE]

Also we have the following bound:

Lemma 1**.**

For $\bm{x}\!\in\!\mathcal{X}_{n}$, $\bm{h}\!\in\!\mathcal{H}_{n}$ and $\lambda\!\geq\!0$,

[TABLE]

Proof.

For any $D\!\in\!\mathcal{B}^{\prime}$, we have

[TABLE]

with c\!=\!\sup_{\xi\geq 0}\big{(}(1\!+\!\xi)^{4}\!-\!2\xi^{4}\big{)}. Then, for $\bm{x}\!=\!(x_{j})_{j=1}^{n}\!\in\!\mathcal{X}_{n}$ and $\bm{h}\!=\!(h_{j})_{j=1}^{n}\!\in\!\mathcal{H}_{n}$,

[TABLE]

where we use the facts that the range of the integration over $y_{j,k}$ can be replaced by $\Lambda\!\cap\!B^{\prime}_{x_{1},\dots,x_{j}}$ here and $|\Lambda\!\cap\!B^{\prime}_{x_{1},\dots,x_{j}}|$ can be bounded by $v_{d}$, the volume of the unit ball in $d$ dimensions. ∎

For $(x_{j})_{j=1}^{n}\!\in\!\mathcal{X}_{n}$ and $\eta\!\in\!\mathcal{T}_{n}$, we have

[TABLE]

with d_{\eta}(j)\!=\!\big{|}\eta^{-1}(\{j\})\big{|} and $\bm{d}_{\eta}\!=\!(d_{\eta}(j))_{j=1}^{n-1}$. Then, by (3), Lemma 1 and the fact $\sup\nolimits_{y\in B_{x_{\eta(j)}}}\!\!C_{x_{j},y}\leq c_{1}e^{2-2|x_{j}-x_{\eta(j)}|}$, we can continue with

[TABLE]

where $U_{x_{1},\dots,x_{n}}^{\phi}(F)$ is an abbreviation for

[TABLE]

We now deal with the three factors $U_{x_{1},\dots,x_{n}}^{\phi}(F)$, $e^{-2\sum_{j=2}^{n}|x_{j}-x_{\eta(j)}|}$ and $\prod_{j=1}^{n-1}d_{\eta}(j)!$ in (3) one by one. First, let us consider $U_{x_{1},\dots,x_{n}}^{\phi}(F)$ with $F\!=\!\phi_{w_{1}}\cdots\phi_{w_{r}}$:

[TABLE]

Before going further, we provide the following lemma, which is needed to bound the expectations $\langle U_{x_{1},\dots,x_{n}}^{\phi}(\phi_{w_{1}}\cdots\phi_{w_{r}})\rangle_{\bm{t};\bm{x}}^{\phi}$.

Lemma 2**.**

For $w_{1},\dots,w_{r}\!\in\!\mathbb{R}^{d}$ and $(x_{1},\dots,x_{n})\!\in\!\mathcal{X}_{n}$,

[TABLE]

Proof.

First, we claim that, for $\sum_{j=1}^{n}\!s_{j}\!=\!2s$ even and $(x_{1},\dots,x_{n})\!\in\!\mathcal{X}_{n}$,

[TABLE]

which is proved by induction on $s$ and is trivial for $s\!=\!0$. Assuming it holds for $s\!=\!\bar{s}\!-\!1$, we consider the case $s\!=\!\bar{s}$ and assume further $s_{1}\!=\!\max_{j=1}^{n}\!s_{j}$ without loss of generality. By performing integration by parts and applying the inductive assumption, we have

[TABLE]

where $\phi_{x}^{-1}$ is temporarily defined as 0. Since

[TABLE]

we can advance the induction for $c$ sufficiently large and thus complete the proof of the claim.

Then, by the Cauchy-Schwartz inequality,

[TABLE]

which yields

[TABLE]

∎

The factor $e^{-2\sum_{j=2}^{n}|x_{j}-x_{\eta(j)}|}$ will play two roles in our derivation. One half of the factor is used later to extract some exponential decay in external points $w_{1},\dots,w_{r}$. The other half is used to control the integrations over $x_{2},\dots,x_{n}$ as follows. By (3) and Lemma 2, we have, for $\bm{t}\!\in\!\mathcal{I}_{n}$ and $\eta\!\in\!\mathcal{T}_{n}$,

[TABLE]

with $c_{5}=\!\int_{\mathbb{R}^{d}}\!dx\,e^{-|x|}$. In the last line of (3), we use the facts that the number of subsets of $\{2,\dots,n\}$ is $2^{n-1}$ and the number of injective maps from $L$ to $\{1,\dots,r\}$ with $|L|\!\leq\!r$ is $\frac{r!}{(r-|L|)!}$. We assume from now on that $\lambda\!<\!(4c_{3}c_{4}c_{5})^{-1}$.

To control the factor $\prod_{j=1}^{n-1}d_{\eta}(j)!$, we need the following lemma:

Lemma 3** (Speer [16]).**

For $n\!\geq\!2$,

[TABLE]

Proof.

For completeness, we give a new proof for this lemma. Let

[TABLE]

Given $\eta\!\in\!\mathcal{T}_{n}$, we have

[TABLE]

where $d_{\eta}(n)$ is defined as 0. Multiplying both sides of (3) by

[TABLE]

performing integrations over $t_{1},\dots,t_{n-1}$ and summing over all $\eta\!\in\!\mathcal{T}_{n}$, we obtain $A_{n}(x)\!-\!B_{n}\!=\!xA_{n+1}(x)$ with

[TABLE]

which yields

[TABLE]

Comparing (3.20) and (3), we complete the proof. ∎

Now we are ready to perform the estimation for $\bm{Z}_{r,\Lambda;w_{1},\dots,w_{r};x_{1},\dots,x_{n}}$ with $n\!\geq\!2$. First, we have

[TABLE]

Using (3), Lemma 3 and the fact $\sum_{j=2}^{n}|x_{j}\!-\!x_{\eta(j)}|\!\geq\!\ell_{x_{1},\dots,x_{n}}$, we obtain that

[TABLE]

Proof of Theorem 1.

For $\bm{w}\!\in\!\mathbb{R}^{dr}$ and $(\bm{x},z_{1})\!\in\!\mathcal{X}_{n+1}$,

[TABLE]

By the inequalities $\ell_{\bm{w};\bm{x}}\geq\ell_{\bm{w};\bm{x},z_{1}}$ and $\ell_{\bm{w};\bm{x},z_{1},\dots,z_{m}}\geq\ell_{\bm{w};\bm{x},z_{1}}-\ell_{z_{1},\dots,z_{m}}$, we have

[TABLE]

where

[TABLE]

and the integration of $|\bm{Z}_{0,(B_{\bm{x}})^{\mathsf{c}};z_{1},\dots,z_{m}}|\,e^{\ell_{z_{1},\dots,z_{m}}}$ is bounded by \big{(}32c_{1}c_{4}c_{5}\lambda\,e^{2+c_{2}\lambda}\big{)}^{m-1}. Thus \|\bm{A}_{0}\|_{r}\leq\tfrac{1}{2}+3c_{1}^{2}v_{d}\lambda+\sum_{m\geq 2}\big{(}64c_{1}c_{4}c_{5}\lambda\,e^{2+c_{2}\lambda}\big{)}^{m-1}\leq\tfrac{3}{4} for $\lambda$ sufficiently small.

Also, for $1\!\leq\!s\!\leq\!r$, $w_{1},\dots,w_{r}\!\in\!\mathbb{R}^{d}$ and $(\bm{x},z_{1})\!\in\!\mathcal{X}_{n+1}$,

[TABLE]

Since $\bm{Z}_{s,(B_{\bm{x}})^{\mathsf{c}};(w_{i})_{i\in I};z_{1},\dots,z_{m}}\!=\!0$ for $\{w_{i}|\,i\!\in\!I\}\!\not\subset\!B_{z_{1},\dots,z_{m}}$ and

[TABLE]

for $\{w_{i}|\,i\!\in\!I\}\!\subset\!B_{z_{1},\dots,z_{m}}$, we have

[TABLE]

for $\lambda$ sufficiently small. ∎

Proof of Theorem 2.

For $1\!\leq\!s\!\leq\!r$ and $w_{1},\dots,w_{r}\!\in\!\mathbb{R}^{d}$,

[TABLE]

where

[TABLE]

for $x_{1}\!=\!w_{1}$ and $\{w_{i}|\,i\!\in\!I\}\!\subset\!B_{x_{1},\dots,x_{n}}$. Then

[TABLE]

for $\lambda$ sufficiently small. ∎

Acknowledgements

The work is partially supported by Wu Wen-Tsun Key Laboratory of Mathematics. The author thanks Professor Zheng Yin for valuable advices and suggestions.