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Infinite Volume Limit for Correlation functions in the Dipole Gas

Tuan Minh Le

TL;DR

This work addresses the long-distance behavior of a classical dipole gas on ${\mathbb{Z}}^d$ ($d\ge 3$) at small activity. It develops a rigorous RG framework with a multiscale, finite-range covariance decomposition and a polymer expansion, extended by an external field to study truncated correlation functions. The main contributions are the existence of infinite-volume limits for correlation functions and explicit decay bounds, proven via a stable-manifold RG flow and careful control of linear and nonlinear RG terms. This provides a mathematically solid foundation for the Coulomb/dipole gas in higher dimensions and yields practical decay estimates that complement prior Mayer-expansion approaches.

Abstract

We study a classical lattice dipole gas with low activity in dimension $d \geq 3$. We investigate long distance properties by a renormalization group analysis. We prove that various correlation functions have an infinite volume limit. We also get estimates on the decay of correlation functions.

Infinite Volume Limit for Correlation functions in the Dipole Gas

TL;DR

This work addresses the long-distance behavior of a classical dipole gas on () at small activity. It develops a rigorous RG framework with a multiscale, finite-range covariance decomposition and a polymer expansion, extended by an external field to study truncated correlation functions. The main contributions are the existence of infinite-volume limits for correlation functions and explicit decay bounds, proven via a stable-manifold RG flow and careful control of linear and nonlinear RG terms. This provides a mathematically solid foundation for the Coulomb/dipole gas in higher dimensions and yields practical decay estimates that complement prior Mayer-expansion approaches.

Abstract

We study a classical lattice dipole gas with low activity in dimension . We investigate long distance properties by a renormalization group analysis. We prove that various correlation functions have an infinite volume limit. We also get estimates on the decay of correlation functions.

Paper Structure

This paper contains 32 sections, 26 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. The main result
  4. Preliminaries
  5. Multiscale decomposition
  6. Renormalization Group Transformation
  7. Local expansion
  8. Basic definitions on the lattice ${\mathbb{Z}}^d$
  9. Local expansion
  10. About norms and their properties
  11. Analysis of the RG Transformation
  12. Coordinates $({_{f}I}_j, {_{f}K}_j)$
  13. Choosing $J$ and Estimating ${\cal L}_1, {\cal L}_2$
  14. Choosing $J$
  15. Estimating ${\cal L}_1, {\cal L}_2$ - the first two linearization parts
  16. ...and 17 more sections

Key Result

Theorem 1

For $|z|$ and $\max_{k} |t_k|$ sufficiently small there is an $\varepsilon = \varepsilon (z)$ close to 1 so that ${_{f}p}_N = | \Lambda_N| ^{-1} \log( {_{f}Z}_N )$ has a limit as $N \to \infty$.In Theorem T1, $f(\phi)$ can be $0$, $\sum_{k=1}^m t_k \partial_{\mu_k} \phi (x_k)$, or $\

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Theorem 4
  • Theorem 5
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 16 more