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Central limit theorems and moderate deviations for additive functionals of SSEP on regular trees

Xiaofeng Xue

TL;DR

This work establishes a central limit theorem and a moderate deviation principle for general additive functionals of the symmetric simple exclusion process on regular trees, with the occupation time as a key special case. It relies on a graphical representation that extends Kipnis’s martingale decomposition to general additive functionals, yielding a uniform $\sqrt{N}$-scaling CLT with variance $\sigma_F^2$. The moderate deviations and the pathwise MDP are proved via an exponential-martingale approach and a sequence of replacement lemmas, supplemented by time-block analyses and heat-kernel bounds on the tree. The results show that fluctuations of additive functionals on $\mathcal{T}_d$ exhibit a dimension-agnostic CLT and a robust, tree-structure-driven moderate-deviation regime, broadening the understanding of fluctuation phenomena for interacting particle systems on non-Euclidean graphs.

Abstract

In this paper, we are concerned with the symmetric simple exclusion process (SSEP) on the regular tree $\mathcal{T}_d$. A central limit theorem and a moderate deviation principle of the additive functional of the process are proved, which include the CLT and the MDP of the occupation time as special cases. A graphical representation of the SSEP plays the key role in proofs of the main results, by which we can extend the martingale decomposition formula introduced in Kipnis (1987) for the occupation time to the case of general additive functionals.

Central limit theorems and moderate deviations for additive functionals of SSEP on regular trees

TL;DR

This work establishes a central limit theorem and a moderate deviation principle for general additive functionals of the symmetric simple exclusion process on regular trees, with the occupation time as a key special case. It relies on a graphical representation that extends Kipnis’s martingale decomposition to general additive functionals, yielding a uniform -scaling CLT with variance . The moderate deviations and the pathwise MDP are proved via an exponential-martingale approach and a sequence of replacement lemmas, supplemented by time-block analyses and heat-kernel bounds on the tree. The results show that fluctuations of additive functionals on exhibit a dimension-agnostic CLT and a robust, tree-structure-driven moderate-deviation regime, broadening the understanding of fluctuation phenomena for interacting particle systems on non-Euclidean graphs.

Abstract

In this paper, we are concerned with the symmetric simple exclusion process (SSEP) on the regular tree . A central limit theorem and a moderate deviation principle of the additive functional of the process are proved, which include the CLT and the MDP of the occupation time as special cases. A graphical representation of the SSEP plays the key role in proofs of the main results, by which we can extend the martingale decomposition formula introduced in Kipnis (1987) for the occupation time to the case of general additive functionals.

Paper Structure

This paper contains 8 sections, 16 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Graphical representation
  4. Proof of Theorem \ref{['theorem 2.1 CLT']}
  5. Proof of Theorem \ref{['theorem 2.2 mdp']}
  6. Outline of the proof
  7. Proofs of Lemmas \ref{['lemma 5.1 exponentially tightness']}, \ref{['lemma 5.2']} and \ref{['lemma 5.3 replacement lemma']}
  8. Proof of Theorem \ref{['theorem 2.3 sample path mdp']}

Key Result

Lemma 1.1

Let $d\geq 2$. For any local function $F: \{0, 1\}^{\mathcal{T}_d}\rightarrow \mathbb{R}$ satisfying equ basic mean zero assumption, there exists $\sigma^2_F<+\infty$ such that

Theorems & Definitions (18)

  • Lemma 1.1
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 5.1
  • Lemma 5.2
  • ...and 8 more