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Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1

Michael Conroy, Sunder Sethuraman

TL;DR

This work analyzes the symmetric simple exclusion process on $\\mathbb{Z}^d$ with $d\\ge 2$ starting from step-like half-space initial data, focusing on the extreme value statistics through $N_t(z)$ and $X_t$. By leveraging self-duality and the stirring representation, it reduces covariance control to a one-dimensional random-walk framework and derives conditions under which $N_t$ converges to a Poisson law, matching the independent-particle limit. For initial data with polynomial growth, it identifies the correct level $z(t)$ and proves a Gumbel limit for the maximal particle position and all order statistics; in dimensions $d\\ge 4$ Poisson convergence follows from mean convergence alone, while in $d=2,3$ an explicit geometric condition is required. Together, these results reveal how higher dimensions yield stronger decorrelation and enable explicit extremal asymptotics, with open questions about broader initial profiles and subexponential growth regimes.

Abstract

We consider the symmetric simple exclusion system on $\mathbb{Z}^d$, $d \ge 2$, starting from a class of ``step'' initial conditions in which particles are constrained within a half-space. One may count the number $N_t$ of particles that have moved beyond a distance $z = z(t)$ into the initially-empty half of $\mathbb{Z}^d$ at time $t$. We show in large generality that when $\lim_{t\to\infty} E[N_t]$ exists, correlations between particles beyond $z$ vanish as $t \to \infty$ so as to allow convergence of $N_t$ to the same Poisson distribution one would get were the particles allowed to move independently. When the initial condition constrains a region of polynomial growth, we identify $z(t)$ and the limit of $E[N_t]$ explicitly. As a consequence of the limit, we obtain a Gumbel limit distribution for the extremal particle position, as well as the limiting distributions of all order statistics.

Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1

TL;DR

This work analyzes the symmetric simple exclusion process on with starting from step-like half-space initial data, focusing on the extreme value statistics through and . By leveraging self-duality and the stirring representation, it reduces covariance control to a one-dimensional random-walk framework and derives conditions under which converges to a Poisson law, matching the independent-particle limit. For initial data with polynomial growth, it identifies the correct level and proves a Gumbel limit for the maximal particle position and all order statistics; in dimensions Poisson convergence follows from mean convergence alone, while in an explicit geometric condition is required. Together, these results reveal how higher dimensions yield stronger decorrelation and enable explicit extremal asymptotics, with open questions about broader initial profiles and subexponential growth regimes.

Abstract

We consider the symmetric simple exclusion system on , , starting from a class of ``step'' initial conditions in which particles are constrained within a half-space. One may count the number of particles that have moved beyond a distance into the initially-empty half of at time . We show in large generality that when exists, correlations between particles beyond vanish as so as to allow convergence of to the same Poisson distribution one would get were the particles allowed to move independently. When the initial condition constrains a region of polynomial growth, we identify and the limit of explicitly. As a consequence of the limit, we obtain a Gumbel limit distribution for the extremal particle position, as well as the limiting distributions of all order statistics.
Paper Structure (16 sections, 22 theorems, 86 equations, 1 figure)

This paper contains 16 sections, 22 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Discussion of results
  3. A step profile in higher dimensions
  4. Overview of results
  5. Proof methods
  6. Open problems
  7. Definitions and preliminaries
  8. The stirring process
  9. Negative association and the Rayleigh property
  10. Notation and conventions
  11. Main results
  12. A Gumbel limit theorem
  13. Proofs of Lemmas \ref{['covintrep']} and \ref{['meanasymp']}
  14. Proof of Theorem \ref{['meta']}
  15. Proofs of Propositions \ref{['ssbound']} and \ref{['mainbound']} and Corollary \ref{['Ccor']}
  16. ...and 1 more sections

Key Result

Lemma 2.1

For any initial profile $\eta \in \{0,1\}^{\mathbb{Z}^d}$ and any $z \in \mathbb{R}$, where $\zeta^{(d)}_t$ is a continuous time simple random walk on $\mathbb{Z}^d$ with $P_j(\cdot) = P(\cdot | \zeta^{(d)}_0 = j)$.

Figures (1)

  • Figure 1: (a) An arbitrary initial profile in $\mathbb{Z}^2$ determined by a nonnegative "shape" function. (b) An initial profile in $\mathbb{Z}^3$ determined by two linear functions.

Theorems & Definitions (32)

  • Lemma 2.1
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Definition 3.7
  • Lemma 3.8
  • Lemma 3.9
  • ...and 22 more