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Inhomogeneous central limit theorems for the voter model occupation times

Xiaofeng Xue

Abstract

In this paper, we extend the functional central limit theorems for the occupation times of the voter models on lattices given in Xue2026 to the case where the initial distribution is a spatially inhomogeneous product measure. The duality relationship between the voter model and the coalescing random walk and the Donsker's invariance principle of the simple random walk play the key roles in the proofs of our main results.

Inhomogeneous central limit theorems for the voter model occupation times

Abstract

In this paper, we extend the functional central limit theorems for the occupation times of the voter models on lattices given in Xue2026 to the case where the initial distribution is a spatially inhomogeneous product measure. The duality relationship between the voter model and the coalescing random walk and the Donsker's invariance principle of the simple random walk play the key roles in the proofs of our main results.
Paper Structure (6 sections, 9 theorems, 124 equations)

This paper contains 6 sections, 9 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['theorem 2.1 main d geq 4']}
  4. Proof of Theorem \ref{['theorem 2.1 main d geq 4']}: The case $d\geq 5$
  5. Proof of Theorem \ref{['theorem 2.1 main d geq 4']}: The case $d=4$
  6. Proof of Theorem \ref{['theorem 2.2 main d=3']}

Key Result

Theorem 2.1

Let $d\geq 4, T>0$, $\rho$ be defined as in Section section one and $h_d$ be defined as in equ 1.3 hdt. If $\eta_0$ is distributed with $\nu_{\rho, N}$, then $\left\{\frac{1}{h_d(N)}\xi_{tN}^{N, d}:~0\leq t\leq T\right\}$ converges weakly, with respect to the uniform topology of $C[0, T]$, to as $N\rightarrow+\infty$, where $\xi_t^{N, d}$ is defined as in equ 1.2 centered occupation time and

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3