Sample path central limit theorem for the occupation time of the voter model on a lattice

TL;DR

This paper extends the central limit theorem for the occupation time of the voter model on a lattice to the sample path case in dimensions three and higher, using duality and resolvent strategies.

Contribution

It introduces a sample path version of the CLT for the voter model's occupation time in higher dimensions, building on previous static results.

Findings

Proves a CLT for occupation time in $d extgreater=3$

Uses duality with coalescing random walks

Provides a conjecture for the $d=2$ case

Abstract

In this paper, we extend the central limit theorem of the occupation time of the voter model on the lattice $\mathbb{Z}^d$ given in \cite{Cox1983} to the sample path case for $d\geq 3$. The proof of our main result utilizes the resolvent strategy and the Poisson flow strategy introduced in previous literatures, where the duality relationship between the voter model and the coalescing random walk plays the key role. For $d=2$ case, we give a conjecture about an analogue result of our main theorem.