Painting a graph with competing random walks

TL;DR

This paper proves a conjecture about the asymptotic variance of sites painted by two competing random walks on high-dimensional tori, revealing how it scales with the size and dimension of the graph.

Contribution

It establishes the asymptotic behavior of the variance of painted sites for competing random walks on $ extbf{Z}_n^d$, confirming a conjecture and identifying explicit constants.

Findings

Variance scales as $n^d$, $n^4$, or $n^4 ext{log} n$ depending on dimension

Explicit formula for the limiting constant $rac{1}{4}eta_d$

As $d o fty$, $eta_d o 1$

Abstract

Let $X_1,X_2$ be independent random walks on $\mathbf{Z}_n^d$, $d\geq3$, each starting from the uniform distribution. Initially, each site of $\mathbf{Z}_n^d$ is unmarked, and, whenever $X_i$ visits such a site, it is set irreversibly to $i$. The mean of $|\mathcal{A}_i|$, the cardinality of the set $\mathcal{A}_i$ of sites painted by $i$, once all of $\mathbf{Z}_n^d$ has been visited, is $\frac{1}{2}n^d$ by symmetry. We prove the following conjecture due to Pemantle and Peres: for each $d\geq3$ there exists a constant $\alpha_d$ such that $\lim_{n\to\infty}\operatorname{Var}(|\mathcal {A}_i|)/h_d(n)=\frac{1}{4}\alpha_d$ where $h_3(n)=n^4$, $h_4(n)=n^4(\log n)$ and $h_d(n)=n^d$ for $d\geq5$. We will also identify $\alpha_d$ explicitly and show that $\alpha_d\to1$ as $d\to\infty$. This is a special case of a more general theorem which gives the asymptotics of $\operatorname{Var}(|\mathcal{A}_i|)$ for a large class of transient, vertex transitive graphs; other examples include the hypercube and the Caley graph of the symmetric group generated by transpositions.