Painting a graph with competing random walks

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.