Painting a graph with competing random walks

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 $α_d$ such that $\lim_{n\to\infty}\operatorname{Var}(|\mathcal {A}_i|)/h_d(n)=\frac{1}{4}α_d$ 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 $α_d$ explicitly and show that $α_d\to1$ 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.

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