  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.
