  Suppose that $\CG$ is a finite, connected graph and $X$ is a lazy random walk
on $\CG$. The lamplighter chain $X^\diamond$ associated with $X$ is the random
walk on the wreath product $\CG^\diamond = \Z_2 \wr \CG$, the graph whose
vertices consist of pairs $(f,x)$ where $f$ is a labeling of the vertices of
$\CG$ by elements of $\Z_2$ and $x$ is a vertex in $\CG$. There is an edge
between $(f,x)$ and $(g,y)$ in $\CG^\diamond$ if and only if $x$ is adjacent to
$y$ in $\CG$ and $f(z) = g(z)$ for all $z \neq x,y$. In each step, $X^\diamond$
moves from a configuration $(f,x)$ by updating $x$ to $y$ using the transition
rule of $X$ and then sampling both $f(x)$ and $f(y)$ according to the uniform
distribution on $\Z_2$; $f(z)$ for $z \neq x,y$ remains unchanged. We give
matching upper and lower bounds on the uniform mixing time of $X^\diamond$
provided $\CG$ satisfies mild hypotheses. In particular, when $\CG$ is the
hypercube $\Z_2^d$, we show that the uniform mixing time of $X^\diamond$ is
$\Theta(d 2^d)$. More generally, we show that when $\CG$ is a torus $\Z_n^d$
for $d \geq 3$, the uniform mixing time of $X^\diamond$ is $\Theta(d n^d)$
uniformly in $n$ and $d$. A critical ingredient for our proof is a
concentration estimate for the local time of random walk in a subset of
vertices.
