  Consider a $d$-ary rooted tree ($d\geq3$) where each edge $e$ is assigned an
i.i.d. (bounded) random variable $X(e)$ of negative mean. Assign to each vertex
$v$ the sum $S(v)$ of $X(e)$ over all edges connecting $v$ to the root, and
assume that the maximum $S_n^*$ of $S(v)$ over all vertices $v$ at distance $n$
from the root tends to infinity (necessarily, linearly) as $n$ tends to
infinity. We analyze the Metropolis algorithm on the tree and show that under
these assumptions there always exists a temperature $1/\beta$ of the algorithm
so that it achieves a linear (positive) growth rate in linear time. This
confirms a conjecture of Aldous [Algorithmica 22 (1998) 388-412]. The proof is
obtained by establishing an Einstein relation for the Metropolis algorithm on
the tree.
