Performance of the Metropolis algorithm on a disordered tree: the Einstein relation

Consider a d-ary rooted tree (d≥ 3) 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/β 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.