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

Abstract

Consider a $d$-ary rooted tree ($d\ge 3$) 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 $S_n^{\ast }$ 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.

Published in at http://dx.doi.org/10.1214/13-AAP972 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Transient Random Walks in Random Environment on a Galton-Watson Tree