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

TL;DR

This paper proves that the Metropolis algorithm on a disordered tree exhibits a linear growth rate at a specific temperature, confirming a conjecture and establishing an Einstein relation for this setting.

Contribution

It demonstrates the existence of a temperature where the Metropolis algorithm achieves linear growth on a disordered tree, confirming a conjecture by Aldous and establishing an Einstein relation.

Findings

Existence of a temperature with linear growth rate for the algorithm

Confirmation of Aldous's conjecture from 1998

Establishment of an Einstein relation for the algorithm on the tree

Abstract

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.