The Annals of Applied Probability

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

Pascal Maillard and Ofer Zeitouni

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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.

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Ann. Appl. Probab., Volume 24, Number 5 (2014), 2070-2090.

First available in Project Euclid: 26 June 2014

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60J22: Computational methods in Markov chains [See also 65C40] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Metropolis algorithm Einstein relation branching random walk random walk in random environment


Maillard, Pascal; Zeitouni, Ofer. Performance of the Metropolis algorithm on a disordered tree: The Einstein relation. Ann. Appl. Probab. 24 (2014), no. 5, 2070--2090. doi:10.1214/13-AAP972.

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