Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Transience in growing subgraphs via evolving sets

Amir Dembo, Ruojun Huang, Ben Morris, and Yuval Peres

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We extend the use of random evolving sets to time-varying conductance models and utilize it to provide tight heat kernel upper bounds. It yields the transience of any uniformly lazy random walk, on $\mathbb{Z}^{d}$, $d\ge3$, equipped with uniformly bounded above and below, independently time-varying edge conductances, of (effectively) non-decreasing in time vertex conductances, thereby affirming part of Conjecture 7.1 (Random walk in changing environment (2015) Preprint).


Nous généralisons la méthode basée sur l’évolution aléatoire d’ensembles au cas de modèles de conductances variant avec le temps. Nous l’utilisons pour prouver des bornes supérieures sur le noyau de la chaleur. Ceci montre la transitivité de n’importe quelle marche aléatoire fainéante, dans $\mathbb{Z}^{d}$, $d\ge3$, avec des conductances par arêtes (bornées uniformément supérieurement et inférieurement) variant indépendamment en temps en fonction des conductances par sites. Ceci répond partiellement à la Conjecture 7.1 (Random walk in changing environment (2015) Preprint).

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1164-1180.

Received: 4 September 2015
Revised: 20 January 2016
Accepted: 10 March 2016
First available in Project Euclid: 21 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K37: Processes in random environments 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Transience Time in-homogeneous Markov chains Heat kernel estimate Growing sub-graphs Conductance models Evolving sets Percolation


Dembo, Amir; Huang, Ruojun; Morris, Ben; Peres, Yuval. Transience in growing subgraphs via evolving sets. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1164--1180. doi:10.1214/16-AIHP751.

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