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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Abstract

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).

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624034

Digital Object Identifier
doi:10.1214/16-AIHP751

Mathematical Reviews number (MathSciNet)
MR3689964

Zentralblatt MATH identifier
1378.60099

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.aihp/1500624034


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