Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On random walk on growing graphs

Ruojun Huang

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Abstract

Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple random walk on slowly growing graphs, upon knowing the volume and Cheeger constant of each graph. For much more specialized cases, we establish matching lower bounds, and deduce sufficient (weak) recurrence criteria. We also address recurrence directly in relation to a universality conjecture of (Electron. J. Probab. 19 (2014) Article ID 106). We answer a related question of (Ann. Probab. 39 (2011) 1161–1203, Problem 1.8) about “inhomogeneous merging” in the negative.

Résumé

Nous considérons un modèle de marche aléatoire sur un graphe dynamique. Pour une suite de graphes finis croissant vers un graphe limite infini, nous montrons une borne supérieure pour la probabilité de transition. Cela donne un critère de transience pour la marche simple, pour des graphes à croissante lente, à partir du volume et de la constante de Cheeger de chaque graphe. Pour des cas plus particuliers, nous montrons une borne inférieure du même ordre et déduisons un critère de récurrence (dans un sens faible). Nous répondons aussi à la question de la récurrence directement, en lien avec une conjecture d’universalité de (Electron. J. Probab. 19 (2014) Article ID 106). Nous répondons aussi négativement à une question reliée de (Ann. Probab. 39 (2011) 1161–1203, Problem 1.8), à propos du « plongement inhomogène ».

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1149-1162.

Dates
Received: 19 April 2017
Revised: 29 April 2018
Accepted: 2 May 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820846

Digital Object Identifier
doi:10.1214/18-AIHP913

Mathematical Reviews number (MathSciNet)
MR3949968

Zentralblatt MATH identifier
07097346

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60K37: Processes in random environments

Keywords
Random walk Time-inhomogeneity Evolving sets Recurrence Transience Heat kernel bounds Merging

Citation

Huang, Ruojun. On random walk on growing graphs. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1149--1162. doi:10.1214/18-AIHP913. https://projecteuclid.org/euclid.aihp/1557820846


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