Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On random walk on growing graphs

Ruojun Huang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • [1] G. Amir, I. Benjamini, O. Gurel-Gurevich and G. Kozma. Random walk in changing environment. Available at arXiv:1504.04870.
  • [2] S. Andres, A. Chiarini, J.-D. Deuschel and M. Slowik. Quenched invariance principle for random walks with time-dependent ergodic degenerate weights. Ann. Probab. 46 (2018) 302–336.
  • [3] M. T. Barlow, R. F. Bass and T. Kumagai. Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan 58 (2006) 485–519.
  • [4] M. T. Barlow, A. Grigor’yan and T. Kumagai. On the equivalence of parabolic Harnack inequalities and heat kernel estimates. J. Math. Soc. Japan 64 (2012) 1091–1146.
  • [5] M. T. Barlow and M. Murugan. Stability of the elliptic Harnack inequality. Ann. of Math. (2) 187 (2018) 777–823.
  • [6] T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoam. 15 (1999) 181–232.
  • [7] A. Dembo, R. Huang, B. Morris and Y. Peres. Transience in growing subgraphs via evolving sets. Ann. Inst. Henri Poincaré B, Probab. Stat. 53 (2017) 1164–1180.
  • [8] A. Dembo, R. Huang and V. Sidoravicius. Walking within growing domains: Recurrence versus transience. Electron. J. Probab. 19 (2014) Article ID 106.
  • [9] A. Dembo, R. Huang and T. Zheng. Random walks among time increasing conductances: Heat kernel estimates. Available at arXiv:1705.07534.
  • [10] G. Giacomin and G. Posta. On recurrent and transient sets of inhomogeneous symmetric random walks. Electron. Commun. Probab. 6 (2001) 39–53.
  • [11] W. Hebisch and L. Saloff-Coste. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 (1993) 673–709.
  • [12] M. Hilário, F. den Hollander, R. S. dos Santos, V. Sidoravicius and A. Teixeira. Random walk on random walks. Electron. J. Probab. 20 (2015) Article ID 95.
  • [13] G. F. Lawler, M. Bramson and D. Griffeath. Internal diffusion limited aggregation. Ann. Probab. 20 (1992) 2117–2140.
  • [14] G. F. Lawler and V. Limic. Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge University Press, Cambridge, 2010.
  • [15] D. A. Levin, Y. Peres and E. L. Wilmer. Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI, 2009.
  • [16] B. Morris and Y. Peres. Evolving sets and mixing. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing 279–286. ACM, New York, 2003.
  • [17] B. Morris and Y. Peres. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2005) 245–266.
  • [18] J.-C. Mourrat and F. Otto. Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments. J. Funct. Anal. 270 (2016) 201–228.
  • [19] L. Saloff-Coste and J. Zúñiga. Merging for inhomogeneous finite Markov chains, part I: Singular values and stability. Electron. J. Probab. 14 (2009) 1456–1494.
  • [20] L. Saloff-Coste and J. Zúñiga. Merging for inhomogeneous finite Markov chains, part II: Nash and log-Sobolev inequalities. Ann. Probab. 39 (2011) 1161–1203.