Abstract
Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. The authors of [3] studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length n on a high-girth n-vertex regular expander, is typically (1+o(1)))log n/log log n, as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes.
In this work, we answer the above question by combining a generalized form of Brun’s sieve with some extensions of the ideas in [3]. Let N t denote the number of vertices visited precisely t times by a non-backtracking random walk of length n on a regular n-vertex expander of fixed degree and girth g. We prove that if g = ω(1), then for any fixed t, N t /n is typically 1/et! + o(1). Furthermore, if g = Ω(log log n), then N t /n is typically 1+o(1)/et! niformly on all t ≤ (1 − o(1)) log n/log log n and 0 for all t ≥ (1 + o(1)) log n/log log n. In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables.
Similar content being viewed by others
References
D. Aldous and J. A. Fill, Reversible Markov Chains and Random Walks on Graphs, in preparation, http://www.stat.berkeley.edu/~aldous/RWG/book.html.
R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lovász and C. Rackoff, Random walks, universal traversal sequences, and the complexity of maze problems, in 20th Annual Symposium on Foundations of Computer Science (San Juan, Puerto Rico, 1979), IEEE, New York, 1979, pp. 218–223.
N. Alon, I. Benjamini, E. Lubetzky and S. Sodin, Non-backtracking random walks mix faster, Communications in Contemporary Mathematics 9 (2007), 585–603.
N. Alon and M. Capalbo, Optimal universal graphs with deterministic embedding, Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), 373–378.
N. Alon and J. H. Spencer, The Probabilistic Method, second edition, Wiley, New York, 2000.
A. D. Barbour, L. Holst and S. Janson, Poisson Approximation, Oxford University Press, New York, 1992.
G. Barnes and U. Feige, Short random walks on graphs, Proc. of 25th STOC, 1993, pp. 728–737. Also in SIAM Journal on Discrete Mathematics 9 (1996), 19–28.
C. E. Bonferroni, Teoria statistica delle classi e calcolo delle probabilitá, Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze 8 (1936), 3–62.
W. Feller, An Introduction to Probability Theory and its Applications, Vol I, Wiley, New York, 1968.
C. Godsil and G. Royle, Algebraic Graph Theory, Volume 207 of Graduate Text in Mathematics, Springer, New York, 2001.
S. Janson, T. Luczak and A. Ruciński, Random Graphs, Wiley, New York, 2000.
M. Krivelevich and B. Sudakov, Pseudo-random graphs, in More Sets, Graphs and Numbers, Bolyai Society Mathematical Studies 15, Springer, New York, 2006, pp. 199–262.
L. Lovász, Random walks on graphs: a survey, in Combinatorics, Paul Erdős is Eighty, Vol. 2 (D. Miklós, V. T. Sós and T. Szőnyi eds.), János Bolyai Mathematical Society, Budapest, 1996, pp. 353–398.
R. M. Meyer, Note on a ‘multivariate’ form of Bonferroni’s inequalities, The Annals of Mathematical Statistics 40 (1969), 692–693.
O. Reingold, Undirected ST-connectivity in log-space, Proc. of 37th STOC (2005), 376–385.
A. Sinclair, Improved bounds for mixing rates of Markov chains and multicommodity flow, Combinatorics, Probability and Computing 1 (1992), 351–370.
N. C. Wormald, Models of random regular graphs, in Surveys in Combinatorics (J. D. Lamb and D. A. Preece eds.), London Mathematical Society Lecture Note Series, Vol. 276, Cambridge University Press, Cambridge, 1999, pp. 239–298.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation, by an ERC advanced grant and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.
Research partially supported by a Charles Clore Foundation Fellowship.
Rights and permissions
About this article
Cite this article
Alon, N., Lubetzky, E. Poisson approximation for non-backtracking random walks. Isr. J. Math. 174, 227–252 (2009). https://doi.org/10.1007/s11856-009-0112-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-009-0112-z