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Poisson approximation for non-backtracking random walks Eyal Lubetzky

Summary: Poisson approximation for non-backtracking random walks
Noga Alon
Eyal Lubetzky
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 Nt 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 = (1), then for any fixed t, Nt/n is typically
et! + o(1). Furthermore, if g = (log log n), then Nt/n is typically 1+o(1)
et! uniformly on all


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics