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Non-Backtracking Random Walks and Cogrowth of Graphs

Published:2007-08-01
Printed: Aug 2007
• Ronald Ortner
• Wolfgang Woess
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Abstract

Let $X$ be a locally finite, connected graph without vertices of degree $1$. Non-backtracking random walk moves at each step with equal probability to one of the forward'' neighbours of the actual state, \emph{i.e.,} it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$. Thus we obtain for infinite $X$ that the $n$-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of old results concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but \emph{small cycles are dense in} $X$, we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen.
 Keywords: graph, oriented line grap, covering tree, random walk, cogrowth, amenability
 MSC Classifications: 05C75 - Structural characterization of families of graphs 60G50 - Sums of independent random variables; random walks 20F69 - Asymptotic properties of groups