Non-concentration of return times

Details Non-concentration of return times Non-concentration of return times

2010-09-08

arxiv.org/abs/1009.1438v1

Mathematics

Probability

23 pages

Scientific paper

We show that the distribution of the first return time tau to the origin, v, of a simple random walk on an infinite recurrent graph is heavy tailed and non-concentrated. More precisely, if d_v is the degree of v then for any t>1 we have P_v(tau \ge t) > c d_v^{-1} t^{-1/2} and P_v(tau = t | tau \ge t) < C log(d_v t) t^{-1}. The first bound is attained for all t when the underlying graph is Z, and as for the second bound, we construct an example of a recurrent graph G for which it is attained for infinitely many t's. Furthermore, we show that in the comb product of G with Z, two independent random walks collide infinitely many times almost surely. This answers negatively a question of Krishnapur and Peres who asked whether any comb product of two infinite recurrent graphs has the finite collision property.

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