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Nonconcentration of return times. (English) Zbl 1268.05183
Summary: 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 \(t\geq 1\) we have \[ {\mathbf P}_v(\tau\geq t)\geq{c\over d_v\sqrt{t}} \] and \[ {\mathbf P}_v(\tau= t\mid \tau\geq t)\leq {C\log(d_v t)\over t} \] for some universal constants \(c> 0\) and \(C<\infty\). The first bound is attained for all \(t\) when the underlying graph is \(\mathbb{Z}\), 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 that graph \(G\) with \(\mathbb{Z}\), two independent random walks collide infinitely many times almost surely. This answers negatively a question of M. Krishnapur and Y. Peres [Electron. Commun. Probab. 9, 72–81 (2004; Zbl 1060.60044)] who asked whether every comb product of two infinite recurrent graphs has the finite collision property.

MSC:
05C81 Random walks on graphs
05C63 Infinite graphs
60G50 Sums of independent random variables; random walks
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References:
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