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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
##### Keywords:
random walks; return times; finite collision property
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##### References:
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