Nonconcentration of return times.

*(English)*Zbl 1268.05183Summary: 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.

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 |

##### References:

[1] | Barlow, M., Peres, Y. and Sousi, P. (2010). Collisions of random walks. Preprint. Available at . · Zbl 1285.60073 · arxiv.org |

[2] | Benjamini, I. and Kozma, G. (2005). A resistance bound via an isoperimetric inequality. Combinatorica 25 645-650. · Zbl 1098.60075 · doi:10.1007/s00493-005-0040-4 |

[3] | Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II , 2nd ed. Wiley, New York. · Zbl 0219.60003 |

[4] | Halmos, P. R. (1963). What does the spectral theorem say? Amer. Math. Monthly 70 241-247. · Zbl 0132.35606 · doi:10.2307/2313117 |

[5] | Krishnapur, M. and Peres, Y. (2004). Recurrent graphs where two independent random walks collide finitely often. Electron. Commun. Probab. 9 72-81 (electronic). · Zbl 1060.60044 · doi:10.1214/ECP.v9-1111 · eudml:124513 |

[6] | Lyons, R. and Peres, Y. (2008). Probability on trees and networks. Unpublished manuscript. Current version available at . · mypage.iu.edu |

[7] | Rudin, W. (1991). Functional Analysis , 2nd ed. McGraw-Hill, New York. · Zbl 0867.46001 |

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