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March 2013 Nonconcentration of return times
Ori Gurel-Gurevich, Asaf Nachmias
Ann. Probab. 41(2): 848-870 (March 2013). DOI: 10.1214/12-AOP785

Abstract

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 nonconcentrated. More precisely, if $d_{v}$ is the degree of $v$, then for any $t\geq1$ we have

\[\mathbf{P} _{v}(\tau\ge t)\ge\frac{c}{d_{v}\sqrt{t}}\]

and

\[\mathbf{P} _{v}(\tau=t\mid\tau\geq t)\leq\frac{C\log(d_{v}t)}{t}\]

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 Krishnapur and Peres [Electron. Commun. Probab. 9 (2004) 72–81] who asked whether every comb product of two infinite recurrent graphs has the finite collision property.

Citation

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Ori Gurel-Gurevich. Asaf Nachmias. "Nonconcentration of return times." Ann. Probab. 41 (2) 848 - 870, March 2013. https://doi.org/10.1214/12-AOP785

Information

Published: March 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1268.05183
MathSciNet: MR3077528
Digital Object Identifier: 10.1214/12-AOP785

Subjects:
Primary: 05C81

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • March 2013
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