## The Annals of Probability

### Nonconcentration of return times

#### 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.

#### Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 848-870.

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.aop/1362750944

Digital Object Identifier
doi:10.1214/12-AOP785

Mathematical Reviews number (MathSciNet)
MR3077528

Zentralblatt MATH identifier
1268.05183

Subjects
Primary: 05C81: Random walks on graphs

#### Citation

Gurel-Gurevich, Ori; Nachmias, Asaf. Nonconcentration of return times. Ann. Probab. 41 (2013), no. 2, 848--870. doi:10.1214/12-AOP785. https://projecteuclid.org/euclid.aop/1362750944

#### References

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