## The Annals of Probability

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

### Nonconcentration of return times

Ori Gurel-Gurevich and Asaf Nachmias

#### 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

**Permanent link to this document**

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

**Keywords**

Random walks return times finite collision property

#### 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