## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 12, 16 pp.

### Intersection and mixing times for reversible chains

Yuval Peres, Thomas Sauerwald, Perla Sousi, and Alexandre Stauffer

#### Abstract

We consider two independent Markov chains on the same finite state space, and study their *intersection time*, which is the first time that the trajectories of the two chains intersect. We denote by $t_I$ the expectation of the intersection time, maximized over the starting states of the two chains. We show that, for any reversible and lazy chain, the total variation mixing time is $O(t_I)$. When the chain is reversible and transitive, we give an expression for $t_I$ using the eigenvalues of the transition matrix. In this case, we also show that $t_I$ is of order $\sqrt{n \mathbb {E}\!\left [I\right ]} $, where $I$ is the number of intersections of the trajectories of the two chains up to the uniform mixing time, and $n$ is the number of states. For random walks on trees, we show that $t_I$ and the total variation mixing time are of the same order. Finally, for random walks on regular expanders, we show that $t_I$ is of order $\sqrt{n} $.

#### Article information

**Source**

Electron. J. Probab., Volume 22 (2017), paper no. 12, 16 pp.

**Dates**

Received: 5 February 2016

Accepted: 28 November 2016

First available in Project Euclid: 3 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1486090892

**Digital Object Identifier**

doi:10.1214/16-EJP18

**Mathematical Reviews number (MathSciNet)**

MR3613705

**Zentralblatt MATH identifier**

1357.60076

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

intersection time random walk mixing time martingale Doob’s maximal inequality

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Peres, Yuval; Sauerwald, Thomas; Sousi, Perla; Stauffer, Alexandre. Intersection and mixing times for reversible chains. Electron. J. Probab. 22 (2017), paper no. 12, 16 pp. doi:10.1214/16-EJP18. https://projecteuclid.org/euclid.ejp/1486090892