## Electronic Journal of Probability

### Intersection and mixing times for reversible chains

#### 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
Accepted: 28 November 2016
First available in Project Euclid: 3 February 2017

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

Digital Object Identifier
doi:10.1214/16-EJP18

Mathematical Reviews number (MathSciNet)
MR3613705

Zentralblatt MATH identifier
1357.60076

#### 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

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