Open Access
2017 Intersection and mixing times for reversible chains
Yuval Peres, Thomas Sauerwald, Perla Sousi, Alexandre Stauffer
Electron. J. Probab. 22: 1-16 (2017). DOI: 10.1214/16-EJP18


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} $.


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Yuval Peres. Thomas Sauerwald. Perla Sousi. Alexandre Stauffer. "Intersection and mixing times for reversible chains." Electron. J. Probab. 22 1 - 16, 2017.


Received: 5 February 2016; Accepted: 28 November 2016; Published: 2017
First available in Project Euclid: 3 February 2017

zbMATH: 1357.60076
MathSciNet: MR3613705
Digital Object Identifier: 10.1214/16-EJP18

Primary: 60J10

Keywords: Doob’s maximal inequality , intersection time , martingale , mixing time , Random walk

Vol.22 • 2017
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