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Collision of random walks

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Regarding his 1920 paper proving recurrence of random walks in Z2, Polya wrote that his motivation was to determine whether 2 independent random walks in Z2 meet infinitely often. Of course, in this case, the problem reduces to the recurrence of a single random walk in Z2, by taking differences. Perhaps surprisingly, however, there exist graphs G where a single random walk is recurrent, yet G has the finite collision property : two independent random walks in G collide only finitely many times almost surely. Some examples were constructed by Krishnapur and Peres (2004), who asked whether critical Galton-Watson trees conditioned on nonextinction also have this property. In this talk I will answer this question as part of a systematic study of the finite collision property. In particular, for two classes of graphs, wedge combs and spherically symmetric trees, we exhibit a phase transition for the finite collision property when growth parameters are varied. I will state the main theorems and give some ideas of the proofs. This is joint work with Martin Barlow and Yuval Peres.

This talk is part of the Optimization and Incentives Seminar series.

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