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Abstract: A recurrent graph $G$ has the infinite collision property if two independentrandom walks on $G$, started at the same point, collide infinitely often a.s.We give a simple criterion in terms of Green functions for a graph to have thisproperty, and use it to prove that a critical Galton-Watson tree with finitevariance conditioned to survive, the incipient infinite cluster in $\Z^d$ with$d \ge 19$ and the uniform spanning tree in $\Z^2$ all have the infinitecollision property. For power-law combs and spherically symmetric trees, wedetermine precisely the phase boundary for the infinite collision property.

Author: Martin T. Barlow, Yuval Peres, Perla Sousi


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