Recurrent Graphs where Two Independent Random Walks Collide Finitely Often

Manjunath Krishnapur,Yuval Peres

doi:10.1214/ecp.v9-1111

Abstract

We present a class of graphs where simple random walk is recurrent, yet two independent walkers meet only finitely many times almost surely. In particular, the comb lattice, obtained from $Z^2$ by removing all horizontal edges off the $x$-axis, has this property. We also conjecture that the same property holds for some other graphs, including the incipient infinite cluster for critical percolation in $Z^2$.

Highlights

In “Two Incidents” [7], George Polya describes the incident that led him to his celebrated results on random walks on Euclidean lattices: “. . . he and his fiancee set out for a stroll in the woods, and suddenly I met them there
Polya formulated the problem of the meeting of two walkers for random walks on a Euclidean lattice; in that case, it reduces to the problem of a single walker returning to his starting point
Say that a graph G has the finite collision property if two independent simple random walks X, Y on G starting from the same vertex meet only finitely many times, i.e., |{n : Xn = Yn}| < ∞, almost surely

Summary

Introduction

In “Two Incidents” [7], George Polya describes the incident that led him to his celebrated results on random walks on Euclidean lattices:. Say that a graph G has the finite collision property if two independent simple random walks X, Y on G starting from the same vertex meet only finitely many times, i.e., |{n : Xn = Yn}| < ∞, almost surely. Liggett [5] has given examples of symmetric recurrent Markov chains for which two independent copies of the chain collide only finitely many times Those examples are not simple random walks on graphs, . If X, Y are independent random walks on a graph starting from a vertex v, the expected number of meetings between them is (p(n)(v, w)), where p(n) is the nnw step transition function. Let X and Y be independent simple random walks (SRWs) on Comb(G) starting from the same vertex (o, 0). This proves that the total number of collisions between the two random walkers on Comb(G) is finite almost surely

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