Recurrent graphs where two independent random walks collide finitely often

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 Z2 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 Z2. © 2004 Rocky Mountain Mathematics Consortium.