Recurrent Graphs where Two Independent Random Walks Collide Finitely Often

Manjunath Krishnapur, Yuval Peres
2004 Electronic Communications in Probability  
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 .
doi:10.1214/ecp.v9-1111 fatcat:pzzbpqvuyfghzgmmicql25xabm