Recurrent Graphs where Two Independent Random Walks Collide Finitely Often

@article{Krishnapur2004RecurrentGW,
title={Recurrent Graphs where Two Independent Random Walks Collide Finitely Often},
author={Manjunath Krishnapur and Y. Peres},
journal={Electronic Communications in Probability},
year={2004},
volume={9},
pages={72-81}
}

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 $Z^2$.