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 $Z^2$.
"Recurrent Graphs where Two Independent Random Walks Collide Finitely Often." Electron. Commun. Probab. 9 72 - 81, 2004. https://doi.org/10.1214/ECP.v9-1111