  A recurrent graph $G$ has the infinite collision property if two independent
random walks on $G$, started at the same point, collide infinitely often a.s.
We give a simple criterion in terms of Green functions for a graph to have this
property, and use it to prove that a critical Galton-Watson tree with finite
variance conditioned to survive, the incipient infinite cluster in $\Z^d$ with
$d \ge 19$ and the uniform spanning tree in $\Z^2$ all have the infinite
collision property. For power-law combs and spherically symmetric trees, we
determine precisely the phase boundary for the infinite collision property.
