Nonconcentration of return times

We show that the distribution of the first return time Τ to the origin, v, of a simple random walk on an infinite recurrent graph is heavy tailed and nonconcentrated. More precisely, if dv is the degree of v, then for any t ≥ 1 we have Pv(Τ ≥ t) ≥ c dv v t and Pv(Τ = t | Τ = t) ≤ C log(dvt) t for some universal constants c >0 andC <8. The first bound is attained for all t when the underlying graph is Z, and as for the second bound, we construct an example of a recurrent graph G for which it is attained for infinitely many t 's. Furthermore, we show that in the comb product of that graph G with Z, two independent random walks collide infinitely many times almost surely. This answers negatively a question of Krishnapur and Peres [Electron. Commun. Probab. 9 (2004) 72-81] who asked whether every comb product of two infinite recurrent graphs has the finite collision property.