NONCONCENTRATION OF RETURN TIMES

We show that the distribution of the first return time tau to the origin, v, of a simple random walk on an infinite recurrent graph is heavy tailed and nonconcentrated. More precisely, if d(v) is the degree of v, then for any t >= 1 we have P-v(tau >= t) >= c/d(v)root t and P-v(tau = t vertical bar tau >= t) <= C log(d(v)t)/t for some universal constants c > 0 and C < infinity. 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.