Nonconcentration of Return Times
Let T be the return time to the origin of a simple random walk on a
recurrent graph. We show that T is heavy tailed and non-concentrated.
More precisely, we have
i) P(T>t) > c/sqrt(t)
ii) P(T=t|T>=t) < C log(t)/t
Inequality i) is attained on Z, and we construct an example
demonstrating the sharpness of ii). We use this example to answer
negatively a question of Peres and Krishnapur about recurrent graphs
with the finite collision property (that is, two independent SRW on
them collide only finitely many times, almost surely).
Joint work with Asaf Nachmias.