  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
$t\geq1$ we have \[\mathbf{P}_v(\tau\ge t)\ge\frac{c}{d_v\sqrt{t}}\] and
\[\mathbf{P}_v(\tau=t\mid\tau\geq t)\leq\frac{C\log(d_vt)}{t}\] for some
universal constants $c>0$ and $C<\infty$. The first bound is attained for all t
when the underlying graph is $\mathbb{Z}$, 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 $\mathbb{Z}$, 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.
