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)≥cdvt√
and
Pv(τ=t∣τ≥t)≤Clog(dvt)t
for some universal constants c>0 and C<∞. 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.
[1]
P. Halmos.
What Does the Spectral Theorem Say
,
1963
.
[2]
Feller William,et al.
An Introduction To Probability Theory And Its Applications
,
1950
.
[3]
William Feller,et al.
An Introduction to Probability Theory and Its Applications
,
1967
.
[4]
Yuval Peres,et al.
Recurrent Graphs where Two Independent Random Walks Collide Finitely Often
,
2004
.
[5]
Perla Sousi,et al.
Collisions of random walks
,
2010,
1003.3255.
[6]
Gady Kozma,et al.
A Resistance Bound Via An Isoperimetric Inequality
,
2005,
Comb..