TY - JOUR

T1 - Nonconcentration of return times

AU - Gurel-Gurevich, Ori

AU - Nachmias, Asaf

PY - 2013/3

Y1 - 2013/3

N2 - 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.

AB - 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.

KW - Finite collision property

KW - Random walks

KW - Return times

UR - http://www.scopus.com/inward/record.url?scp=84878982342&partnerID=8YFLogxK

U2 - 10.1214/12-AOP785

DO - 10.1214/12-AOP785

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AN - SCOPUS:84878982342

SN - 0091-1798

VL - 41

SP - 848

EP - 870

JO - Annals of Probability

JF - Annals of Probability

IS - 2

ER -