Nonconcentration of return times

• Autores: Ori Gurel-Gurevich, Asaf Nachmias
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 2, 2013, págs. 848-870
• Idioma: inglés
• DOI: 10.1214/12-AOP785
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• Resumen

  • 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.