TY - JOUR

T1 - Recurrence of planar graph limits

AU - Gurel-Gurevich, Ori

AU - Nachmias, Asaf

PY - 2013/3

Y1 - 2013/3

N2 - We prove that any distributional limit of finite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform infinite planar triangulation and quadrangulation (UIPT and UIPQ) are almost surely recurrent, resolving a conjecture of Angel, Benjamini and Schramm. We also settle another related problem of Benjamini and Schramm. We show that in any bounded degree, finite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for T steps is at most C/log T.

AB - We prove that any distributional limit of finite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform infinite planar triangulation and quadrangulation (UIPT and UIPQ) are almost surely recurrent, resolving a conjecture of Angel, Benjamini and Schramm. We also settle another related problem of Benjamini and Schramm. We show that in any bounded degree, finite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for T steps is at most C/log T.

UR - http://www.scopus.com/inward/record.url?scp=84874806715&partnerID=8YFLogxK

U2 - 10.4007/annals.2013.177.2.10

DO - 10.4007/annals.2013.177.2.10

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AN - SCOPUS:84874806715

SN - 0003-486X

VL - 177

SP - 761

EP - 781

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -