Recurrence of planar graph limits

By Ori Gurel-Gurevich, Asaf Nachmias. 2012

119 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/dee04b56be6311d56be293b3a6c961693250f85e

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points (2012, background): “Acknowledgments: We are grateful to Christophe Garban for a very stimulating discussion and for useful remarks.”
Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps (2014, methodology): “Our principal tool for the proof will be the main result of [11]: since (M∞, E + ∞) is the limit in distribution of ( (Mn, E + n ), n ∈ N ) , and since E+ n is chosen according to the stationary distribution on Mn (that is, a vertex is chosen with probability proportional to its degree, i.”
Percolation on uniform infinite planar maps (2013, background): “( m,m′ ) = ( 1 + sup { r : B?r (m) = B?r (m′) })−1 for all rooted maps m, m′, where, for r > 1, we denote by B?r (m) the planar map obtained as the union of all faces of m that have at least one vertex at distance strictly smaller than r from the root.”
Planar stochastic hyperbolic triangulations (2016, methodology): “The special case κ = 2 27 has already received a lot of attention: it is known that the UIPT is recurrent [27] and subdiffusive in the quadrangular case [12].”
Hyperbolic and Parabolic Unimodular Random Maps (2016, background): “See [4] for an alternative proof, and [39, 6] for related results establishing recurrence in certain unbounded degree cases.”
Random geometry on the sphere (2014, methodology): “The recurrence of simple random walk on the UIPT or the UIPQ has been obtained recently by Gurel-Gurevich and Nachmias [26].”
A view from infinity of the uniform infinite planar quadrangulation (2012, background): “8 and its corollaries also answer positively the three conjectures raised by Krikun (2008). Note that the existence of the limit in (∗) was shown in Krikun (2008). It also follows from our fine study of the geodesics and their coalescence properties in the UIPQ, see Proposition 3.”