Recurrence of bipartite planar maps

Abstract

This paper concerns random bipartite planar maps which are defined by assigning weights to their faces. The paper presents a threefold contribution to the theory. Firstly, we prove the existence of the local limit for all choices of weights and describe it in terms of an infinite mobile. Secondly, we show that the local limit is in all cases almost surely recurrent. And thirdly, we show that for certain choices of weights the local limit has exactly one face of infinite degree and has in that case spectral dimension $4/3$ (the latter requires a mild moment condition).

47 pages, 6 figures. Revised version

References in corpus (4)

• The topological structure of scaling limits of large planar maps
• Local limit of labeled trees and expected volume growth in a random quadrangulation
• The spectral dimension of generic trees
• Condensation in nongeneric trees

Cited by in corpus (9)

• Basic properties of the infinite critical-FK random map
• Infinite random planar maps related to Cauchy processes
• The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)
• Simply generated non-crossing partitions
• Planar stochastic hyperbolic infinite triangulations
• Infinite stable Boltzmann planar maps are subdiffusive
• How fast planar maps get swallowed by a peeling process
• Uniform infinite half-planar quadrangulations with skewness
• Markovian explorations of random planar maps are roundish