The topology of scaling limits of positive genus random quadrangulations

Abstract

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given g, we consider, for every $n\ge 1$, a random quadrangulation ${\mathfrak{q}}_n$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus g with n faces. We view it as a metric space by endowing its set of vertices with the graph metric. As n tends to infinity, this metric space, with distances rescaled by the factor $n^{-1/4}$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the limiting space is almost surely homeomorphic to the genus g-torus.

Published in at http://dx.doi.org/10.1214/11-AOP675 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• The topological structure of scaling limits of large planar maps

Cited by in corpus (11)

• Scaling limits of random trees and planar maps
• Liouville Quantum Gravity on the complex tori
• A bijection for rooted maps on general surfaces
• Stability of geodesics in the Brownian map
• Computational topology of graphs on surfaces
• Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves
• Geodesics in Brownian surfaces (Brownian maps)
• The Brownian map is the scaling limit of uniform random plane quadrangulations
• Compact Brownian surfaces I. Brownian disks
• On tessellations of random maps and the $t_g$-recurrence
• Planar maps and random partitions