The scaling limit of random simple triangulations and random simple quadrangulations

Abstract

Let $M_n$ be a simple triangulation of the sphere $S^2$, drawn uniformly at random from all such triangulations with n vertices. Endow $M_n$ with the uniform probability measure on its vertices. After rescaling graph distance on $V(M_n)$ by $(3/(4n){)}^{1/4}$, the resulting random measured metric space converges in distribution, in the Gromov-Hausdorff-Prokhorov sense, to the Brownian map. In proving the preceding fact, we introduce a labelling function for the vertices of $M_n$. Under this labelling, distances to a distinguished point are essentially given by vertex labels, with an error given by the winding number of an associated closed loop in the map. We establish similar results for simple quadrangulations.

47 pages, 10 figures Revised argument in section 6, section 4 rewritten

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• Compact Brownian surfaces I. Brownian disks
• The hull process of the Brownian plane
• Joint convergence of random quadrangulations and their cores
• The Brownian plane with minimal neck baby universe
• Random maps: proceeding of the Journ{é}es MAS 2014