Scaling limits for random triangulations on the torus

2019/05/06 by Vincent Beffara, Beffara, Vincent, Cong Bang Huynh +3
Computer Science · Mathematics · #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1905.01873

openalex publication_date 2019/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the scaling limit of essentially simple triangulations on the torus. We consider, for every n≥ 1, a uniformly random triangulation G_n over the set of (appropriately rooted) essentially simple triangulations on the torus with n vertices. We view G_n as a metric space by endowing its set of vertices with the graph distance denoted by d_Gn and show that the random metric space (V(G_n),n^-1/4d_Gn) converges in distribution in the Gromov-Hausdorff sense when n goes to infinity, at least along subsequences, toward a random metric space. One of the crucial steps in the argument is to construct a simple labeling on the map and show its convergence to an explicit scaling limit. We moreover show that this labeling approximates the distance to the root up to a uniform correction of order o(n^1/4).

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