Scaling limits for random triangulations on the torus

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 −1/4 d 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 ).