On the metric structure of random planar maps and SLE-decorated Liouville quantum gravity

Abstract

A random planar map is a graph embedded in the sphere, viewed modulo orientation-preserving homeomorphisms. Random planar maps are the discrete analogues of random fractal surfaces called [gamma]-Liouville quantum gravity (LQG) surfaces with parameter [gamma] E (0, 2]. We study the large-scale structure of random planar maps (and statistical mechanics models on them) viewed as metric measure spaces equipped with the graph distance and the counting measure on vertices. In particular, we show that uniform random planar maps (which correspond to the case [gamma]= [square root of]8/3) decorated by a self-avoiding walk or a critical percolation interface converge in the scaling limit to [square root of]8/3- LQG surfaces decorated by SLE8/3 and SLE6, respectively, with respect to a generalization of the Gromov-Hausdorff topology. We also introduce an approach for analyzing certain random planar maps belonging to the [gamma]-LQG universality class for general [gamma] E (0, 2) and use this approach to prove several estimates for graph distances in such maps.