Scaling limit of first passage percolation geodesics on planar maps

TL;DR

This paper investigates the scaling limits of geodesics in first passage percolation on random planar maps, revealing new insights into their geometric structure and providing bounds on map diameters.

Contribution

It introduces the scaling limit of geodesics and their associated trees, using a stochastic flow approach, and relates these to the geometry of large random maps.

Findings

Scaling limit of the number of faces along geodesics

Comparison between first passage percolation and dual graph distance

Upper bounds for the diameter of large random maps

Abstract

We establish the scaling limit of the geodesics to the root for the first passage percolation distance on random planar maps. We first describe the scaling limit of the number of faces along the geodesics. This result enables us to compare the metric balls for the first passage percolation and the dual graph distance. It also enables us to give an upper bound for the diameter of large random maps. Then, we describe the scaling limit of the tree of first passage percolation geodesics to the root via a stochastic coalescing flow of pure jump diffusions. Using this stochastic flow, we also construct some random metric spaces which we conjecture to be the scaling limits of random planar maps with high degrees. The main tool in this work is a time-reversal of the uniform peeling exploration.