# First-passage percolation in random planar maps and Tutte's bijection

## TL;DR

This paper investigates the behavior of first-passage percolation distances on large random planar maps, demonstrating their proportionality to graph distances and analyzing the metric effects of Tutte's bijection.

## Contribution

It establishes the large-scale equivalence of first-passage percolation and graph distances in random planar maps and explores the metric implications of Tutte's bijection.

## Key findings

- First-passage percolation distances scale linearly with graph distances.
- Graph distances on quadrangulations and their Tutte-bijection maps are asymptotically equivalent.
- Method applies to both quadrangulations and general planar maps.

## Abstract

We consider large random planar maps and study the first-passage percolation distance obtained by assigning independent identically distributed lengths to the edges. We consider the cases of quadrangulations and of general planar maps. In both cases, the first-passage percolation distance is shown to behave in large scales like a constant times the usual graph distance. We apply our method to the metric properties of the classical Tutte bijection between quadrangulations with $n$ faces and general planar maps with $n$ edges. We prove that the respective graph distances on the quadrangulation and on the associated general planar map are in large scales equivalent when $n \to \infty$.