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

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 $n\to \mathrm{\infty }$.