Some results on the statistics of hull perimeters in large planar triangulations and quadrangulations

TL;DR

This paper derives explicit probability distributions for hull perimeters at various distances in large random planar triangulations and quadrangulations, revealing universal behaviors and underlying recursive structures.

Contribution

It introduces a new recursive approach to analyze hull perimeter statistics in large planar maps, providing explicit formulas and showing universality across triangulations and quadrangulations.

Findings

Explicit probability densities for hull perimeters at distance d.

Joint probability densities for hull perimeters at two distances.

Universal laws applicable to both triangulations and quadrangulations.

Abstract

The hull perimeter at distance d in a planar map with two marked vertices at distance k from each other is the length of the closed curve separating these two vertices and lying at distance d from the first one (d<k). We study the statistics of hull perimeters in large random planar triangulations and quadrangulations as a function of both k and d. Explicit expressions for the probability density of the hull perimeter at distance d, as well as for the joint probability density of hull perimeters at distances d1 and d2, are obtained in the limit of infinitely large k. We also consider the situation where the distance d at which the hull perimeter is measured corresponds to a finite fraction of k. The various laws that we obtain are identical for triangulations and for quadrangulations, up to a global rescaling. Our approach is based on recursion relations recently introduced by the author which determine the generating functions of so-called slices, i.e. pieces of maps with appropriate distance constraints. It is indeed shown that the map decompositions underlying these recursion relations are intimately linked to the notion of hull perimeters and provide a simple way to fully control them.