Refined universal laws for hull volumes and perimeters in large planar maps

TL;DR

This paper derives universal laws describing the statistical properties of hull volumes and perimeters in large random planar maps, revealing two distinct regimes depending on the relative distances within the maps.

Contribution

It introduces refined universal laws for hull statistics in large planar maps, covering arbitrary distances and regimes, extending previous understanding of map geometry.

Findings

Identifies two regimes: finite hull volume with infinite complement, and infinite hull with finite complement.

Calculates probabilities for each regime based on the ratio d/k.

Provides universal statistical laws for hull perimeter and volume in large maps.

Abstract

We consider ensembles of planar maps with two marked vertices at distance k from each other and look at the closed line separating these vertices and lying at distance d from the first one (d<k). This line divides the map into two components, the hull at distance d which corresponds to the part of the map lying on the same side as the first vertex and its complementary. The number of faces within the hull is called the hull volume and the length of the separating line the hull perimeter. We study the statistics of the hull volume and perimeter for arbitrary d and k in the limit of infinitely large planar quadrangulations, triangulations and Eulerian triangulations. We consider more precisely situations where both d and k become large with the ratio d/k remaining finite. For infinitely large maps, two regimes may be encountered: either the hull has a finite volume and its complementary is infinitely large, or the hull itself has an infinite volume and its complementary is of finite size. We compute the probability for the map to be in either regime as a function of d/k as well as a number of universal statistical laws for the hull perimeter and volume when maps are conditioned to be in one regime or the other.