Distance statistics in quadrangulations with no multiple edges and the geometry of minbus

TL;DR

This paper calculates the distance-dependent two-point function for quadrangulations without multiple edges, analyzes their large-scale geometry, and explores the structure of minimal neck baby universes (minbus) in these maps.

Contribution

It provides a detailed analytical computation of the two-point function and characterizes the geometry of minbus in quadrangulations with no multiple edges, extending understanding of their universal properties.

Findings

Universal scaling function for large distances

Explicit characterizations of minbus geometry

Distance laws within minbus and to the mother universe

Abstract

We present a detailed calculation of the distance-dependent two-point function for quadrangulations with no multiple edges. Various discrete observables measuring this two-point function are computed and analyzed in the limit of large maps. For large distances and in the scaling regime, we recover the same universal scaling function as for general quadrangulations. We then explore the geometry of "minimal neck baby universes" (minbus), which are the outgrowths to be removed from a general quadrangulation to transform it into a quadrangulation with no multiple edges, the "mother universe". We give a number of distance-dependent characterizations of minbus, such as the two-point function inside a minbu or the law for the distance from a random point to the mother universe.