Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop

TL;DR

This paper derives explicit generating functions for quadrangulations with boundaries, analyzing their scaling limits and universal behaviors, including cases with self-avoiding loops and different boundary regimes.

Contribution

It provides new explicit formulas for distance statistics in quadrangulations with boundaries and introduces universal scaling functions for various boundary regimes and loop configurations.

Findings

Explicit generating functions for boundary and bulk distances.

Identification of universal scaling functions in different regimes.

Discovery of new scaling functions for self-avoiding loops.

Abstract

We consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual distance in the map. For large maps, this yields explicit formulas for the bulk-boundary and boundary-boundary correlators in the various encountered scaling regimes: a small boundary, a dense boundary and a critical boundary regime. The critical boundary regime is characterized by a one-parameter family of scaling functions interpolating between the Brownian map and the Brownian Continuum Random Tree. We discuss the cases of both generic and self-avoiding boundaries, which are shown to share the same universal scaling limit. We finally address the question of the bulk-loop distance statistics in the context of planar quadrangulations equipped with a self-avoiding loop. Here again, a new family of scaling functions describing critical loops is discovered.