Statistics of geodesics in large quadrangulations

TL;DR

This paper analyzes the statistical behavior of shortest paths in large random quadrangulations, revealing properties of geodesics and confluent paths, with explicit enumeration and scaling limits.

Contribution

It extends Schaeffer's bijection to quadrangulations with marked geodesics, providing generating functions and analyzing geodesic confluence and exceptional points.

Findings

Average number of geodesics between two points is 3*2^i

Confluent geodesics tend to stay close with many contacts

Exceptional points linked by truly distinct geodesics

Abstract

We study the statistical properties of geodesics, i.e. paths of minimal length, in large random planar quadrangulations. We extend Schaeffer's well-labeled tree bijection to the case of quadrangulations with a marked geodesic, leading to the notion of "spine trees", amenable to a direct enumeration. We obtain the generating functions for quadrangulations with a marked geodesic of fixed length, as well as with a set of "confluent geodesics", i.e. a collection of non-intersecting minimal paths connecting two given points. In the limit of quadrangulations with a large area n, we find in particular an average number 3*2^i of geodesics between two fixed points at distance i>>1 from each other. We show that, for generic endpoints, two confluent geodesics remain close to each other and have an extensive number of contacts. This property fails for a few "exceptional" endpoints which can be linked by truly distinct geodesics. Results are presented both in the case of finite length i and in the scaling limit i ~ n^(1/4). In particular, we give the scaling distribution of the exceptional points.