Distance statistics in large toroidal maps

TL;DR

This paper derives universal scaling functions for distance statistics in large bipartite toroidal quadrangulations, including distributions of shortest non-contractible loops and distances between random points, using combinatorial map coding techniques.

Contribution

It provides explicit formulas for distance distributions in large genus-one maps using well-labeled 1-tree representations, a novel combinatorial approach.

Findings

Explicit probability distribution for shortest non-contractible loop length

Distribution for distance between two random points in large maps

Scaling limits expressed via well-labeled 1-tree generating functions

Abstract

We compute a number of distance-dependent universal scaling functions characterizing the distance statistics of large maps of genus one. In particular, we obtain explicitly the probability distribution for the length of the shortest non-contractible loop passing via a random point in the map, and that for the distance between two random points. Our results are derived in the context of bipartite toroidal quadrangulations, using their coding by well-labeled 1-trees, which are maps of genus one with a single face and appropriate integer vertex labels. Within this framework, the distributions above are simply obtained as scaling limits of appropriate generating functions for well-labeled 1-trees, all expressible in terms of a small number of basic scaling functions for well-labeled plane trees.