Distances and isoperimetric inequalities in random triangulations of high genus

TL;DR

This paper demonstrates that high-genus uniform random triangulations typically have logarithmic diameter and small pairwise distances, supported by a novel isoperimetric inequality relating perimeter and volume.

Contribution

It introduces a new isoperimetric inequality for high-genus triangulations and establishes probabilistic bounds on their diameter and distance distribution.

Findings

Diameter of order log n with high probability

Distances between most pairs differ by at most a constant

Large parts have perimeter proportional to volume

Abstract

We prove that uniform random triangulations whose genus is proportional to their size $n$ have diameter of order $\log n$ with high probability. We also show that in such triangulations, the distances between most pairs of points differ by at most an additive constant. Our main tool to prove those results is an isoperimetric inequality of independent interest: any part of the triangulation whose size is large compared to $\log n$ has a perimeter proportional to its volume.