Uniform mixing time and bottlenecks in uniform finite quadrangulations

TL;DR

This paper establishes bounds on bottleneck sizes in uniform quadrangulations and uses these to derive upper bounds on the mixing time of lazy random walks on these structures and their duals, involving explicit Laplace transform computations.

Contribution

It provides the first simultaneous scale lower bounds on bottlenecks and upper bounds on mixing times in uniform quadrangulations, advancing understanding of their geometric and probabilistic properties.

Findings

Lower bounds on bottleneck sizes at all scales

Upper bounds on mixing times for random walks on quadrangulations

Explicit Laplace transform calculations for face counts in truncated hulls

Abstract

We prove a lower bound on the size of bottlenecks in uniform quadrangulations, valid at all scales simultaneously. We use it to establish upper bounds on the uniform mixing time of the lazy random walk on uniform quadrangulations, as well as on their dual. The proofs involve an explicit computation of the Laplace transform of the number of faces in truncated hulls of the uniform infinite plane quadrangulation.