On the mixing time of the flip walk on triangulations of the sphere

TL;DR

This paper establishes a lower bound of n^{5/4} on the mixing time for a Markov chain that samples uniform sphere triangulations via edge flips, providing insights into the efficiency of this sampling method.

Contribution

It offers the first non-trivial lower bound on the mixing time of the flip walk for sphere triangulations, advancing understanding of Markov chain convergence in geometric combinatorics.

Findings

Lower bound of n^{5/4} on mixing time

Highlights slow mixing behavior for large n

Provides theoretical foundation for sampling algorithms

Abstract

A simple way to sample a uniform triangulation of the sphere with a fixed number $n$ of vertices is a Monte-Carlo method: we start from an arbitrary triangulation and flip repeatedly a uniformly chosen edge. We give a lower bound in $n^{5/4}$ on the mixing time of this Markov chain.