Rapid mixing and Markov bases

TL;DR

This paper investigates the mixing times of Markov chain random walks on lattice points of polytopes, revealing limitations of fixed Markov bases and proposing an adaptive approach to improve mixing efficiency.

Contribution

It demonstrates that fixed Markov bases do not ensure rapid mixing under dilation and introduces a method to adapt Markov bases for faster mixing.

Findings

Fixed Markov bases do not mix rapidly under dilation

Adding more moves does not improve mixing asymptotically

Adaptive Markov bases can achieve faster mixing

Abstract

The mixing behaviour of random walks on lattice points of polytopes using Markov bases is examined. It is shown that under a dilation of the underlying polytope, these random walks do not mix rapidly when a fixed Markov basis is used. We also show that this phenomenon does not disappear after adding more moves to the Markov basis. Avoiding rejections by sampling applicable moves does also not lead to an asymptotic improvement. As a way out, a method of how to adapt Markov bases in order to achieve the fastest mixing behaviour is introduced.