Expanders from Markov bases

TL;DR

This paper introduces a method to construct expander graphs from Markov bases, enabling rapid mixing of Markov chains in large state spaces, which was previously limited by slow mixing times.

Contribution

It presents a novel approach to create expanders from Markov bases, improving mixing times for large state space Markov chains derived from algebraic methods.

Findings

Constructed expanders from Markov bases for faster mixing

Demonstrated rapid mixing in large state spaces

Enhanced algebraic Markov chain methods

Abstract

Diaconis and Sturmfels introduced an influential method to construct Markov chains using commutative algebra. One major point of their method is that infinite families of graphs are simultaneously proved to be connected by a single algebraic calculation. For large state spaces in the infinite families these Markov chains are not rapidly mixing and only ad hoc methods have been available to improve their mixing times. We provide a method to get rapid mixing by constructing expanders for the Diaconis-Sturmfels type Markov chains.