Heat-bath random walks with Markov bases

TL;DR

This paper investigates the properties of heat-bath random walks on lattice point graphs, establishing bounds on graph diameter and conditions for rapid mixing and expansion, with implications for sampling and combinatorial optimization.

Contribution

It provides new bounds on graph diameter and explicit conditions for heat-bath random walks to be expanders, generalizing Glauber dynamics for lattice point graphs.

Findings

Diameter of graphs bounded by a constant

Heat-bath random walks can be expanders under certain conditions

Conditions for rapid mixing of the Markov chains

Abstract

Graphs on lattice points are studied whose edges come from a finite set of allowed moves of arbitrary length. We show that the diameter of these graphs on fibers of a fixed integer matrix can be bounded from above by a constant. We then study the mixing behaviour of heat-bath random walks on these graphs. We also state explicit conditions on the set of moves so that the heat-bath random walk, a generalization of the Glauber dynamics, is an expander in fixed dimension.