Random walks on the BMW monoid: an algebraic approach

TL;DR

This paper introduces an algebraic framework using representation theory to analyze random walks on BMW monoid basis elements, providing insights into their convergence behavior.

Contribution

It develops a novel algebraic approach employing trace functions and representation theory to study the mixing times of random walks on BMW monoid elements.

Findings

Established a trace-based norm for analyzing convergence

Connected random walks to algebraic multiplication operators

Provided tools for studying mixing times in BMW algebra context

Abstract

We consider Metropolis-based systematic scan algorithms for generating Birman-Murakami-Wenzl (BMW) monoid basis elements of the BMW algebra. As the BMW monoid consists of tangle diagrams, these scanning strategies can be rephrased as random walks on links and tangles. We translate these walks into left multiplication operators in the corresponding BMW algebra. Taking this algebraic perspective enables the use of tools from representation theory to analyze the walks; in particular, we develop a norm arising from a trace function on the BMW algebra to analyze the time to stationarity of the walks.