Polynomials Associated with Finite Markov Chains

TL;DR

This paper explores algebraic properties of polynomials derived from finite Markov chains, focusing on minors of lifted transition matrices and proposing a conjecture about their factorization.

Contribution

It introduces a novel analysis of minors of lifted transition matrices and conjectures a general factorization property for these polynomials.

Findings

Identified a simple case with a clear factorization of minors

Proposed a conjecture for a general factorization pattern

Provided algebraic insights into Markov chain liftings

Abstract

Given a finite Markov chain, we investigate the first minors of the transition matrix of a lifting of this Markov chain to covering trees. In a simple case we exhibit a nice factorisation of these minors, and we conjecture that it holds more generally.