Combinatorial considerations on the invariant measure of a stochastic matrix

TL;DR

This paper provides a simple combinatorial proof of the Markov tree theorem, which explicitly represents the invariant measure of a stochastic matrix, with simplified cases under detailed balance.

Contribution

It introduces a straightforward combinatorial proof of the Markov tree theorem, enhancing understanding of invariant measures in stochastic matrices.

Findings

Explicit combinatorial proof of the Markov tree theorem

Simplified proof in the detailed balance case

Enhanced understanding of invariant measures in Markov processes

Abstract

The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit representation of the invariant measure of a stochastic matrix. In this note, we given a simple and purely combinatorial proof of the Markov tree theorem. In the symmetric case of detailed balance, the statement and the proof simplifies even more.