Explicit formulae for stochastic equilibria

TL;DR

The paper addresses the problem of computing stochastic equilibria for finite-state Markov chains by solving $\pi=\pi P$ for row-stochastic $P$ and proposes analytic, broadly applicable constructions. Starting with explicit formulas for small matrices ($2\times2$, $3\times3$, $4\times4$), it shows how adjugates and principal minors yield direct expressions for the equilibrium vector in the unique (irreducible) case. It then generalizes to $n\times n$ matrices, proving that, when the equilibrium is unique, $\pi_n$ is proportional to the vector of principal minors $M_{ii}(I_n-P_n)$, with a compact adjugate-based justification, and discusses exceptional reducible cases. The method extends to random walks on graphs, where $P=D^{-1}A$ leads to an integer-minor formula involving $D-A$, providing a practical, exact route to equilibria and a clear pedagogical tool for linear algebra and stochastic processes.

Abstract

Finding the stochastic equilibria for finite-state stochastic matrices amounts to solving an eigen\-vector problem $π= πP$. Various techniques for doing so are known, some extremely computationally intensive. Herein we shall aim to extract a number of relatively simple analytic results that shed light on this problem. It is very easy to find an explicit general formula for the equilibrium vector (when it is unique) of a $2\times 2$ stochastic matrix. The corresponding explicit general formula for the equilibrium vector (when it is unique) of a $3\times 3$ stochastic matrix is a somewhat messier four-line result. (Though with a bit of work you can shoe-horn it into one line of text.) An explicit general formula for the equilibrium vector (when it is unique) of a $4\times 4$ stochastic matrix requires a paragraph of text. Ultimately, for $n\times n$ stochastic matrices a general and fully explicit construction of the equilibrium vector (when it is unique) can be developed in terms of a suitable adjugate (classical adjoint) matrix, and can subsequently be reduced to the computation of $n$ principal matrix minors. Finally, an application to random walks on graphs is presented.