Explicit formulae for stochastic equilibria

Finding the stochastic equilibria for finite-state stochastic matrices amounts to solving an eigen\-vector problem $\pi = \pi P$. 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.