Mean recurrence times of Markov chains and spanning tree invariants.

Ricardo Gómez

Linear Algebra and its Applications 433 (2010) 1714-1718.

Abstract. We show that the mean recurrence times of (countable state) irreducible and positively recurrent Markov chains are the spanning tree invariants of the first return loop systems. Then, by the Perron-Frobenius Theorem, the spanning tree invariants of the first return loop systems of a finite state Markov chain are all equal if and only if the process is doubly stochastic, settling a conjecture on a question in [GS] where it was verified for matrices of size at most three.