Tree formulas, mean first passage times and Kemeny’s constant of a Markov chain

Abstract

This paper offers some probabilistic and combinatorial insights into tree formulas for the Green function and hitting probabilities of Markov chains on a finite state space. These tree formulas are closely related to loop-erased random walks by Wilson’s algorithm for random spanning trees, and to mixing times by the Markov chain tree theorem. Let mij be the mean first passage time from i to j for an irreducible chain with finite state space S and transition matrix (pij; i, j ? S). It is well known that mjj = 1/pj = (1)/j, where p is the stationary distribution for the chain, j is the tree sum, over nn-2 trees t spanning S with root j and edges i ? k directed towards j, of the tree product i?k?t pik, and (1) := j?S j. Chebotarev and Agaev (Linear Algebra Appl. 356 (2002) 253-274) derived further results from Kirchhoff’s matrix tree theorem. We deduce that for i = j, mij = ij /j, where ij is the sum over the same set of nn-2 spanning trees of the same tree product as for j, except that in each product the factor pkj is omitted where k = k(i, j, t) is the last state before j in the path from i to j in t. It follows that Kemeny’s constant j?S mij /mjj equals (2)/(1), where (r) is the sum, over all forests f labeled by S with r directed trees, of the product of pij over edges i ? j of f. We show that these results can be derived without appeal to the matrix tree theorem. A list of relevant literature is also reviewed.