Kemeny’s constant and the Lemoine point of a simplex

Abstract

Kemeny's constant is an invariant of discrete-time Markov chains, equal to the expected number of steps between two states sampled from the stationary distribution. It appears in applications as a concise characterization of the mixing properties of a Markov chain and has many alternative definitions. In this short article, we derive a new geometric expression for Kemeny's constant, which involves the distance between two points in a simplex associated with the Markov chain: the circumcenter and the Lemoine point. Our proof uses an expression due to Wang, Dubbeldam, and Van Mieghem of Kemeny's constant in terms of effective resistances and Fiedler's interpretation of effective resistances as edge lengths of a simplex.