Involutive random walks on total orders and the anti-diagonal eigenvalue property

Abstract

This paper studies a family of random walks defined on the finite ordinals using their order reversing involutions. Starting at x∈{0,1,…,n−1}, an element y⩽x is chosen according to a prescribed probability distribution, and the walk then steps to n−1−y. We show that under very mild assumptions these walks are irreducible, recurrent and ergodic. We then find the invariant distributions, eigenvalues and eigenvectors of a distinguished subfamily of walks whose transition matrices have the global anti-diagonal eigenvalue property studied in earlier work by Ochiai, Sasada, Shirai and Tsuboi. We prove that this subfamily of walks is characterised by their reversibility. As a corollary, we obtain the invariant distributions and rate of convergence of the random walk on the set of subsets of {1,…,m} in which steps are taken alternately to subsets and supersets, each chosen equiprobably. We then consider analogously defined random walks on the real interval [0,1] and use techniques from the theory of self-adjoint compact operators on Hilbert spaces to prove analogues of the main results in the discrete case.