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

This paper studies a family of random walks defined on the finite ordinals using their order reversing involutions. Starting at x in {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 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.