首页|Involutive random walks on total orders and the anti-diagonal eigenvalue property
Involutive random walks on total orders and the anti-diagonal eigenvalue property
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NSTL
Elsevier
This paper studies a family of random walks defined on the finite ordinals using their order reversing involutions. Starting at x is an element of {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. (C) 2022 Elsevier Inc. All rights reserved.
Random walkInvolutionEigenvectorEigenvalueAnti-diagonal eigenvalue propertyBinomial transformSpectrum
NSTL
Elsevier
