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Lookup NU author(s): Dr John Britnell
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).
© 2022 Elsevier Inc. 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.
Author(s): Britnell JR, Wildon M
Publication type: Article
Publication status: Published
Journal: Linear Algebra and Its Applications
Year: 2022
Volume: 641
Pages: 1-47
Print publication date: 15/05/2022
Online publication date: 01/02/2022
Acceptance date: 29/01/2022
Date deposited: 01/02/2023
ISSN (print): 0024-3795
ISSN (electronic): 1873-1856
Publisher: Elsevier Inc.
URL: https://doi.org/10.1016/j.laa.2022.01.018
DOI: 10.1016/j.laa.2022.01.018
ePrints DOI: 10.57711/gd0z-q102
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