ePrints

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

Lookup NU author(s): Dr John Britnell

Downloads

• Accepted version [.pdf]

Licence

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).

Abstract

© 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.

Publication metadata

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

Altmetrics

Share