**Involutive random walks on total orders and the anti-diagonal eigenvalue property.** / Britnell, John R.; Wildon, Mark.

Research output: Contribution to journal › Article › peer-review

Published

**Involutive random walks on total orders and the anti-diagonal eigenvalue property.** / Britnell, John R.; Wildon, Mark.

Research output: Contribution to journal › Article › peer-review

Britnell, JR & Wildon, M 2022, 'Involutive random walks on total orders and the anti-diagonal eigenvalue property', *Linear Algebra and Its Applications*, vol. 641, pp. 1-47. https://doi.org/10.1016/j.laa.2022.01.018

Britnell, J. R., & Wildon, M. (2022). Involutive random walks on total orders and the anti-diagonal eigenvalue property. *Linear Algebra and Its Applications*, *641*, 1-47. https://doi.org/10.1016/j.laa.2022.01.018

Britnell JR, Wildon M. Involutive random walks on total orders and the anti-diagonal eigenvalue property. Linear Algebra and Its Applications. 2022 May 15;641:1-47. https://doi.org/10.1016/j.laa.2022.01.018

@article{0cefc9491a8144d386dfc3ae8eb1869f,

title = "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 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.",

keywords = "Random walk, involution, eigenvector, eigenvalue, anti-diagonal eigenvalue property, binomial transform, spectrum",

author = "Britnell, {John R.} and Mark Wildon",

year = "2022",

month = may,

day = "15",

doi = "10.1016/j.laa.2022.01.018",

language = "English",

volume = "641",

pages = "1--47",

journal = "Linear Algebra and Its Applications",

issn = "0024-3795",

publisher = "Elsevier Inc.",

}

TY - JOUR

T1 - Involutive random walks on total orders and the anti-diagonal eigenvalue property

AU - Britnell, John R.

AU - Wildon, Mark

PY - 2022/5/15

Y1 - 2022/5/15

N2 - 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.

AB - 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.

KW - Random walk

KW - involution

KW - eigenvector

KW - eigenvalue

KW - anti-diagonal eigenvalue property

KW - binomial transform

KW - spectrum

U2 - 10.1016/j.laa.2022.01.018

DO - 10.1016/j.laa.2022.01.018

M3 - Article

VL - 641

SP - 1

EP - 47

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -