Random walks on rings and modules

Ayyer, Arvind; Steinberg, Benjamin

doi:10.5802/alco.94

Ayyer, Arvind ¹; Steinberg, Benjamin ²

¹ Department of Mathematics Indian Institute of Science Bangalore 560012 India
² Department of Mathematics City College of New York Convent Avenue at 138th Street New York, New York 10031 USA

Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 309-329.

Abstract

We consider two natural models of random walks on a module $V$ over a finite commutative ring $R$ driven simultaneously by addition of random elements in $V$ , and multiplication by random elements in $R$ . In the coin-toss walk, either one of the two operations is performed depending on the flip of a coin. In the affine walk, random elements $a \in R, b \in V$ are sampled independently, and the current state $x$ is taken to $a x + b$ . For both models, we obtain the complete spectrum of the transition matrix from the representation theory of the monoid of all affine maps on $V$ under a suitable hypothesis on the measure on $V$ (the measure on $R$ can be arbitrary).

Received: 2018-08-14
Revised: 2019-08-08
Accepted: 2019-08-09
Published online: 2020-04-01

DOI: 10.5802/alco.94

Classification: 60J10, 20M30, 13M99, 05E10, 60C05
Keywords: Random walks, rings, modules, monoids, representation theory

Author's affiliations:

Ayyer, Arvind ¹; Steinberg, Benjamin ²

¹ Department of Mathematics Indian Institute of Science Bangalore 560012 India
² Department of Mathematics City College of New York Convent Avenue at 138th Street New York, New York 10031 USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Ayyer, Arvind; Steinberg, Benjamin. Random walks on rings and modules. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 309-329. doi : 10.5802/alco.94. https://alco.centre-mersenne.org/articles/10.5802/alco.94/

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