Random walks on rings and modules
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 309-329.

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 aR,bV are sampled independently, and the current state x is taken to ax+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).

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DOI: 10.5802/alco.94
Classification: 60J10, 20M30, 13M99, 05E10, 60C05
Keywords: Random walks, rings, modules, monoids, representation theory
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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