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).

Received: 2018-08-14
Revised: 2019-08-08
Accepted: 2019-08-09
Published online: 2020-04-01
DOI: https://doi.org/10.5802/alco.94
Classification: 60J10,  20M30,  13M99,  05E10,  60C05
Keywords: Random walks, rings, modules, monoids, representation theory
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     author = {Ayyer, Arvind and Steinberg, Benjamin},
     title = {Random walks on rings and modules},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {2},
     year = {2020},
     pages = {309-329},
     doi = {10.5802/alco.94},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_2_309_0/}
}
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/item/ALCO_2020__3_2_309_0/

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