{
"id": "1708.04223",
"version": "v1",
"published": "2017-08-14T17:39:53.000Z",
"updated": "2017-08-14T17:39:53.000Z",
"title": "Random walks on rings and modules",
"authors": [
"Arvind Ayyer",
"Benjamin Steinberg"
],
"categories": [
"math.CO",
"math.GR",
"math.PR",
"math.RA",
"math.RT"
],
"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 $ax+b$. 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).",
"revisions": [
{
"version": "v1",
"updated": "2017-08-14T17:39:53.000Z"
}
],
"analyses": {
"subjects": [
"60J10",
"20M30",
"13M99",
"05E10",
"60C05"
],
"keywords": [
"random walks",
"random elements",
"representation theory",
"transition matrix",
"natural models"
],
"note": {
"typesetting": "TeX",
"pages": 0,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}