Random walks on a finite group and the Frobenius-Schur theorem

2023/07/09 by Olexandr Vyshnevetskiy, Vyshnevetskiy, Olexandr, Alexander Bendikov +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #60B15 (Primary) 20C15 (Secondary) #DNA and Biological Computing #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Probability (math.PR) #Representation Theory (math.RT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2307.04164

openalex publication_date 2023/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider random walk on a finite group G as follows. We can consider G as a group of substitutions. Randomly (i.e. with probability U(g)=|G|^-1 ) we choose a substitution g ∈ G and execute it twice in a row, i.e. execute a substitution g² ∈ G . Then the set of squares of elements of the group G be a carrier of a probability P(g)=(r(g))/(|G|) (g ∈ G) , where r(g) is a number of elements h ∈ G such that h² = g . Using well-known Frobenius-Schur theorem we find speed of convergence of n-fold convolution of P to the uniform probability U and conditions for the convergence.

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