A survey of results on random random walks on finite groups

Abstract

A number of papers have examined various aspects of "random random walks" on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss some of the arguments and results in this work. This article also provides a number of exercises. Some exercises involve straightforward computations; others involve proving details in proofs or extending results proved in the article. This article also describes some problems for further study.

Some things we’ve learned (about Markov chain Monte Carlo)

“….. In a series of extensions, Martin Hildebrand [37,39] has shown this result is quite robust to variations: p need not be prime, the probability distribution of ε i can be fairly general, the multiplier 2 can be replaced by a general a and even a n+1 chosen randomly (e.g., 2 or 1/2 (mod p) with probability 1/2). The details vary and the arguments require new ideas.…”

Random Multiplicative Walks on the Residues Modulo

“…Inserting this back into (9) in place of the integral, and reindexing so that the summation runs over the exponent on the logarithm we get that the main term is…”

“…The problem of studying random walks on groups has been studied extensively, see for example [1], [9]. Many common random walks can in fact be viewed as walks on groups, for example the walk on the infinite 1 line (Z) or on a cycle (Z/nZ under addition).…”

The cutoff phenomenon for random birth and death chains

“…That is, do typical sequences of Markov chains exhibit cutoff? There has been a great deal of work on this question for random walks on groups, going back to work of Dou, Hildebrand and Wilson [11] [28] and described in the survey paper [14]. More recently, this has progressed to other random walks with uniform stationary distribution [20] [9].…”