Most Words are Geometrically Almost Uniform

TL;DR

This paper proves that for most words in multiple letters, evaluating them on large finite simple groups results in distributions that are nearly uniform, confirming a conjecture about their asymptotic behavior.

Contribution

It establishes that the proportion of words producing nearly uniform distributions approaches 1 as the group size increases, resolving a previously open question.

Findings

Most words yield almost uniform distributions on large finite simple groups.

The proportion of such words approaches 100% as the group order tends to infinity.

Confirms the conjecture that almost all words are geometrically almost uniform.

Abstract

If w is a word in d>1 letters and G is a finite group, evaluation of w on a uniformly randomly chosen d-tuple in G gives a random variable with values in G, which may or may not be uniform. It is known that if G ranges over finite simple groups of given root system and characteristic, a positive proportion of words w give a distribution which approaches uniformity in the limit as |G| goes to infinity. In this paper, we show that the proportion is in fact 1.