Central limit theorems for Coxeter systems and Artin systems of extra large type

Abstract

Given a Coxeter system of large type we prove a non--commutative central limit theorem: After normalisation with the square root of n the characteristic function of the set of the first n generators tends in distribution to Wigners semi--circle law. If one chooses the group presentation at random, now admitting a certain amount of commutation among the generators, the distributional limit will be the q--deformed semi--circle law, $1\leq q \leq 1$, almost surely.