Combinatorial aspects of Connes's embedding conjecture and asymptotic distribution of traces of products of unitaries

Florin Radulescu

paper

Combinatorial aspects of Connes's embedding conjecture and asymptotic distribution of traces of products of unitaries

arXiv:math/0404308

Abstract

In this paper we study the asymptotic distribution of the moments of (non-normalized) traces $\Tr (w_1), \Tr(w_2), ..., \Tr(w_r)$, where $w_{1}, w_{2}, > ..., w_{r}$ are reduced words in unitaries in the group $\cU(N)$. We prove that as $N \to \infty$ these variables are distributed as normal gaussian variables $j_{1} Z_{1}, ..., Z_{r}$ , where $j_{1}, ..., j_{r}$ are the number of cyclic rotations of the words $w_{1}, ..., w_{s}$ leaving them invariant. This extends a previous result by Diaconis (\cite{Diac}), where this it was proved, that $\Tr(U), \Tr(U^2), ...,$ $\Tr(U^p)$ are asymptotically distributed as $Z_{1}, 2 Z_{2}, ..., p Z_{p}$ . We establish a combinatorial formula for $\int |\Tr (w_1)|^2...| \Tr(w_p)|^2$. In our computation we reprove some results from \cite{BC}.

9 pages