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

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 $\sqrt{j_1}{Z}_1,...,\sqrt{Z_r}$, 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 $Z_1,\sqrt{2}{Z}_2,...,\sqrt{p}{Z}_p$. 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

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