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

In this paper we study the asymptotic distribution of the moments of (non-normalized) traces $\text{Tr}(w_1),\text{Tr}(w_2),\mathrm{.}\mathrm{.}\mathrm{.},\text{Tr}(w_r)$, where $ w_1, w_2,

…, w_r$arereducedwordsinunitariesinthegroup$\cU(N)$. We prove that as $N\to \mathrm{\infty }$ these variables are distributed as normal gaussian variables $\sqrt{j_1}{Z}_1,\mathrm{.}\mathrm{.}\mathrm{.},\sqrt{Z_r}$, where $j_1,\mathrm{.}\mathrm{.}\mathrm{.},j_r$ are the number of cyclic rotations of the words $w_1,\mathrm{.}\mathrm{.}\mathrm{.},w_s$ leaving them invariant. This extends a previous result by Diaconis (\cite{Diac}), where this it was proved, that $\text{Tr}(U),\text{Tr}(U^2),\mathrm{.}\mathrm{.}\mathrm{.},$ $\text{Tr}(U^p)$ are asymptotically distributed as $Z_1,\sqrt{2}{Z}_2,\mathrm{.}\mathrm{.}\mathrm{.},\sqrt{p}{Z}_p$. 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}.