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 (nonnormalized) traces Tr(w1),Tr(w2), . . . ,Tr(wr), where w1, w2, . . . , wr are reduced words in unitaries in the group U(N). We prove that as N → ∞ these variables are distributed as normal gaussian variables √ j1Z1, . . . , √ Zr , where j1, . . . , jr are the number of cyclic rotations of the words w1, . . . , ws leaving them invariant. This extends a previous result by Diaconis ([4]), where this it was proved, that Tr(U),Tr(U), . . . , Tr(U) are asymptotically distributed as Z1, √ 2Z2, . . . , √ pZp. We establish a combinatorial formula for ∫ |Tr(w1)| · · · |Tr(wp)|. In our computation we reprove some results from [1].