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(U2), . . . , Tr(Up) are asymptotically distributed as Z1,√2Z2, . . . ,√pZp. We establish a combinatorial formula for R |Tr(w1)|2 · · · |Tr(wp)|2. In our computation we reprove some results from [1].