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

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In this paper we study the asymptotic distribution of the moments of (non-normalized) traces (w₁), (w₂),..., (wᵣ), where w₁, w₂, >..., wᵣ are reduced words in unitaries in the group (N). We prove that as N→ ∞ these variables are distributed as normal gaussian variables √ j₁ Z₁,..., √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 (Diac), where this it was proved, that (U), (U²),..., (Uᵖ) are asymptotically distributed as Z₁, √ 2 Z₂,..., √ p Zₚ. We establish a combinatorial formula for ∫ | (w₁)|²...| (wₚ)|². In our computation we reprove some results from BC.