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

TL;DR

This paper investigates the asymptotic distribution of traces of products of unitaries, showing they tend to Gaussian variables and providing combinatorial formulas for trace moments, extending previous results in the field.

Contribution

It introduces a new combinatorial approach to analyze the asymptotic distribution of traces of words in unitaries, generalizing earlier findings by Diaconis.

Findings

Traces of products of unitaries become Gaussian in the limit

Provides a combinatorial formula for trace moment integrals

Extends Diaconis's asymptotic trace distribution results

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}.