A Central Limit Theorem for Star-Generators of ${S}_{\infty }$, Which Relates to the Law of a GUE Matrix

Abstract

It is well known that, on a purely algebraic level, a simplified version of the central limit theorem (CLT) can be proved in the framework of a non-commutative probability space, under the hypotheses that the sequence of non-commutative random variables we consider is exchangeable and obeys a certain vanishing condition of some of its joint moments. In this approach (which covers versions for both the classical CLT and the CLT of free probability), the determination of the resulting limit law has to be addressed on a case-by-case basis. In this paper we discuss an instance of the above theorem that takes place in the framework of the group algebra ${{\mathbb {C}}}[ S_{\infty } ]$ of the infinite symmetric group: The exchangeable sequence is provided by the star-generators of $S_{\infty }$, and the expectation functional used on ${{\mathbb {C}}}[ S_{\infty } ]$ depends in a natural way on a parameter $d \in {{\mathbb {N}}}$. We identify precisely the limit distribution $\mu _d$ for this special instance of CLT, via a connection that $\mu _d$ turns out to have with the average empirical eigenvalue distribution of a random $d \times d$ GUE matrix. Moreover, we put into evidence a multivariate version of this result which follows from the observation that, on the level of calculations with pair-partitions, the (non-centered) star-generators are related to a (centered) exchangeable sequence of GUE matrices with independent entries.