The notion of non-commutative probability space offers a convenient framework for proving interesting limit theorems where one follows the blueprint of the Central Limit Theorem (CLT), but where the arising limit law may no longer be the familiar Gaussian law. A well-known result of this kind is the CLT of free probability, where the limit law is Wigner's semicircle law. Another example of such a CLT-style limit theorem appeared in a 1995 paper of Biane, where the star-generators $(1,2), (1,3),...,(1,n),...$ of the infinite symmetric group $S_{\infty}$ were viewed as a sequence of non-commutative random variables, and where the corresponding limit law was once again found to be Wigner's semicircle law.

Wigner's law usually pops up in connection to $d \to \infty$ limits for $d \times d$ random matrix models, e.g. in the well-known 'GUE' model. One may then suspect there is a parameter $d$ to be considered in relation to the CLT for star-generators, where the above mentioned result of Biane is obtained for $d \to \infty$.

In my talk, I will present a joint work with Jacob Campbell and Claus Koestler (International J. Math 2022, also available as arXiv:2203.01763), where we fix a $d$ and some weights $w_1, \ldots, w_d$ of sum 1, and we use an expectation functional on $S_{\infty}$ that is naturally defined by this data, via a classification scheme known from the 1960's and due to Thoma. We prove a CLT-style theorem where the limit law turns out to be the law of a $d \times d$ 'traceless CCR-GUE' matrix, an analogue of the traceless GUE where the off-diagonal entries satisfy certain canonical commutation relations dictated by the weights $w_i$. The special case when all the $w_i$'s are equal to $1/d$ yields the law of a bona fide traceless GUE matrix, and we retrieve a result of Koestler-Nica from 2021, which in turn retrieves for $d \to \infty$ the 1995 result of Biane.