Fluctuations of Wigner-type random matrices associated with symmetric spaces of class DIII and CI

TL;DR

This paper develops a mathematical framework to analyze the fluctuations of Wigner-type random matrices associated with symmetric spaces of classes DIII and CI, providing explicit central limit theorems for these cases.

Contribution

It introduces a calculus of patterns to control asymptotic contributions of non-crossing pair partitions, extending previous work to new symmetric space classes.

Findings

Derived explicit CLTs for classes DIII and CI

Controlled asymptotic behavior of non-crossing pair partitions

Extended the framework for Wigner matrices with symmetries

Abstract

Wigner-type randomizations of the tangent spaces of classical symmetric spaces can be thought of as ordinary Wigner matrices on which additional symmetries have been imposed. In particular, they fall within the scope of a framework, due to Schenker and Schulz-Baldes, for the study of fluctuations of Wigner matrices with additional dependencies among their entries. In this contribution, we complement the results of these authors in that we develop a calculus of patterns which makes it possible to control the asymptotic contributions of dihedral non-crossing pair partitions for the Cartan classes DIII and CI, thus obtaining explicit CLTs for these cases.