Quasimodular Asymptotics of Spherical Integrals

TL;DR

This paper demonstrates that the spherical integral of the Circular Unitary Ensemble converges to Euler's generating function for partitions, with subsequent corrections characterized by quasimodular forms, revealing deep connections between random matrix theory and number theory.

Contribution

It establishes the quasimodular nature of subleading corrections to the spherical integral's asymptotics in high dimensions, linking random matrix integrals to modular forms.

Findings

Convergence of the spherical integral to Euler's generating function.

Identification of subleading corrections as quasimodular forms.

Insight into the structure of high-dimensional random matrix integrals.

Abstract

We show that the spherical integral of the Circular Unitary Ensemble converges in expectation to Euler's generating function for integer partitions, and that subleading corrections to this high-dimensional limit are quasimodular forms.