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Quasimodular Asymptotics of Spherical Integrals

Jonathan Novak

TL;DR

This work analyzes the spherical integral $K_N(q,A,B)$ of the Circular Unitary Ensemble by viewing it as a spectral observable, proving that its expected value tends to Euler's generating function for partitions $H_1(q)=\prod_{n\ge1}(1-q^n)^{-1}$ for $|q|<e^{-1}$. It develops a full $N\to\infty$ asymptotic expansion of the principal log $L_N(q)=\log K_N(q)$ whose subleading terms are quasimodular forms, with an explicit first correction expressed in Eisenstein series. The approach connects random matrix theory to monotone Hurwitz numbers and Hurwitz theory on elliptic curves, providing both a stable (genus-$g$) expansion via $F_g^d$ and a concentration framework for normalized integrals. A dual interpretation emerges: the asymptotics describe an annealed partition function over partitions and covers, linking spectral observables to topological counting problems and modular object structures. The results establish a concrete bridge between high-dimensional random matrices and number-theoretic objects, highlighting the emergence of modular and quasimodular phenomena in large-N limits.

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.

Quasimodular Asymptotics of Spherical Integrals

TL;DR

This work analyzes the spherical integral of the Circular Unitary Ensemble by viewing it as a spectral observable, proving that its expected value tends to Euler's generating function for partitions for . It develops a full asymptotic expansion of the principal log whose subleading terms are quasimodular forms, with an explicit first correction expressed in Eisenstein series. The approach connects random matrix theory to monotone Hurwitz numbers and Hurwitz theory on elliptic curves, providing both a stable (genus-) expansion via and a concentration framework for normalized integrals. A dual interpretation emerges: the asymptotics describe an annealed partition function over partitions and covers, linking spectral observables to topological counting problems and modular object structures. The results establish a concrete bridge between high-dimensional random matrices and number-theoretic objects, highlighting the emergence of modular and quasimodular phenomena in large-N limits.

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.

Paper Structure

This paper contains 14 sections, 15 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. First Order Asymptotics
  3. Convergence of derivatives
  4. Uniform boundedness
  5. Uniform convergence
  6. Stability and Concentration
  7. Stable derivatives
  8. Stable limits
  9. Hurwitz theory
  10. Concentration inequality
  11. Higher Order Asymptotics
  12. Convergence of derivatives
  13. Uniform boundedness
  14. Uniform convergence

Key Result

Theorem 1.1

For any $\varepsilon \in [0,e^{-1})$ and any $m \in \mathbb{N}$ we have where $F_g(q) = \sum_{d=1}^\infty \frac{q^d}{d!} F_g^d$ is an analytic function on $\mathbb{D}$ and $F_g^d$ counts configurations $(\pi_1,\pi_2,\tau_1,\dots,\tau_{2g-2})$ of permutations from $\mathrm{S}^d=\operatorname{Aut}\{1,\dots,d\}$ such that:

Theorems & Definitions (24)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2: Hunter
  • Theorem 3.3
  • proof
  • ...and 14 more