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Moments of C$β$E field partition function, $\mathsf{Sine}_β$ correlations and stochastic zeta

Theodoros Assiotis, Joseph Najnudel

TL;DR

This work resolves the Fyodorov–Keating prediction for supercritical moments of the Circular $β$-ensemble by linking them to the Hua-Pickrell stochastic zeta function $\boldsymbol{\xi}^{β,δ}$, obtained as a scaling limit of normalized characteristic polynomials through a novel coupling with circular Jacobi beta ensembles. It also provides the first complete, all-order expression for the $m$-point correlations of the universal $\mathsf{Sine}_β$ process for every $β>0$, expressed via expectations of powers of the Hua-Pickrell zeta and a universal constant. The primary machinery is a unified probabilistic framework built on random orthogonal polynomials on the unit circle, enabling a sharp, uniform moment bound that drives both the moment asymptotics and the correlation formulae. These results deepen the connection between random matrix theory, Gaussian multiplicative chaos, and stochastic zeta functions, with potential implications for extreme value problems and zeta-function universality.

Abstract

We prove a conjecture of Fyodorov and Keating on the supercritical moments of the partition function of the C$β$E field or equivalently the supercritical moments of moments of the characteristic polynomial of the C$β$E ensemble for general $β>0$ and general real moment exponents. Moreover, we give the first expression for all correlation functions of the $\mathsf{Sine}_β$ point process for all $β>0$. The main object behind both results is the Hua-Pickrell stochastic zeta function introduced by Li and Valkó.

Moments of C$β$E field partition function, $\mathsf{Sine}_β$ correlations and stochastic zeta

TL;DR

This work resolves the Fyodorov–Keating prediction for supercritical moments of the Circular -ensemble by linking them to the Hua-Pickrell stochastic zeta function , obtained as a scaling limit of normalized characteristic polynomials through a novel coupling with circular Jacobi beta ensembles. It also provides the first complete, all-order expression for the -point correlations of the universal process for every , expressed via expectations of powers of the Hua-Pickrell zeta and a universal constant. The primary machinery is a unified probabilistic framework built on random orthogonal polynomials on the unit circle, enabling a sharp, uniform moment bound that drives both the moment asymptotics and the correlation formulae. These results deepen the connection between random matrix theory, Gaussian multiplicative chaos, and stochastic zeta functions, with potential implications for extreme value problems and zeta-function universality.

Abstract

We prove a conjecture of Fyodorov and Keating on the supercritical moments of the partition function of the CE field or equivalently the supercritical moments of moments of the characteristic polynomial of the CE ensemble for general and general real moment exponents. Moreover, we give the first expression for all correlation functions of the point process for all . The main object behind both results is the Hua-Pickrell stochastic zeta function introduced by Li and Valkó.
Paper Structure (7 sections, 23 theorems, 175 equations)

This paper contains 7 sections, 23 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Moments of the partition function of the C$\beta$E field
  3. $\mathsf{Sine}_{\beta}$ correlations
  4. Main results
  5. On the proof
  6. Proofs of main results
  7. Proof of the main bound

Key Result

Theorem 1.7

Let $k\in \mathbb{N}, s\in \mathbb{R}_+$, $\beta >0$ such that $2ks^2>\beta$. Then, as $N \to \infty$, where $\mathcal{F}^{(\beta)}_{k;s}$ is given explicitly The same result holds for $k \in \mathbb{R}_{\ge 1}$ whenever both $2s^2(2k-\lceil k \rceil)>\beta$ and, for $k>1$, $4\lceil k-1\rceil(k-\lceil k-1\rceil)s^2>\beta$.

Theorems & Definitions (52)

  • Definition 1.1
  • Conjecture 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Definition 2.1
  • ...and 42 more