Bulk asymptotics of the Gaussian $\beta$-ensemble characteristic polynomial

TL;DR

This paper provides a detailed asymptotic analysis of the Gaussian β-ensemble's characteristic polynomial, capturing both local fluctuations and global log-correlated structures, with implications for stochastic zeta functions and CLTs.

Contribution

It offers a comprehensive asymptotic description of the GβE characteristic polynomial in the bulk, unifying local and global behaviors with new corollaries and corrections.

Findings

Convergence of ratios to the stochastic zeta function

Martingale approximation of the log-characteristic polynomial

Description of order one correction via stochastic Airy function

Abstract

The Gaussian $\beta$-ensemble (G$\beta$E) is a fundamental model in random matrix theory. In this paper, we provide a comprehensive asymptotic description of the characteristic polynomial of the G$\beta$E anywhere in the bulk of the spectrum that simultaneously captures both local-scale fluctuations (governed by the Sine-$\beta$ point process) and global/mesoscopic log-correlated Gaussian structure, which is accurate down to vanishing errors as $N\to\infty$. As immediate corollaries, we obtain several important results: (1) convergence of characteristic polynomial ratios to the stochastic zeta function, extending known results from Valko and Virag to the G$\beta$E; (2) a martingale approximation of the log-characteristic polynomial which immediately recovers the central limit theorem from Bourgade, Mody and Pain; (3) a description of the order one correction to the martingale in terms of the stochastic Airy function.