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

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.