高斯 $\beta$-ensemble特征多项式的整体渐近分析
摘要
The Gaussian -ensemble (GE) is a fundamental model in random matrix theory. In this paper, we provide a comprehensive asymptotic description of the characteristic polynomial of the GE anywhere in the bulk of the spectrum that simultaneously captures both local-scale fluctuations (governed by the Sine- point process) and global/mesoscopic log-correlated Gaussian structure, which is accurate down to vanishing errors as . 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 GE; (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.
引用
@article{arxiv.2508.01458,
title = {Bulk asymptotics of the Gaussian $\beta$-ensemble characteristic polynomial},
author = {Gaultier Lambert and Elliot Paquette},
journal= {arXiv preprint arXiv:2508.01458},
year = {2025}
}