Quenched universality for deformed Wigner matrices

• 成果类型: Article
• 署名作者: Cipolloni, Giorgio; Erdos, Laszlo; Schroeder, Dominik
• 署名单位: Institute of Science & Technology - Austria; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01156-7
• 发表日期: 2023
• 页码: 1183-1218
• 关键词: fixed-energy universality; local spectral statistics; generalized wigner; spacing distribution; edge; eigenvalues; ensembles; bulk

摘要

Following E. Wigner's original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix H yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices H + x A with a deterministic Hermitian matrix A and a fixed Wigner matrix H, just using the randomness of a single scalar real random variable x. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble.