DIMENSION RESULTS FOR THE SPECTRAL MEASURE OF THE CIRCULAR β ENSEMBLES

• 成果类型: Article
• 署名作者: Alberts, Tom; Normand, Raoul
• 署名单位: Utah System of Higher Education; University of Utah; New York University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1798
• 发表日期: 2022
• 页码: 4642-4680
• 关键词: additive set functions; large deviations; SUM-RULES; matrices; conjecture; PROOF; cmv

摘要

We study the dimension properties of the spectral measure of the circular beta-ensembles. For beta >= 2 it was previously shown by Simon that the spec-tral measure is almost surely singular continuous with respect to Lebesgue measure on partial differential D and the dimension of its support is 1 - 2/beta. We reprove this result with a combination of probabilistic techniques and the so-called Jitomirskaya-Last inequalities. Our method is simpler in nature and mostly self-contained, with an emphasis on the probabilistic aspects rather than the analytic. We also extend the method to prove a large deviations principle for norms involved in the Jitomirskaya-Last analysis.