The law of large numbers for the maximum of almost Gaussian log-correlated fields coming from random matrices

• 成果类型: Article
• 署名作者: Lambert, Gaultier; Paquette, Elliot
• 署名单位: University of Zurich; University System of Ohio; Ohio State University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0832-2
• 发表日期: 2019
• 页码: 157-209
• 关键词: universality

摘要

We compute the leading asymptotics as N of the maximum of the field QN(q)=log|q-AN|,qC, for any unitarily invariant Hermitian random matrix AN associated to a non-critical real-analytic potential. Hence, we verify the leading order in a conjecture ofFyodorov and Simm (Nonlinearity 29:2837, 2016. arXiv:1503.07110 [math-ph]) formulated for the GUE. The method relies on a classical upper-bound and a more sophisticated lower-bound based on a variant of the second-moment method which exploits the hyperbolic branching structure of the field QN(q),qH. Specifically, we compare QN to an idealized Gaussian field by means of exponential moments. In principle, this method could also be applied to random fields coming from other point processes provided that one can compute certain mixed exponential moments. For unitarily invariant ensembles, we show that these assumptions follow from the Fyodorov-Strahov formula(Fyodorov and Strahov in J Phys A 36(12):3203-3213, 2003. 10.1088/0305-4470/36/12/320) and asymptotics of orthogonal polynomials derived inDeift et al. (Commun Pure Appl Math 52(11):1335-1425, 1999. 10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1).