Random Hermitian matrices and Gaussian multiplicative chaos

• 成果类型: Article
• 署名作者: Berestycki, Nathanael; Webb, Christian; Wong, Mo Dick
• 署名单位: University of Cambridge; Aalto University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0806-9
• 发表日期: 2018
• 页码: 103-189
• 关键词: orthogonal polynomials; strong asymptotics; unitary ensemble; QUANTUM-GRAVITY; random surfaces; determinants; toeplitz; maximum; hankel; limit

摘要

We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called L2-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher-Hartwig singularities. Using Riemann-Hilbert methods, we prove a rather general Fisher-Hartwig formula for one-cut regular unitary invariant ensembles.