On the ubiquity of the Cauchy distribution in spectral problems

• 成果类型: Article
• 署名作者: Aizenman, Michael; Warzel, Simone
• 署名单位: Princeton University; Princeton University; Technical University of Munich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0587-3
• 发表日期: 2015
• 页码: 61-87
• 关键词: bulk universality conjecture; wave chaos; systems; statistics; shifts

摘要

We consider the distribution of the values at real points of random functions which belong to the Herglotz-Pick (HP) class of analytic mappings of the upper half plane into itself. It is shown that under mild stationarity assumptions the individual values of HP functions with singular spectra have a Cauchy type distribution. The statement applies to the diagonal matrix elements of random operators, and holds regardless of the presence or not of level repulsion, i.e. applies to both random matrix and Poisson-type spectra.