From Gumbel to Tracy-Widom

• 成果类型: Article
• 署名作者: Johansson, K.
• 署名单位: Royal Institute of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-006-0012-7
• 发表日期: 2007
• 页码: 75-112
• 关键词: wigner random matrices; brownian-motion model; Poisson; UNIVERSALITY; statistics; distributions; ensembles; GROWTH; dyson; edge

摘要

The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution exp(- exp(-x)), the Gumbel distribution, and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density exp(-x) and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, M = M-0 + root 2SV, with Mo diagonal with independent elements and V from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.