Paths in Weyl chambers and random matrices

• 成果类型: Article
• 署名作者: Bougerol, P; Jeulin, T
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Cite; Sorbonne Universite; Universite Paris Cite; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400200221
• 发表日期: 2002
• 页码: 517-543
• 关键词: brownian-motion; LIMIT-THEOREMS; REPRESENTATION; eigenvalues

摘要

Baryshnikov [4] and Gravner, Tracy & Widom (15) have shown that the largest eigenvalue of a random matrix of the G.U.E. of order d has the same distribution as max(1greater than or equal tot1greater than or equal to...greater than or equal totd-1greater than or equal to0) [W-1(1) - W-1(t(1)) + W-2(t(1)) - W-2(t(2)) + ... + W-d(t(d-1))], where W = (W-1,...,W-d) is a d-dimensional Brownian motion. We provide a generalization of this formula to all the eigenvalues and give a geometric interpretation. For any Weyl chamber a(+) of an Euclidean finite-dimensional space a, we define a natural continuous path transformation T which associates to a path w in a a path Tw in (a) over bar (+). This transformation occurs in the description of the asymptotic behaviour of some deterministic dynamical systems on the symmetric space G/K where G is the complex group with chamber a(+). When a = R-d, a(+) = {(x(1),..., x(d)); x(1) > x(2) >...> x(d)} and if W is the Euclidean Brownian motion on a then TW is the process of the eigenvalues of the Dyson Brownian motion on the set of Hermitian matrices and (TW)(1) is distributed as the eigenvalues of the G.U.E.