Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum

• 成果类型: Article
• 署名作者: Adler, M; van Moerbeke, P
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2661373
• 发表日期: 2001
• 页码: 149-189
• 关键词: differential-equations; matrix; POLYNOMIALS; INTEGRALS; toda

摘要

Given the Hermitian, symmetric and symplectic ensembles, it is shown that the probability that the spectrum belongs to one or several intervals satisfies a nonlinear PDE. This is done for the three classical ensembles: Gaussian, Laguerre and Jacobi. For the Hermitian ensemble, the PDE (in the boundary points of the intervals) is related to the Toda lattice and the KP equation, whereas for the symmetric and symplectic ensembles the PDE is an inductive equation, related to the so-called Pfaff-KP equation and the Pfaff lattice. The method consists of inserting time-variables in the integral and showing that this integral satisfies integrable lattice equations and Virasoro constraints.