Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum
Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum
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DOI：
10.2307/2661373
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发表时间：
2000-09
影响因子：
4.9
通讯作者：
M. Adler;P. Moerbeke
中科院分区：
文献类型：
--
作者：
M. Adler;P. Moerbeke
给定埃尔米特系综、对称系综和辛系综，表明谱属于一个或多个区间的概率满足非线性偏微分方程。这是针对三个经典系综进行的：高斯系综、拉盖尔系综和雅可比系综。对于 Hermitian 系综，PDE（在区间的边界点）与 Toda 晶格和 KP 方程相关，而对于对称和辛系综，PDE 是一个归纳方程，与所谓的 Pfafi-KP 方程和 Pfafi 晶格相关。该方法包括在积分中插入时间变量并表明该积分满足可积晶格方程和 Virasoro 约束。
Given the Hermitian, symmetric and symplectic ensembles, it is shown that the probability that the spectrum belongs to one or several intervals satisfles 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 Pfafi-KP equation and the Pfafi lattice. The method consists of inserting time-variables in the integral and showing that this integral satisfles integrable lattice equations and Virasoro constraints.