Dean David S, Majumdar Satya N
Laboratoire de Physique Théorique (UMR 5152 du CNRS), Université Paul Sabatier, 118, Route de Narbonne, Toulouse Cedex 4, France.
Phys Rev Lett. 2006 Oct 20;97(16):160201. doi: 10.1103/PhysRevLett.97.160201.
We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N x N) random matrix are positive (negative) decreases for large N as approximately exp[-betatheta(0)N2] where the parameter beta characterizes the ensemble and the exponent theta(0)=(ln3)/4=0.274 653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number zeta, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at zeta.
我们通过解析计算属于高斯正交系综、酉系综和辛系综的随机矩阵最大（最小）特征值与其均值的大偏差概率。特别地，我们表明对于大的(N)，一个((N×N))随机矩阵所有特征值为正（负）的概率近似以(\exp[-\beta\theta(0)N^2])的形式减小，其中参数(\beta)表征系综，指数(\theta(0)=(\ln{3})/4 = 0.274653\cdots)是通用的。我们还精确计算了特征值被限制为大于固定数(\zeta)的矩阵中的态密度平均值，从而推广了著名的维格纳半圆律。态密度通常在(\zeta)处呈现平方根倒数奇点。
