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摘要翻译：
我们计算了属于高斯正交系综、酉系综和辛系综的随机矩阵的最大（最小）特征值出现大偏差的概率的精确渐近结果。特别地，我们证明了当N大时，NxN随机矩阵的所有特征值为正（负）的概率减小，其中Dyson指数β表征系综，指数θ(0)=(ln3)/4=0.274653..是普遍的。通过计算特征值位于区间[\zeta_1,\zeta_2]内的概率，可以计算最小和最大特征值的联合概率分布。作为副产品，我们还精确地得到了特征值限制在[zeta_1,zeta_2]区间内的高斯系综的平均态密度，从而将著名的Wigner半圆定律推广到这些限制系综。研究发现，在势垒位置，态密度一般呈现一个逆平方根奇异性。数值模拟证实了这些结果。
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英文标题：
《Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices》
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作者：
David S. Dean and Satya N. Majumdar
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最新提交年份：
2008
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分类信息：
一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
We compute exact asymptotic results for the probability of the occurrence of large deviations 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 (NxN) random matrix are positive (negative) decreases for large N as ~\exp[-\beta \theta(0) N^2] where the Dyson index \beta characterizes the ensemble and the exponent \theta(0)=(\ln 3)/4=0.274653... is universal. We compute the probability that the eigenvalues lie in the interval [\zeta_1,\zeta_2] which allows us to calculate the joint probability distribution of the minimum and the maximum eigenvalue. As a byproduct, we also obtain exactly the average density of states in Gaussian ensembles whose eigenvalues are restricted to lie in the interval [\zeta_1,\zeta_2], thus generalizing the celebrated Wigner semi-circle law to these restricted ensembles. It is found that the density of states generically exhibits an inverse square-root singularity at the location of the barriers. These results are confirmed by numerical simulations.
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PDF链接：
https://arxiv.org/pdf/801.173
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