PDEs satisfied by extreme eigenvalues distributions of GUE and LUE
PDEs satisfied by extreme eigenvalues distributions of GUE and LUE
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DOI：
10.1142/s2010326311500031
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发表时间：
2011-02
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影响因子：
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通讯作者：
E. Basor;Yang Chen;Lun Zhang
中科院分区：
文献类型：
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作者：
E. Basor;Yang Chen;Lun Zhang
本文研究有限元n元酉系综本征值都落在区间元(a，b)内的概率。这与最大本征值小于$b$和最小本征值大于$a$的概率相同。证明了一个与$\extsf{prob}(n，a，b)$有关的量，即$$H_n(a，b)：=\Left[\frac{\Partial}{\Partial a}+\FRAC{\Partial}{\Partial b}\Right]\ln\Textsf{prob}(n，a，b)，$$在高斯酉系中(GUE)和$$H_n(a，B)：=\Left[a\frac{\Partial}{\Partial a}+b\FRAC{\Partial}{\Partial b}\Right]\ln\Textsf{prob}(n，a，b)，$$满足固定$n$的某些非线性偏微分方程组，将$H_n(a，b)$解释为$a$和$b$的函数。这些偏微分方程组可以分别看作是Painlev‘{e}V系统和Painlev’{e}V系统的两个变量推广。作为我们结果的一个应用，我们给出了一个解析证明，当中心和尺度适当时，GUE和LUE的极值是渐近独立的。
In this paper we study, $\textsf{Prob}(n,a,b),$ the probability that all the eigenvalues of finite $n$ unitary ensembles lie in the interval $(a,b)$. This is identical to the probability that the largest eigenvalue is less than $b$ and the smallest eigenvalue is greater than $a$. It is shown that a quantity allied to $\textsf{Prob}(n,a,b)$, namely, $$ H_n(a,b):=\left[\frac{\partial}{\partial a}+\frac{\partial}{\partial b}\right]\ln\textsf{Prob}(n,a,b),$$ in the Gaussian Unitary Ensemble (GUE) and $$ H_n(a,b):=\left[a\frac{\partial}{\partial a}+b\frac{\partial}{\partial b}\right]\ln \textsf{Prob}(n,a,b),$$ in the Laguerre Unitary Ensemble (LUE) satisfy certain nonlinear partial differential equations for fixed $n$, interpreting $H_n(a,b)$ as a function of $a$ and $b$. These partial differential equations maybe considered as two variable generalizations of a Painlev\'{e} IV and a Painlev\'{e} V system, respectively. As an application of our result, we give an analytic proof that the extreme eigenvalues of the GUE and the LUE, when suitably centered and scaled, are asymptotically independent.