Tracy-Widom method for Janossy density and joint distribution of extremal eigenvalues of random matrices
Tracy-Widom method for Janossy density and joint distribution of extremal eigenvalues of random matrices
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随机矩阵极值特征值 Janossy 密度和联合分布的 Tracy-Widom 方法
DOI：
10.1093/ptep/ptab123
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发表时间：
2021
影响因子：
3.5
通讯作者：
Shinsuke M Nishigaki
中科院分区：
文献类型：
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作者：
Kang Kyungkeun;Miura Hideyuki;Tsai Tai-Peng;Masami Sakai;Terashima Seiji;亀子 正喜;亀子 正喜;亀子 正喜;Shinsuke M Nishigaki
确定性点过程的Jánossy密度是一个区间包含除指定位点以外的所有点的概率密度。与可积核相关的Jánossy密度可以表示为变换核的Fredholm行列式。我们观察到，如果存在，则满足Tracy和Widom的准则，因为该结构映射为协变常数截面之间的亚纯规范变换。这一观察结果使Tracy-Widom方法能够应用于Jánossy密度，以区间端点的微分方程系统的解表示。我们的方法并没有明确地提到与前面工作中使用的painlevew方程相关的等同构系统。作为说明性的例子，我们计算了与随机厄米矩阵的两个最大特征值和随机复矩阵的两个最小奇异值的联合分布有关的Airy和Bessel核的Jánossy密度。
The Jánossy density for a determinantal point process is the probability density that an intervalcontains exactlypoints except for those atdesignated loci. The Jánossy density associated with an integrable kernelis shown to be expressed as a Fredholm determinantof a transformed kernel. We observe thatsatisfies Tracy and Widom’s criteria ifdoes, because of the structure that the mapis a meromorphicgauge transformation between covariantly constant sections. This observation enables application of the Tracy–Widom method to Jánossy densities, expressed in terms of a solution to a system of differential equations in the endpoints of the interval. Our approach does not explicitly refer to isomonodromic systems associated with Painlevé equations employed in the preceding works. As illustrative examples we compute Jánossy densities withfor Airy and Bessel kernels, related to the joint distributions of the two largest eigenvalues of random Hermitian matrices and of the two smallest singular values of random complex matrices.