Title: Tracy-Widom method for Janossy density and joint distribution of extremal eigenvalues of random matrices Show Chinese title

Title: Janossy 密度和随机矩阵极值特征值的联合分布的 Tracy-Widom 方法

Abstract: The J\'{a}nossy density for a determinantal point process is the probability density that an interval $I$ contains exactly $p$ points except for those at $k$ designated loci. The J\'{a}nossy density associated with an integrable kernel $\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$ is shown to be expressed as a Fredholm determinant $\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of a transformed kernel $\tilde{\mathbf{K}}\doteq (\tilde{\varphi}(x)\tilde{\psi}(y)-\tilde{\psi}(x)\tilde{\varphi}(y))/(x-y)$. We observe that $\tilde{\mathbf{K}}$ satisfies Tracy and Widom's criteria if $\mathbf{K}$ does, because of the structure that the map $(\varphi, \psi)\mapsto (\tilde{\varphi}, \tilde{\psi})$ is a meromorphic $\mathrm{SL}(2,\mathbb{R})$ gauge transformation between covariantly constant sections. This observation enables application of the Tracy--Widom method to J\'{a}nossy 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 Painlev\'{e} equations employed in the preceding works. As illustrative examples we compute J\'{a}nossy densities with $k=1, p=0$ for 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. Show Chinese abstract

Abstract: 确定点过程的Jánossy密度是概率密度，它表示区间$I$恰好包含$p$个点的概率，但排除了位于$k$个指定位置上的点。 与可积核$\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$相关的Jánossy密度被证明可以表示为经过变换后的核$\tilde{\mathbf{K}}\doteq (\tilde{\varphi}(x)\tilde{\psi}(y)-\tilde{\psi}(x)\tilde{\varphi}(y))/(x-y)$的Fredholm行列式$\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$。 我们注意到，如果$\mathbf{K}$满足 Tracy 和 Widom 的标准，则$\tilde{\mathbf{K}}$也会满足，这是由于映射$(\varphi, \psi)\mapsto (\tilde{\varphi}, \tilde{\psi})$是协变常数截面之间的亚纯$\mathrm{SL}(2,\mathbb{R})$规范变换的结构特性。 这一观察结果使得 Tracy-Widom 方法能够应用于 Jánossy 密度，这些密度可以用区间端点处一个微分方程组解的形式来表达。 我们的方法并未明确涉及前人工作中使用的与 Painlevé 方程相关的等单性系统。 作为说明性例子，我们计算了与随机 Hermite 矩阵的两个最大特征值联合分布和随机复矩阵的两个最小奇异值联合分布相关的 Airy 核和 Bessel 核的 Jánossy 密度，其中参数为$k=1, p=0$。