We present the conditional determinantal point process (DPP) approach to obtain new (mostly Fredholm determinantal) expressions for various eigenvalue statistics in random matrix theory. It is well-known that many (especially $\beta=2$) eigenvalue $n$-point correlation functions are given in terms of $n\times n$ determinants, i.e., they are continuous DPPs. We exploit a derived kernel of the conditional DPP which gives the $n$-point correlation function conditioned on the event of some eigenvalues already existing at fixed locations. Using such kernels we obtain new determinantal expressions for the joint densities of the $k$ largest eigenvalues, probability density functions of the $k^\text{th}$ largest eigenvalue, density of the first eigenvalue spacing, and more. Our formulae are highly amenable to numerical computations and we provide various numerical experiments. Several numerical values that required hours of computing time could now be computed in seconds with our expressions, which proves the effectiveness of our approach. We also demonstrate that our technique can be applied to an efficient sampling of DR paths of the Aztec diamond domino tiling. Further extending the conditional DPP sampling technique, we sample Airy processes from the extended Airy kernel. Additionally we propose a sampling method for non-Hermitian projection DPPs.

翻译：我们提出了条件行列式点过程（DPP）方法，用于获得随机矩阵理论中各种特征值统计的新表达式（主要是Fredholm行列式形式）。众所周知，许多（尤其是β=2）特征值的n点相关函数可表示为n×n行列式，即它们是连续DPP。我们利用条件DPP的导出核，该核给出了在已存在某些特征值于固定位置的事件条件下，n点相关函数的表达式。通过此类核，我们获得了k个最大特征值的联合密度、第k个最大特征值的概率密度函数、首个特征值间距的密度等的新行列式表达式。我们的公式高度适用于数值计算，并提供了多种数值实验。此前需要数小时计算时间的若干数值值，现在通过我们的表达式可在数秒内完成计算，这证明了我们方法的有效性。我们同时还展示了该技术可有效应用于阿兹特克钻石多米诺骨牌平铺中DR路径的采样。进一步扩展条件DPP采样技术，我们通过扩展Airy核实现了Airy过程的采样。此外，我们提出了一种针对非厄米投影DPP的采样方法。