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：我们提出条件行列式点过程（DPP）方法，以获取随机矩阵理论中多种特征值统计量的新（主要是Fredholm行列式）表达式。众所周知，许多（尤其是β=2的）特征值n点关联函数由n×n行列式给出，即它们是连续DPP。我们利用条件DPP的一个导出核，该核给出了在已存在某些固定位置特征值的事件条件下n点关联函数。利用此类核，我们获得了前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、首个特征值间隔的密度等新的行列式表达式。我们的公式高度适用于数值计算，并提供了多种数值实验。多个原本需要数小时计算时间的数值值，现在可通过我们的表达式在数秒内计算完成，这证明了我们方法的有效性。我们还证明了该技术可有效应用于Aztec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，我们从扩展Airy核中采样了Airy过程。此外，我们还提出了一种适用于非厄米投影DPP的采样方法。