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:本文提出了基于条件determinantal point process(DPP)的方法,用于生成随机矩阵理论中各种特征值统计量的新型(主要为Fredholm determinantal)表达式。众所周知,许多(特别是$\beta=2$)特征值$n$-点关联函数都是用$n\times n$行列式表示的,即连续的DPP。我们利用条件DPP所得到的内核,从一些特征值在固定位置上已存在的情况下,得到$n$-点关联函数的条件概率密度函数。使用这些内核,我们可以获得最大的$k$个特征值的联合概率密度函数、第$k$个特征值概率密度函数、第一个特征值间距的概率密度函数等新的determinantal表达式。我们的公式非常适合用于数值计算,并提供了各种数值实验。各种需要数小时计算时间才能计算得出的数值,现在可以用我们的表达式在几秒钟内计算出来,这证明了我们的方法的有效性。我们还演示了我们的技术可以应用于有效采样Aztec diamond domino tiling的DR路径。进一步扩展条件DPP采样技术,我们从扩展的Airy kernel中采样Airy processes。此外,我们还提出了一种用于非Hermitian投影DPP采样的方法。