论文标题
与Bergman内核相关的确定点过程：构建和限制定理
Determinantal point processes associated with Bergman kernels: construction and limit theorems
论文作者
论文摘要
我们研究了确定点过程，其相关核是在紧凑的复杂歧管上具有正遗传式霍尔米克式线束的高功率的伯格曼内核。我们类似于随机矩阵理论中的正交合奏，其中相关内核是著名的ChristOffel-Darboux内核。使用Bergman内核的几乎基因扩展，我们证明了这些点过程的缩放限制是通过无限ginibre集合的多维概括给出的。作为一种应用，我们获得了其经验措施概率与与复杂Monge-ampère方程相关的平衡度量的趋同。我们最终为这些过程的加权版本建立了一个较大的偏差原理，其速率函数是mabuchi功能的传奇式触发器变换。
We study determinantal point processes whose correlation kernel is the Bergman kernel of a high power of a positive Hermitian holomorphic line bundle over a compact complex manifold. We construct such processes in analogy to the orthogonal ensembles in random matrix theory, where the correlation kernel is the famous Christoffel-Darboux kernel. Using a near-diagonal expansion of the Bergman kernel, we prove that the scaling limit of these point processes is given by a multidimensional generalization of the infinite Ginibre ensemble. As an application, we obtain a convergence in probability of their empirical measures to an equilibrium measure related to the complex Monge-Ampère equations. We finally establish a large deviation principle for weighted versions of these processes, whose rate function is the Legendre-Fenchel transform of the Mabuchi functional.
