Two-Dimensional Elliptic Determinantal Point Processes and Related Systems
Two-Dimensional Elliptic Determinantal Point Processes and Related Systems
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DOI：
10.1007/s00220-019-03351-5
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发表时间：
2018-07
影响因子：
2.4
通讯作者：
M. Katori
中科院分区：
文献类型：
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作者：
M. Katori
本文在复平面上引入了一类新的行列式点过程（DPPs），并将其按照既约仿射根系RN=AN− 1，BN，，CN，，BCN，DN，分成七类.它们的多元概率密度具有双周期性，周期为（L，iW），。本文证明了Rosengren和Schlosser引入的-theta函数在基本域上的二重积分的正交关系。在常数密度和常数W的标度极限下，我们得到了四类具有无穷多个点的DPPs，它们具有周期性。在常数的进一步限制下，它们退化为三个无限维的DPP。其中一个系统在上是一致的，与随机矩阵理论中的Ginibre点过程等价，另两个系统是关于原点旋转对称的，但在上是不一致的.我们证明AN-1型椭圆DPP与Forrester研究的二维单组分等离子体精确可解模型中减去背景效应后得到的粒子截面是一致的.另外两个精确可解的单组元等离子体模型分别与CN型和DN型椭圆型等离子体相关联。讨论了这三种精确可解等离子体模型与环面上高斯自由场的关系。
We introduce new families of determinantal point processes (DPPs) on a complex plane, which are classified into seven types following the irreducible reduced affine root systems,RN=AN−1,BN,,CN,,BCN,DN,. Their multivariate probability densities are doubly periodic with periods (L,iW),,. The construction is based on the orthogonality relations with respect to the double integrals over the fundamental domain,, which are proved in this paper for the-theta functions introduced by Rosengren and Schlosser. In the scaling limitwith constant densityand constantW, we obtain four types of DPPs with an infinite number of points on, which have periodicity with periodiW. In the further limitwith constant, they are degenerated into three infinite-dimensional DPPs. One of them is uniform onand equivalent with the Ginibre point process studied in random matrix theory, while other two systems are rotationally symmetric around the origin, but non-uniform on. We show that the elliptic DPP of typeAN-1is identified with the particle section, obtained by subtracting the background effect, of the two-dimensional exactly solvable model for one-component plasma studied by Forrester. Other two exactly solvable models of one-component plasma are constructed associated with the elliptic DPPs of typesCNandDN. Relationship to the Gaussian free field on a torus is discussed for these three exactly solvable plasma models.