Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

Abstract

We introduce new families of determinantal point processes (DPPs) on a complex plane $\mathbb{C}$, which are classified into seven types following the irreducible reduced affine root systems, $R_N=A_{N-1}$, $B_N$, $B_N^{\vee }$, $C_N$, $C_N^{\vee }$, $B{C}_N$, $D_N$, $N\in \mathbb{N}$. Their multivariate probability densities are doubly periodic with periods $(L,iW)$, $0<L,W<\infty$, $i=\sqrt{-1}$. The construction is based on the orthogonality relations with respect to the double integrals over the fundamental domain, $[0,L)\times i[0,W)$, which are proved in this paper for the $R_N$-theta functions introduced by Rosengren and Schlosser. In the scaling limit $N\to \infty ,L\to \infty$ with constant density $\rho =N/(LW)$ and constant $W$, we obtain four types of DPPs with an infinite number of points on $\mathbb{C}$, which have periodicity with period $iW$. In the further limit $W\to \infty$ with constant $\rho$, they are degenerated into three infinite-dimensional DPPs. One of them is uniform on $\mathbb{C}$ and 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 $\mathbb{C}$. We show that the elliptic DPP of type $A_{N-1}$ is 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 types $C_N$ and $D_N$. Relationship to the Gaussian free field on a torus is discussed for these three exactly solvable plasma models.

v2:AMS-LaTeX, 34 pages, no figure

References in corpus (3)

• The random normal matrix model: insertion of a point charge
• Filtering with Wavelet Zeros and Gaussian Analytic Functions
• Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes

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