Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

TL;DR

This paper introduces new elliptic determinantal point processes on the complex plane classified by root systems, explores their scaling limits, and connects them to models in random matrix theory and plasma physics.

Contribution

It develops a classification of elliptic DPPs based on root systems, proves orthogonality relations for R_N-theta functions, and links these processes to exactly solvable plasma models and the Gaussian free field.

Findings

Four types of infinite DPPs with periodicity on the complex plane.

Identification of the type A_{N-1} DPP with a particle section of a solvable plasma model.

Construction of two additional plasma models associated with types C_N and D_N.

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, $R_N=A_{N-1}$, $B_N$, $B^{\vee}_N$, $C_N$, $C^{\vee}_N$, $BC_N$, $D_N$, $N \in {\mathbb{N}}$. 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 $i W$. 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.