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, ${B^{\vee}_N}$, 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 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.