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Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

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 $ρ=N/(LW)$ 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 $ρ$, 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.