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We introduce new families of determinantal point processes (DPPs) on acomplex plane ${\mathbb{C}}$, which are classified into seven types followingthe irreducible reduced affine root systems, $R_N=A_{N-1}$, $B_N$,$B^{\vee}_N$, $C_N$, $C^{\vee}_N$, $BC_N$, $D_N$, $N \in {\mathbb{N}}$. Theirmultivariate probability densities are doubly periodic with periods $(L, iW)$,$0 < L, W < \infty$, $i=\sqrt{-1}$. The construction is based on theorthogonality relations with respect to the double integrals over thefundamental domain, $[0, L) \times i [0, W)$, which are proved in this paperfor the $R_N$-theta functions introduced by Rosengren and Schlosser. In thescaling 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 ofpoints on ${\mathbb{C}}$, which have periodicity with period $i W$. In thefurther limit $W \to \infty$ with constant $\rho$, they are degenerated intothree infinite-dimensional DPPs. One of them is uniform on ${\mathbb{C}}$ andequivalent with the Ginibre point process studied in random matrix theory,while other two systems are rotationally symmetric around the origin, butnon-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 backgroundeffect, of the two-dimensional exactly solvable model for one-component plasmastudied by Forrester. Other two exactly solvable models of one-component plasmaare 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 threeexactly solvable plasma models.

Two-Dimensional Elliptic Determinantal Point Processes and Related Systems