Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

We introduce new families of determinantal point processes (DPPs) on a complex plane C, which are classified into seven types following the irreducible reduced affine root systems, R-N = A(N-1), B-N, B-N, C-N, C. , BCN, D-N, N is an element of N. Their multivariate probability densities are doubly periodic with periods (L, iW), 0 < L, W < infinity, i = root-1. The construction is based on the orthogonality relations with respect to the double integrals over the fundamental domain, [0, L) x i [0, W), which are proved in this paper for the RN -theta functions introduced by Rosengren and Schlosser. In the scaling limit N. 8, L. 8 with constant density = N/(LW) and constant W, we obtain four types of DPPs with an infinite number of points on C, which have periodicity with period iW. In the further limit W. 8 with constant., they are degenerated into three infinite-dimensional DPPs. One of them is uniform on 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 C. We show that the elliptic DPP of type AN-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 CN and DN. Relationship to the Gaussian free field on a torus is discussed for these three exactly solvable plasma models.