2018/07/31 by Makoto Katori
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Combinatorics #Complex plane #Constant (computer programming) #Domain (mathematical analysis) #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Physics #Point process #Quantum mechanics #Random Matrices and Applications #Random matrix #Scaling #Scaling limit #Statistics #Stochastic processes and statistical mechanics #Type (biology) #cond-mat.stat-mech #math-ph #math.CA #math.MP #math.PR #nlin.SI
paper · pdf · doi:10.1007/s00220-019-03351-5
published as Commun. Math. Phys. 371, 1283-1321 (2019) · v2:AMS-LaTeX, 34 pages, no figure
arxiv created 2018/11/20 · openalex publication_date 2019/02/18 · arxiv updated 2020/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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, RN=AN-1, BN, B\veeN, CN, C\veeN, BCN, DN, N ∈ ℕ. Their multivariate probability densities are doubly periodic with periods (L, iW), 0 < L, W < ∞, i=√(-1). The construction is based on the orthogonality relations with respect to the double integrals over the fundamental domain, [0, L) × i [0, W), which are proved in this paper for the RN-theta functions introduced by Rosengren and Schlosser. In the scaling limit N → ∞, L → ∞ with constant density ρ=N/(LW) and constant W, we obtain four types of DPPs with an infinite number of points on ℂ, which have periodicity with period i W. In the further limit W → ∞ with constant ρ, 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.