Family Of Determinantal Point Processes Research Articles

Entanglement entropy and hyperuniformity of Ginibre and Weyl–Heisenberg ensembles

We show that, for a class of planar determinantal point processes (DPP) {mathcal {X}}, the growth of the entanglement entropy S({mathcal {X}}(Omega )) of {mathcal {X}} on a compact region Omega subset {mathbb {R}}^{2d}, is related to the variance {mathbb {V}}left( {mathcal {X}}(Omega )right) as follows: VX(Ω)≲SX(Ω)≲VX(Ω).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathbb {V}}\\left( {\\mathcal {X}}(\\Omega )\\right) \\lesssim S\\left( \\mathcal {X(} \\Omega \\mathcal {)}\\right) \\lesssim {\\mathbb {V}}\\left( {\\mathcal {X}}(\\Omega )\\right) . \\end{aligned}$$\\end{document}Therefore, such DPPs satisfy an area lawSleft( {mathcal {X}}_{g} mathcal {(}Omega mathcal {)}right) lesssim left| partial {Omega } right| , where partial {Omega } is the boundary of Omega if they are of Class I hyperuniformity ({mathbb {V}}left( {mathcal {X}} (Omega )right) lesssim left| partial {Omega }right| ), while the area law is violated if they are of Class II hyperuniformity (as Lrightarrow infty , {mathbb {V}}left( {mathcal {X}} (LOmega )right) sim C_{Omega }L^{d-1}log L). As a result, the entanglement entropy of Weyl–Heisenberg ensembles (a family of DPPs containing the Ginibre ensemble and Ginibre-type ensembles in higher Landau levels), satisfies an area law, as a consequence of its hyperuniformity.

Determinantal point processes in the flat limit

Determinantal point processes (DPPs) are repulsive point processes where the interaction between points depends on the determinant of a positive-semi definite matrix. In this paper, we study the limiting process of L-ensembles based on kernel matrices, when the kernel function becomes flat (so that every point interacts with every other point, in a sense). We show that these limiting processes are best described in the formalism of extended L-ensembles and partial projection DPPs, and the exact limit depends mostly on the smoothness of the kernel function. In some cases, the limiting process is even universal, meaning that it does not depend on specifics of the kernel function, but only on its degree of smoothness. Since flat-limit DPPs are still repulsive processes, this implies that practically useful families of DPPs exist that do not require a spatial length-scale parameter.

Local number variances and hyperuniformity of the Heisenberg family of determinantal point processes

The bulk scaling limit of eigenvalue distribution on the complex plane of the complex Ginibre random matrices provides a determinantal point process (DPP). This point process is a typical example of disordered hyperuniform system characterized by an anomalous suppression of large-scale density fluctuations. As extensions of the Ginibre DPP, we consider a family of DPPs defined on the D-dimensional complex spaces , , in which the Ginibre DPP is realized when D = 1. This one-parameter family () of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szegő kernels for the reduced Heisenberg group. For each D, using the modified Bessel functions, an exact and useful expression is shown for the local number variance of points included in a ball with radius R in . We prove that any DPP in the Heisenberg family is in the hyperuniform state of class I, in the sense that the number variance behaves as R 2D−1 as R → ∞. Our exact results provide asymptotic expansions of the number variances in large R.

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 $\rho=N/(LW)$ and constant $W$, we obtain four types of DPPs with an infinite number of points on ${\mathbb{C}}$, which have periodicity with period $i W$. 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.

Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes

Rosengren and Schlosser introduced notions of RN-theta functions for the seven types of irreducible reduced affine root systems, RN = AN−1, BN, BN∨, CN, CN∨, BCN, DN, N∈N, and gave the Macdonald denominator formulas. We prove that if the variables of the RN-theta functions are properly scaled with N, they construct seven sets of biorthogonal functions, each of which has a continuous parameter t ∈ (0, t*) with given 0 &amp;lt; t* &amp;lt; ∞. Following the standard method in random matrix theory, we introduce seven types of one-parameter (t ∈ (0, t*)) families of determinantal point processes in one dimension, in which the correlation kernels are expressed by the biorthogonal theta functions. We demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions. In the scaling limits associated with N → ∞, we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter t ∈ (0, t*). We give new expressions for the Macdonald denominators using the Karlin–McGregor–Lindström–Gessel–Viennot determinants for noncolliding Brownian paths and show the realization of the associated elliptic determinantal point processes as noncolliding Brownian brides with a time duration t*, which are specified by the pinned configurations at time t = 0 and t = t*.

A quantization of the harmonic analysis on the infinite-dimensional unitary group

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(∞). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski [31]). The required decomposition is governed by certain probability measures on an infinite-dimensional space Ω, which is a dual object to U(∞). A way to describe those measures is to convert them into determinantal point processes on the real line; it turned out that their correlation kernels are computable in explicit form — they admit a closed expression in terms of the Gauss hypergeometric function F12 (Borodin and Olshanski [8]). In the present work we describe a (nonevident) q-discretization of the whole construction. This leads us to a new family of determinantal point processes. We reveal its connection with an exotic finite system of q-discrete orthogonal polynomials — the so-called pseudo big q-Jacobi polynomials. The new point processes live on a double q-lattice and we show that their correlation kernels are expressed through the basic hypergeometric function ϕ12. A crucial novel ingredient of our approach is an extended version G of the Gelfand–Tsetlin graph (the conventional graph describes the Gelfand–Tsetlin branching rule for irreducible representations of unitary groups). We find the q-boundary of G, thus extending previously known results (Gorin [17]).