Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative equidistribution of eigenvalues of Random Normal Matrices in the Wasserstein distance

P. García Arias

Abstract

The object of study in this paper is the expected $2$-Wasserstein distance between the empirical measures of several point processes and their respective limit. For this, the main tool developed is a smoothing procedure in Euclidean spaces using the heat equation with Neumann boundary conditions. It is applied to the spectrum of Random Normal Matrices with \textit{reasonable} assumptions, as well as to several families of Homogeneous Point Processes such as the infinite Ginibre ensemble, the Bessel ensemble, and the zero set of the planar Gaussian Analytic Function.

Quantitative equidistribution of eigenvalues of Random Normal Matrices in the Wasserstein distance

Abstract

The object of study in this paper is the expected -Wasserstein distance between the empirical measures of several point processes and their respective limit. For this, the main tool developed is a smoothing procedure in Euclidean spaces using the heat equation with Neumann boundary conditions. It is applied to the spectrum of Random Normal Matrices with \textit{reasonable} assumptions, as well as to several families of Homogeneous Point Processes such as the infinite Ginibre ensemble, the Bessel ensemble, and the zero set of the planar Gaussian Analytic Function.
Paper Structure (16 sections, 19 theorems, 106 equations, 3 figures)

This paper contains 16 sections, 19 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Wasserstein distance
  4. Point Processes
  5. Main results
  6. Proofs
  7. Heat Smoothing inequality
  8. Homogeneous Point Processes
  9. Case 1:
  10. Case 2:
  11. Random normal matrices
  12. Bounding the last measure:
  13. Bounding intermediate measures:
  14. Case 1:
  15. Case 2:
  16. ...and 1 more sections

Key Result

Theorem 2

Let $Q$ be a potential satisfying the regularity assumptions of Remark remark:regularity and whose droplet $S$ is convex. If we denote by $z_1, z_2, \dots, z_N$ the points given by the RNM model, then

Figures (3)

  • Figure 1: To the left, a sample from the zero set of the planar GAF, restricted to the unit square, and intensity $L=150$. To the right, a sample from the infinite Ginibre ensemble restricted to the unit square, and intensity $L=150$.
  • Figure 2: Sample of 500 points of the Ginibre ensemble (left) and the Elliptic Ginibre ensemble (right).
  • Figure 3: Representation of the edge (in blue) and the bulk (in orange), and the relevant boundaries.

Theorems & Definitions (40)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Theorem 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 30 more