Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal transport, determinantal point processes and the Bergman kernel

William Driot, Laurent Decreusefond

Abstract

We study the Bergman determinantal point process from a theoretical point of view motivated by its simulation. We construct restricted and restricted-truncated variants of the Bergman kernel and show optimal transport inequalities involving the Kantorovitch-Rubinstein Wasserstein distance to show to what extent it is fair to truncate the restriction of this point process to a compact ball of radius $1 - \varepsilon $. We also investigate the deviation of the number of points of the restricted Bergman determinantal point process, indicate which number of points looks like an optimal choice, and provide upper bounds on its deviation, providing an answer to an open question asked in [5]. We also consider restrictions to other regions and investigate the choice of such regions for restriction. Finally, we provide general results as to the deviation of the number of points of any determinantal point process.

Optimal transport, determinantal point processes and the Bergman kernel

Abstract

We study the Bergman determinantal point process from a theoretical point of view motivated by its simulation. We construct restricted and restricted-truncated variants of the Bergman kernel and show optimal transport inequalities involving the Kantorovitch-Rubinstein Wasserstein distance to show to what extent it is fair to truncate the restriction of this point process to a compact ball of radius . We also investigate the deviation of the number of points of the restricted Bergman determinantal point process, indicate which number of points looks like an optimal choice, and provide upper bounds on its deviation, providing an answer to an open question asked in [5]. We also consider restrictions to other regions and investigate the choice of such regions for restriction. Finally, we provide general results as to the deviation of the number of points of any determinantal point process.

Paper Structure

This paper contains 5 sections, 17 theorems, 86 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Determinantal Point Processes
  3. The Bergman DPP on a disc
  4. The Bergman DPP on other regions
  5. General deviations results for determinantal point processes

Key Result

Theorem 9

Under Hypothesis hyp:main-hypothesis, consider Mercer's decomposition of the kernel $k$ of the determinantal point process $\eta$: Here, the eigenvalues are all in $[0,1]$ and can all be chosen in $(0,1]$ ; $n$ is equal to the rank of $K$, which can be either finite or infinite. The $(\phi_n)$ form a Hilbert basis is the space $L^2(E, \lambda)$. Consider then a sequence of independent Bernoulli r

Figures (2)

  • Figure 1: Simulation of the Bergman DPP restricted to a radius of 0.9995, using MorozSoftware. Number of points : 985.
  • Figure 2: A zoom on the boundary of the simulation on the left.

Theorems & Definitions (53)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 8
  • Theorem 9
  • Corollary 10
  • Definition 11
  • ...and 43 more