Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A pluricomplex error-function kernel at the edge of polynomial Bergman kernels

L. D. Molag

Abstract

We consider polynomial Bergman kernels with respect to exponentially varying weights $e^{-n \mathscr Q(z)}$ depending on a potential $\mathscr Q:\mathbb C^d\to\mathbb R$. We use these kernels to construct determinantal point processes on $\mathbb C^d$. Under mild conditions on the potential, the points are known to accumulate on a compact set $S_{\mathscr Q}$ called the droplet. We show that the local behavior of the kernel in the vicinity of the edge $\partial S_{\mathscr Q}$ is described in two different ways by universal limiting kernels. One of these limiting kernels is the error-function kernel, which is ubiquitous in random matrix theory, while the other limiting kernel is a new universal object: a multivariate version of the error-function kernel. We prove the universality in two qualitatively different settings: (i) the tensorized case where $\mathscr Q$ decomposes as a sum of planar potentials, and (ii) the case where $\mathscr Q$ is rotational symmetric. We also explicitly identify the subspace of the Bargmann-Fock space where the multivariate error-function kernel is reproducing. To treat regular edge points that exhibit a certain type of bulk degeneracy, we also find the behavior of the planar kernel with number of terms of order $o(n)$ instead of $n$. Lastly, we prove an edge scaling limit for counting statistics.

A pluricomplex error-function kernel at the edge of polynomial Bergman kernels

Abstract

We consider polynomial Bergman kernels with respect to exponentially varying weights depending on a potential . We use these kernels to construct determinantal point processes on . Under mild conditions on the potential, the points are known to accumulate on a compact set called the droplet. We show that the local behavior of the kernel in the vicinity of the edge is described in two different ways by universal limiting kernels. One of these limiting kernels is the error-function kernel, which is ubiquitous in random matrix theory, while the other limiting kernel is a new universal object: a multivariate version of the error-function kernel. We prove the universality in two qualitatively different settings: (i) the tensorized case where decomposes as a sum of planar potentials, and (ii) the case where is rotational symmetric. We also explicitly identify the subspace of the Bargmann-Fock space where the multivariate error-function kernel is reproducing. To treat regular edge points that exhibit a certain type of bulk degeneracy, we also find the behavior of the planar kernel with number of terms of order instead of . Lastly, we prove an edge scaling limit for counting statistics.

Paper Structure

This paper contains 14 sections, 24 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Polynomial Bergman kernels on $\mathbb C^d$
  3. Local scaling limits
  4. Local edge universality conjectures
  5. Summary of the main results
  6. A factorization into planar weights
  7. Preparation: some planar potential theory
  8. Preparation: some pluripotential theory
  9. Local edge scaling limits of the kernel
  10. Rotational symmetric weights
  11. Local edge scaling limits of the kernel
  12. An edge scaling limit for counting statistics
  13. Edge point bulk degeneracy: kernels with $o(n)$ terms
  14. Appendix: a Gaussian integral identity

Key Result

Theorem 1

Suppose that $\mathscr Q:\mathbb C^d\to\mathbb R$ decomposes as a sum of $[0,1]$-admissible planar potentials. Assume that the droplet $S_{\mathscr Q}$ has a smooth boundary. Then the following statements are true. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (55)

  • Definition 1
  • Conjecture 1
  • Conjecture 2
  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 45 more