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

TL;DR

This paper introduces a new multivariate error-function kernel for polynomial Bergman kernels at the edge of droplet sets, demonstrating universality in different potential settings and identifying its reproducing subspace.

Contribution

It constructs a novel multivariate error-function kernel as a universal limit at the edge of polynomial Bergman kernels, extending known universality results in random matrix theory.

Findings

The multivariate error-function kernel is universal at the droplet edge.

Explicit identification of the kernel's reproducing subspace in Bargmann-Fock space.

Edge scaling limits for counting statistics are established.

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.