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

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 $\mathbb C<sup>d$. 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.

• The paper identifies two distinct edge scaling limits—including a classical erfc and a multivariate erfc kernel—and rigorously proves their universality in polynomial Bergman spaces.
• It applies advanced techniques from pluripotential theory, tensorized potentials, and rotational symmetry to convert multi-index sums into Gaussian integrals for precise asymptotic analysis.
• The work extends one-dimensional edge kernel results to higher dimensions, clarifying reproducing kernel properties via the Bargmann transform and impacting random matrix theory.

Pluricomplex Error-Function Kernels at the Edge of Polynomial Bergman Kernels

Introduction and Scope

The paper addresses the local scaling limits of polynomial Bergman kernels on $\mathbb{C}^d$ with exponentially varying weights, focusing in particular on their behavior near the edge of the so-called "droplet" that supports the limiting distribution of points in associated determinantal point processes (DPPs). While the universality of bulk scaling limits (e.g., Ginibre kernels) is relatively well-understood in several complex variables, the extension of the classic error-function (erfc) edge kernel to the multivariate (pluricomplex) setting remained unresolved except in highly specific examples. This work offers a systematic analysis, identifying universal edge kernels and demonstrating their appearance under general conditions on the weight in two settings: decomposable (tensorized) potentials and rotationally symmetric cases.

Setting and Background

Consider the Hilbert space of multivariate polynomials of degree less than $n$ on $\mathbb{C}^d$, equipped with the weighted $L^2$-norm using weight $e^{-n \mathscr{Q}(z)}$, where $\mathscr{Q}:\mathbb{C}^d \to \mathbb{R}$ is a confining potential satisfying suitable growth and regularity. The corresponding (weighted) polynomial Bergman kernel gives rise to an $N_n^d$-point DPP with correlation structure dictated by the kernel. When $n\to\infty$, the empirical measure of particles accumulates on a compact "droplet" $S_{\mathscr{Q}}$ determined by the pluripotential envelope $\check{\mathscr{Q}}$.

Established results for $n$0 (bulk and edge) relate the local scaling limit of the kernel in the interior (resp. at the boundary) to the Ginibre (resp. erfc) kernel, with implications for universality of local statistics in random normal matrices and 2D Coulomb gases. For $n$1, while bulk universality was known (Berman), the corresponding edge results lacked generality, with only a handful of explicit models (notably, the elliptic Ginibre ensemble) previously analyzed.

Main Results

Universality of Pluricomplex Edge Kernels

The central achievement is the identification and rigorous proof of the universality of two distinct types of edge scaling limits:

1. Classical erfc edge kernel: At regular boundary points where a distinguished normal direction exists, the local kernel asymptotically matches the one-dimensional error-function kernel (with explicit coordinate adaptation), i.e.,

$n$2

2. Pluricomplex (multivariate) erfc kernel: In directions corresponding to edge points where several coordinates are simultaneously at the boundary (not reducible to a one-dimensional problem), the local scaling limit is governed by a genuinely multivariate kernel,

$n$3

This object is new to the literature and generalizes the familiar erfc kernel to several complex variables.

The two types of universality are demonstrated in:

• A factorized (tensorized) setting: $n$4, with $n$5 satisfying suitable admissibility, capturing a broad class of non-interacting but anisotropic models.
• A rotationally invariant setting: $n$6, admitting droplets that are balls and limiting kernels adapted to radial symmetry.

Edge Kernel as a Reproducing Kernel

The paper identifies the Hilbert subspace of the Fock-Bargmann space for which the (multivariate) erfc edge kernel is reproducing. Specifically, for a unit vector $n$7, the kernel

$n$8

is the reproducing kernel of the image under the Bargmann transform of $n$9 functions supported on the half-space $\mathbb{C}^d$0. This result clarifies the functional-analytic structure underlying edge universality in several variables.

Bulk Degeneracy and Partial Bergman Kernels

A further analysis considers "degenerate" edge points, where some coordinates are at the minimum of the potential (bulk) and others at the edge. The analysis establishes uniform asymptotics for the corresponding partial Bergman kernels (with $\mathbb{C}^d$1 terms), extending existing approaches based on H\"ormander $\mathbb{C}^d$2-methods, but offering control uniform in the bulk.

Counting Statistics at the Edge

For rotationally invariant weights, the paper establishes an explicit edge scaling limit for the variance of counting statistics (number variance) in microscopic neighborhoods near the boundary, generalizing known results in $\mathbb{C}^d$3 to higher dimensions. The limiting variance matches that of the one-dimensional case up to a dimensional factor.

Technical Approach

The proofs blend techniques from several complex variables, pluripotential theory, precise asymptotics for planar orthogonal polynomials (and their multivariate tensorizations), analytic continuations (hermitian-analytic off-diagonal estimates), and functional analysis via the Bargmann transform. Especially noteworthy is the careful Riemann sum analysis converting sums over multi-indices into Gaussian integrals, and the treatment of "bulk degeneracy" cases where limit transitions interpolate between bulk and edge regimes.

Analytic reductions leverage the regularity of the potential and its decomposition, allowing for explicit identification of the obstacle function and Monge-Ampère measure in both tensorized and radial cases. The extension to partial Bergman kernels via pointwise extremal characterizations and Lagrange multiplier analysis yields strong uniform control.

Implications and Potential Developments

On the theoretical side, this work resolves conjectures on universality of edge statistics for a broad class of pluripotential DPPs, advancing the understanding of local spectral behavior in high-dimensional non-Hermitian random matrices, and strengthening the foundational links between complex probabilistic models, analysis, and geometry.

The explicit identification of the pluricomplex erfc kernel as a universal object suggests new research directions, particularly in classifying universality classes for local statistics in various models, including possible breakdowns at singular boundary points, and exploring analogous behavior in yet more general (e.g., Kähler or non-compact) complex manifolds.

Practically, these results offer analytic tools for the study of correlated point processes in multivariate complex systems—relevant for random polynomials, quantum gases, and random matrix ensembles—in higher dimensions, especially those lacking direct potential-theoretic minimization structures.

The paper also raises intriguing analytic challenges: extending the results to non-symmetric or singular potentials, understanding the implications for associated extremal problems, and further exploring the connection to functional analysis and harmonic analysis in the setting of generalized Fock spaces and partial Bergman kernels.

Conclusion

This work introduces and rigorously establishes the universality of a multivariate error-function ("pluricomplex erfc") kernel as the edge scaling limit for polynomial Bergman kernels in several complex variables with a large class of potentials. It links advanced pluripotential theory with precise random matrix statistics, unifies previous partial results, and clarifies the analytic and probabilistic structure at the spectral edge in higher dimensions. The identification of the related Hilbert space and the handling of degenerate cases complete a comprehensive analysis that forms a new foundation for future investigations in pluripotential DPPs.