arXiv subjects

L. D. Molag

Publications and source records attributed to L. D. Molag.

7 recordsLinked to original sources

arXiv2026

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

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.

math.PR↗

arXiv2026

Universality for fluctuations of counting statistics of random normal matrices

We consider the fluctuations of the number of eigenvalues of $n\times n$ random normal matrices depending on a potential $Q$ in a given set $A$. These eigenvalues are known to form a determinantal point process, and are known to accumulate on a compact set called the droplet under mild conditions on $Q$. When $A$ is a Borel set strictly inside the droplet, we show that the variance of the number of eigenvalues $N_A^{(n)}$ in $A$ has a limiting behavior given by \begin{align*} \lim_{n\to\infty} \frac1{\sqrt n}\operatorname{Var } N_A^{(n)} = \frac{1}{2π\sqrtπ}\int_{\partial_* A} \sqrt{ΔQ(z)} \, d\mathcal H^1(z), \end{align*} where $\partial_* A$ is the measure theoretic boundary of $A$, $d\mathcal H^1(z)$ denotes the one-dimensional Hausdorff measure, and $Δ= \partial_z \overline{\partial_z}$. We also consider the case where $A$ is a microscopic dilation of the droplet and fully generalize a result by Akemann, Byun and Ebke for arbitrary potentials. In this result $d\mathcal H^1(z)$ is replaced by the harmonic measure at $\infty$ associated with the exterior of the droplet. This second result is proved by strengthening results due to Hedenmalm-Wennman and Ameur-Cronvall on the asymptotic behavior of the associated correlation kernel near the droplet boundary.

math.PR↗

arXiv2026

Fluctuations in Various Regimes of Non-Hermiticity and a Holographic Principle

The variance of the number of particles in a set is an important quantity in understanding the statistics of non-interacting fermionic systems in low dimensions. An exact map of their ground state in a harmonic trap in one and two dimensions to the classical Gaussian unitary and complex Ginibre ensemble, respectively, allows to determine the counting statistics at finite and infinite system size. We will establish two new results in this setup. First, we uncover an interpolating central limit theorem between known results in one and two dimensions, for linear statistics of the elliptic Ginibre ensemble. We find an entire range of interpolating weak non-Hermiticity limits, given by a two-parameter family for the mesoscopic scaling regime. Second, we considerably generalize the proportionality between the number variance and the entanglement entropy between Fermions in a set $A$ and its complement in two dimensions. Previously known only for rotationally invariant sets and external potentials, we prove a holographic principle for general non-rotationally invariant sets and random normal matrices. It states that both number variance and entanglement entropy are proportional to the circumference of $A$.

math-ph↗

arXiv2023

Edge Behavior of Higher Complex-Dimensional Determinantal Point Processes

As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density of the eigenvalues, and the correlation kernel, where the Faddeeva plasma kernel emerges. These results are neatly expressed with the help of the outward unit normal vector on the edge. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in a specific ``elliptic'' class of determinantal point processes defined on $\mathbb C^d$, which are higher dimensional generalizations of the determinantal point processes describing the eigenvalues of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. These models describe a system of particles in $\mathbb C^d$ with mutual repulsion, that are confined to the origin by an external field $\mathscr V(z) = |z|^2 - τ\operatorname{Re}(z_1^2+\ldots+z_d^2)$, where $0\leq τ<1$. Their average density of particles converges to a uniform law on a $2d$-dimensional ellipsoidal region. It is on the hyperellipsoid bounding this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of our knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. The results provide evidence for a possible edge universality theorem for determinantal point processes on $\mathbb C^d$.

math-ph↗

arXiv2022

The Elliptic Ginibre Ensemble: A Unifying Approach to Local and Global Statistics for Higher Dimensions

The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows to interpolate between the rotational invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field, at inverse temperature $β=2$. Furthermore, it represents a determinantal point process in the complex plane with corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a know correspondence between non-interacting fermions in a trap in $d$ real dimensions $\mathbb{R}^d$ and the $d$-dimensional harmonic oscillator. We present a rigorous proof for the local $d$-dimensional bulk (sine-) and edge (Airy-) kernel first defined by Dean et al., complementing recent results by Deleporte and Lambert. Using the same relation to the $d$-dimensional harmonic oscillator in $d$ complex dimensions $\mathbb{C}^d$, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in $d$ real and $d$ complex dimensions. For $\mathbb{C}^d$ with $d=1$ this corresponds to non-interacting fermions in a rotating trap.

math-ph↗

arXiv2020

The local universality of Muttalib-Borodin ensembles when the parameter $θ$ is the reciprocal of an integer

The Muttalib-Borodin ensemble is a probability density function for $n$ particles on the positive real axis that depends on a parameter $θ$ and a weight $w$. We consider a varying exponential weight that depends on an external field $V$. In a recent article, the large $n$ behavior of the associated correlation kernel at the hard edge was found for $θ=\frac{1}{2}$, where only few restrictions are imposed on $V$. In the current article we generalize the techniques and results of this article to obtain analogous results for $θ=\frac{1}{r}$, where $r$ is a positive integer. The approach is to relate the ensemble to a type II multiple orthogonal polynomial ensemble with $r$ weights, which can then be related to an $(r+1)\times (r+1)$ Riemann-Hilbert problem. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

math.CA↗

arXiv2018

The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $θ= \frac{1}{2}$

The Muttalib-Borodin biorthogonal ensemble is a probability density function for $n$ particles on the positive real line that depends on a parameter $θ$ and an external field $V$. For $θ=\frac{1}{2}$ we find the large $n$ behavior of the associated correlation kernel with only few restrictions on $V$. The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a $3\times 3$ Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the origin, with the help of Meijer G-functions, and its matching condition with a global parametrix. We will present a new iterative technique to obtain the matching condition, which we expect to be applicable in more general situations as well.

math.CA↗