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Yacin Ameur

Publications and source records attributed to Yacin Ameur.

At least 19 recordsLinked to original sources
arXiv2026

Szegő type correlations for two-dimensional outpost ensembles

We consider two-dimensional Coulomb systems for which the coincidence set contains an outpost in the form of a suitable Jordan curve. We study asymptotics for correlations along the union of the outpost and the outer boundary of the droplet. These correlations turn out to have a universal character and are given in terms of the reproducing kernel for a certain Hilbert space of analytic functions, generalizing the Szegő type edge correlations obtained recently by Ameur and Cronvall. There are several additional results, for example on the effect of insertion of an exterior point charge in the presence of an outpost.

math.CV↗
arXiv2026

On fluctuations of Coulomb systems and universality of the Heine distribution

We consider a class of external potentials on the complex plane $\mathbb{C}$ for which the coincidence set to the obstacle problem contains a Jordan curve in the exterior of the droplet. We refer to this curve as a spectral outpost. We study the corresponding Coulomb gas at $β=2$. Generalizing recent work in the radially symmetric case, we prove that the number of particles which fall near the spectral outpost has an asymptotic Heine distribution, as the number of particles $n\to\infty$. We also consider a class of potentials with disconnected droplets whose connected components are separated by a ring-shaped spectral gap. We prove that the fluctuations of the number of particles that fall near a given component has an asymptotic discrete normal distribution, which depends on $n$. For the case of disconnected droplets we also consider fluctuations of general smooth linear statistics and show that they tend to distribute as the sum of a Gaussian field and an independent, oscillatory, discrete Gaussian field. Our techniques involve a new asymptotic formula on the norm of monic orthogonal polynomials in the bifurcation regime and a variant of the method of limit Ward identities of Ameur, Hedenmalm, and Makarov.

math-ph↗
arXiv2025

A formula for the edge density $\sqrt{n}$-correction for two-dimensional Coulomb systems

In connection with recent work on smallest gaps, C. Charlier proves that the 1-point function of a suitable planar Coulomb system $\{z_j\}_1^n$, in the determinantal case with respect to an external potential $Q(z)$, admits the expansion, as $n\to\infty$, $$R_n\bigg(z_0+\frac t {\sqrt{2n\partial\bar{\partial} Q(z_0)}}ν(z_0)\bigg)=n\partial\bar{\partial} Q(z_0)\frac {\operatorname{erfc} t}2+\sqrt{n\partial\bar{\partial} Q(z_0)}\,C(z_0;t)+\mathcal{O}(\log^3 n).$$ Here $t$ is a real parameter, $z_0$ is a regular boundary point of the (connected) Coulomb droplet and $ν(z_0)$ is the outwards unit normal; the coefficient $C(z_0;t)$ has an apriori structure depending on a number of parameters. In this note we identify the parameters and obtain a formula for $C(z_0;t)$ in potential theoretic and geometric terms. Our formula holds for a large class of potentials such that the droplet is connected with smooth boundary. Our derivation uses the well known expectation of fluctuations formula.

math-ph↗
arXiv2025

The two-dimensional Coulomb gas: fluctuations through a spectral gap

We study a class of radially symmetric Coulomb gas ensembles at inverse temperature $β=2$, for which the droplet consists of a number of concentric annuli, having at least one bounded ``gap'' $G$, i.e., a connected component of the complement of the droplet, which disconnects the droplet. Let $n$ be the total number of particles. Among other things, we deduce fine asymptotics as $n \to \infty$ for the edge density and the correlation kernel near the gap, as well as for the cumulant generating function of fluctuations of smooth linear statistics. We typically find an oscillatory behaviour in the distribution of particles which fall near the edge of the gap. These oscillations are given explicitly in terms of a discrete Gaussian distribution, weighted Szegő kernels, and the Jacobi theta function, which depend on the parameter $n$.

math-ph↗
arXiv2025

Free energy and fluctuations in the random normal matrix model with spectral gaps

We study large $n$ expansions for the partition function of a Coulomb gas $$Z_n=\frac 1 {π^n}\int_{\mathbb{C}^n}\prod_{1\le i<j\le n}|z_i-z_j|^2\prod_{i=1}^n e^{-nQ(z_i)}\, d^2 z_i,$$ where $Q$ is a radially symmetric confining potential on the complex plane $\mathbb{C}$. The droplet is not assumed to be connected, but may consist of a number of disjoint connected annuli and possibly a central disk. The boundary condition is ``soft edge'', i.e., $Q$ is smooth in a $\mathbb{C}$-neighbourhood of the droplet. We include the following possibilities: (i) existence of ``outposts'', i.e., components of the coincidence set which falls outside of the droplet, (ii) a conical (or Fisher-Hartwig) singularity at the origin, (iii) perturbations $Q-\frac h n$ where $h$ is a smooth radially symmetric test-function. In each case, the free energy $\log Z_n$ admits a large $n$ expansion of the form \begin{equation*}\log Z_n=C_1n^2+C_2n\log n+C_3 n+C_4\log n+C_5+\mathcal{G}_{n}+o(1)\end{equation*} where $C_1,\ldots,C_5$ are certain geometric functionals. The $n$-dependent term $\mathcal{G}_n$ is bounded as $n\to\infty$; it arises in the presence of spectral gaps. We use the free energy expansions to study the distribution of fluctuations of linear statistics. We prove that the fluctuations are well approximated by the sum of a Gaussian and certain independent terms which provide the displacement of particles from one component to another. This displacement depends on $n$ and is expressed in terms of the Heine distribution. We also prove (under suitable assumptions) that the number of particles which fall near a spectral outpost converges to a Heine distribution.

math.PR↗
arXiv2025

Remarks on the one-point density of Hele-Shaw $β$-ensembles

In this note we prove equicontinuity for the family of one-point densities with respect to a two-dimensional Coulomb gas at an inverse temperature $β\ge 1/2$ confined by an external potential of Hele-Shaw (or quasi-harmonic) type. As a consequence, subsequential limiting Lipschitz continuous densities are defined on the microscopic scale. There are several additional results, for example comparing the one-point density with the thermal equilibrium density.

math.PR↗
arXiv2024

Gaussian beta ensembles: the perfect freezing transition and its characterization in terms of Beurling-Landau densities

The Gaussian $β$-ensemble is a real $n$-point configuration $\{x_j\}_1^n$ picked randomly with respect to the Boltzmann factor $e^{-\fracβ2H_n}$, $H_n=\sum_{i\ne j}\log\frac 1{|x_i-x_j|}+n\sum_{i=1}^n\tfrac 12x_i^2.$ The point process $\{x_j\}_1^n$ tends to follow the semicircle law $σ(x)=\tfrac 1{2π}\sqrt{(4-x^2)_+}$ in certain average senses. A Fekete configuration (minimizer of $H_n$) is spread out in a much more uniform way in the interval $[-2,2]$ with respect to the regularization $σ_n(x)=\max\{σ(x),n^{-\frac 1 3}\}$ of the semicircle law. In particular, Fekete configurations are "equidistributed" with respect to $σ_n(x)$, in a certain technical sense of Beurling-Landau densities. We consider the problem of characterizing sequences $β_n$ of inverse temperatures, which guarantee almost sure equidistribution as $n\to\infty$. We find that a necessary and sufficient condition is that $β_n$ grows at least logarithmically in $n$: $$β_n\gtrsim \log n.$$ We call this growth rate the perfect freezing regime. We give several further results on the distribution of particles when $β_n\gtrsim\log n$, for example on minimal spacing, discrepancies, and sampling and interpolation for weighted polynomials. The condition $β_n\gtrsim\log n$ was introduced by some of the authors in the context of two-dimensional Coulomb gas ensembles, where it is shown to be sufficient for equidistribution. Although the technical implementation requires some considerable modifications, the strategy from dimension two adapts well to prove sufficiency also for one-dimensional Gaussian ensembles. On a technical level, we use estimates for weighted polynomials due to Levin, Lubinsky, Gustavsson and others. The other direction (necessity) involves estimates due to Ledoux and Rider on the distribution of particles which fall near or outside the boundary.

math.PR↗
arXiv2024

Random normal matrices: eigenvalue correlations near a hard wall

We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant $Γ=2$ and that the number $n$ of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order $1/n$ from the hard edge. At distances much larger than $1/\sqrt{n}$, the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the ``semi-hard edge''. More precisely, we provide asymptotics for the correlation kernel $K_{n}(z,w)$ as $n\to\infty$ in two microscopic regimes (with either $|z-w| = \mathcal{O} (1/\sqrt{n})$ or $|z-w| = \mathcal{O} (1/n)$), as well as in three macroscopic regimes (with $|z-w| \asymp 1$). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels.

math-ph↗
arXiv2023

Szegő type asymptotics for the reproducing kernel in spaces of full-plane weighted polynomials

In this work we find and discuss an asymptotic formula, as $n\to\infty$, for the reproducing kernel $K_n(z,w)$ in spaces of full-plane weighted polynomials $W(z)=P(z)\cdot e^{-\frac 12nQ(z)},$ where $P(z)$ is a holomorphic polynomial of degree at most $n-1$ and $Q(z)$ is a fixed, real-valued function termed "external potential". The kernel $K_n$ corresponds precisely to the canonical correlation kernel in the theory of random normal matrices. As is well-known, the large $n$ behaviour of $K_n(z,w)$ must depend crucially on the position of the points $z$ and $w$ relative to the droplet $S$, i.e., the support of Frostman's equilibrium measure in external potential $Q$. In the particular case when $z$ and $w$ are at the edge and $z\ne w$, we prove the formula $K_n(z,w)\sim\sqrt{2πn}\,ΔQ(z)^{\frac 1 4}ΔQ(w)^{\frac 14}\,S(z,w)$ where $S(z,w)$ is the Szegő kernel associated with the Hardy space $H^2_0(U)$ of analytic functions on unbounded component $U$ of $\hat{\mathbb{C}}\setminus S$ which vanish at infinity. This gives a rigorous description of the slow decay of correlations at the boundary, which was predicted by Forrester and Jancovici in 1996, in the context of elliptic Ginibre ensembles.

math-ph↗
arXiv2023

Exponential moments for disk counting statistics at the hard edge of random normal matrices

We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let $n$ be the number of points. We focus on two regimes: (a) the ``hard edge regime" where all disk boundaries are at a distance of order $\frac{1}{n}$ from the hard wall, and (b) the ``semi-hard edge regime" where all disk boundaries are at a distance of order $\frac{1}{\sqrt{n}}$ from the hard wall. As $n \to + \infty$, we prove that the moment generating function enjoys asymptotics of the form \begin{align*} & \exp \bigg(C_{1}n + C_{2}\ln n + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})\bigg), & & \mbox{for the hard edge}, \\ & \exp \bigg(C_{1}n + C_{2}\sqrt{n} \hspace{0.12cm} + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}\bigg(\frac{(\ln n)^{4}}{n}\bigg)\bigg), & & \mbox{for the semi-hard edge}. \end{align*} In both cases, we determine the constants $C_{1},\dots,C_{4}$ explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the ``bulk", ``soft edge" and ``semi-hard edge" regimes, the second and higher order cumulants of the disk counting function in the ``hard edge" regime are proportional to $n$ and not to $\sqrt{n}$.

math-ph↗
arXiv2023

Almost-Hermitian random matrices and bandlimited point processes

We study the distribution of eigenvalues of almost-Hermitian random matrices associated with the classical Gaussian and Laguerre unitary ensembles. In the almost-Hermitian setting, which was pioneered by Fyodorov, Khoruzhenko and Sommers in the case of GUE, the eigenvalues are not confined to the real axis, but instead have imaginary parts which vary within a narrow ``band'' about the real line, of height proportional to $\tfrac 1 N$, where $N$ denotes the size of the matrices. We study vertical cross-sections of the 1-point density as well as microscopic scaling limits, and we compare with other results which have appeared in the literature in recent years. Our approach uses Ward's equation and a property which we call ``cross-section convergence'', which relates the large-$N$ limit of the cross-sections of the density of eigenvalues with the equilibrium density for the corresponding Hermitian ensemble: the semi-circle law for GUE and the Marchenko-Pastur law for LUE. As an application of our approach, we prove the bulk universality of the almost-circular ensembles.

math.PR↗
arXiv2023

Eigenvalues of truncated unitary matrices: disk counting statistics

Let $T$ be an $n\times n$ truncation of an $(n+α)\times (n+α)$ Haar distributed unitary matrix. We consider the disk counting statistics of the eigenvalues of $T$. We prove that as $n\to + \infty$ with $α$ fixed, the associated moment generating function enjoys asymptotics of the form \begin{align*} \exp \big( C_{1} n + C_{2} + o(1) \big), \end{align*} where the constants $C_{1}$ and $C_{2}$ are given in terms of the incomplete Gamma function. Our proof uses the uniform asymptotics of the incomplete Beta function.

math-ph↗
arXiv2022

Disk counting statistics near hard edges of random normal matrices: the multi-component regime

We consider a two-dimensional point process whose points are separated into two disjoint components by a hard wall, and study the multivariate moment generating function of the corresponding disk counting statistics. We investigate the ``hard edge regime" where all disk boundaries are a distance of order $\frac{1}{n}$ away from the hard wall, where $n$ is the number of points. We prove that as $n \to + \infty$, the asymptotics of the moment generating function are of the form \begin{align*} & \exp \bigg(C_{1}n + C_{2}\ln n + C_{3} + \mathcal{F}_{n} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})\bigg), \end{align*} and we determine the constants $C_{1},\dots,C_{4}$ explicitly. The oscillatory term $\mathcal{F}_{n}$ is of order $1$ and is given in terms of the Jacobi theta function. Our theorems allow us to derive various precise results on the disk counting function. For example, we prove that the asymptotic fluctuations of the number of points in one component are of order $1$ and are given by an oscillatory discrete Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by the Weierstrass $\wp$-function.

math-ph↗
arXiv2022

The planar low temperature Coulomb gas: separation and equidistribution

We consider planar Coulomb systems consisting of a large number $n$ of repelling point charges in the low temperature regime, where the inverse temperature $β$ grows at least logarithmically in $n$ as $n \longrightarrow \infty$, i.e., $β\gtrsim \log n$. Under suitable conditions on an external potential we prove results to the effect that the gas is with high probability uniformly separated and equidistributed with respect to the corresponding equilibrium measure (in the given external field). Our results generalize earlier results about Fekete configurations, i.e., the case $β=\infty$. There are also several auxiliary results which could be of independent interest. For example, our method of proof of equidistribution (a variant of "Landau's method") works for general families of configurations which are uniformly separated and which satisfy certain sampling and interpolation inequalities.

math.PR↗
arXiv2021

The random normal matrix model: insertion of a point charge

In this article, we study microscopic properties of a two-dimensional eigenvalue ensemble near a conical singularity arising from insertion of a point charge in the bulk of the support of eigenvalues. In particular, we characterize all rotationally symmetric scaling limits ('Mittag-Leffler fields') and obtain universality of them when the underlying potential is algebraic. Applications include a result on the asymptotic distribution of $\log|p_n(ζ)|$ where $p_n$ is the characteristic polynomial of an $n$:th order random normal matrix.

math-ph↗
arXiv2021

On the Uniqueness Problem for Quadrature Domains

We study questions of existence and uniqueness of quadrature domains using computational tools from real algebraic geometry. These problems are transformed into questions about the number of solutions to an associated real semi-algebraic system, which is analyzed using the method of real comprehensive triangular decomposition.

math.CV↗
arXiv2021

A localization theorem for the planar Coulomb gas in an external field

We examine a Coulomb gas consisting of $n$ identical repelling point charges at an arbitrary inverse temperature $β$, subjected to a suitable external field. We prove that the gas is effectively localized to a small neighbourhood of the "droplet" -- the support of the equilibrium measure determined by the external field. More precisely, we prove that the distance between the droplet and the vacuum is with very high probability at most proportional to $$\sqrt{\dfrac {\log n}{βn}}.$$ This order of magnitude is known to be "tight" when $β=1$ and the external field is radially symmetric. In addition, we prove estimates for the one-point function in a neighbourhood of the droplet, proving in particular a fast uniform decay as one moves beyond a distance roughly of the order $\sqrt{\frac {\log n}{βn}}$ from the droplet.

math.PR↗