Related papers: Critical point asymptotics for Gaussian random wav…

We study monochromatic random waves on $\mathbb{R}^n$ defined by Gaussian variables whose variances tend to zero sufficiently fast. This has the effect that the Fourier transform of the monochromatic wave is an absolutely continuous measure…

Let $f:\mathbb{R}^d \to \mathbb{R}^k$ be a smooth centered stationary Gaussian field and $\mathcal{B} \subset \mathbb{R}^d$ be a bounded Borel set. In this paper, we determine the asymptotics as $R \to \infty$ of all the cumulants of the…

In large dimension, we study the asymptotic behavior of the mean number of critical points with index k below a level u for an isotropic centered Gaussian random field defined on a family of subsets of $R^d$ depending on d. We prove the…

We establish a general criterion for the positivity of the variance of a chaotic component of local functionals of stationary vector-valued Gaussian fields. This criterion is formulated in terms of the spectral properties of the covariance…

This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves $\phi_\lambda$ of frequency $\lambda$ on a compact, smooth, Riemannian manifold $(M,g)$ as $\lambda \rightarrow \infty$. We prove that…

We study the high-frequency Fourier asymptotics of imaginary Gaussian multiplicative chaos on the unit circle, a complex-valued random distribution formally given by $\mathrm M_{\mathrm i\beta}=\exp(\mathrm i\beta X)$, where $X$ is a…

In this paper we study the asymptotic behavior of the angular bispectrum of spherical random fields. Here, the asymptotic theory is developed in the framework of fixed-radius fields, which are observed with increasing resolution as the…

We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for…

We describe the asymptotic behavior of critical points of $\int_{\Omega} [(1/2)|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon\to 0$. Here, $W$ is a Ginzburg-Landau type potential, vanishing on a simple closed curve $\Gamma$. Unlike the…

We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere . We show that the asymptotic behaviour of these functionals as…

The Gauss circle problem asks for an approximation to the number of lattice points of $\mathbb{Z}^2$ contained in $B_r$, the disk of radius $r$ centered at the origin. Upper, lower, and average bounds have been established for this…

We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and…

If $p:\mathbb{C} \to \mathbb{C}$ is a non-constant polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when $p$ is a random polynomial of degree $n$ with…

We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk of the spectrum and near the spectral edge. This…

We study inelastic resonant scattering of a Gaussian wave packet with the parameters close to a zero of the complex scattering coefficient. We demonstrate, both theoretically and experimentally, that such near-zero scattering can result in…

Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for…

In the past decades, the central limit theorem (CLT) has been generalized to non-Euclidean data spaces. Some years ago, it was found that for some random variables on the circle, the sample Fr\'echet mean fluctuates around the population…

We study here the random fluctuations in the number of critical points with values in an interval $I\subset \mathbb{R}$ for Gaussian spherical eigenfunctions $\left\{f_{\ell }\right\} $, in the high energy regime where $\ell \rightarrow…

The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree $N$ in an open set $U \subset C$ as the degree $N \to \infty$, and more generally for the zeros of…

Core partitions have attracted much attention since Anderson's work (2002) on the number of $(s,t)$-core partitions for coprime $s,t$. Recently, there has been a growing interest in studying the limiting distributions of the sizes of random…