arXiv subjects

Louis Gass

Publications and source records attributed to Louis Gass.

9 recordsLinked to original sources

arXiv2026

Scars in random waves and the FGF 1/2 universality class

We study the large-domain asymptotics of geometric observables in Berry's random wave model on $\mathbb{R}^d$. We show that, in sharp contrast with the behavior of stationary random fields with absolutely continuous spectral measures, any observable whose fluctuations are asymptotically fully correlated with its second Wiener chaos projection belongs to a common universality class governed by a fractional Gaussian field with Hurst index $H=(1-d)/2$. This class also includes the classical stationary Poisson line process in $\mathbb{R}^d$. Our findings show that suitable raw observables of Berry's random wave (such as critical point counts or non-nodal level set volumes) have large-domain fluctuations that become arbitrarily close -- in the sense of random tempered distributions -- to those generated by a (possibly noisy) Poisson line process. This probabilistic approximation provides evidence that the large-scale filamentary patterns observed in numerical simulations of random waves -- often referred to as "scars" or "scarlets" following the numerical investigations of Heller, O'Connor and Gehlen (1987)-- may admit a natural probabilistic interpretation. In the second part of our work, we characterize the scaling limit -- in a distributional sense -- of suitable quadratic transformations of the Radon--Fourier coefficients associated with a large class of stationary fields. We show that random waves are characterized by the property that such a scaling limit is a generalized random field obtained by composing white noise on the affine Grassmannian of lines with a dimension-dependent deterministic operator. As an application of our main results, we derive explicit conditions ensuring that quadratic functionals of pullback monochromatic waves on compact Riemannian manifolds exhibit distributional limits in the fractional Gaussian universality class described above.

math.PR↗

arXiv2025

Tightness of Stationary Nodal Measures

We study the rescaled nodal volume field $\xi_R$ associated with a smooth, stationary Gaussian field on $[0,R]^d$, whose covariance satisfies adequate integrability conditions. Our main theorem shows that, as $R \to \infty$, the process $\xi_R$ converges in distribution, in an appropriate space of c\`adl\`ag mappings, to a standard Brownian sheet. The proof relies on a recent finite-dimensional CLT by Ancona, Gass, Letendre, and Stecconi (2025), as well as on a multidimensional Kolmogorov--Chentsov criterion for tightness due to Bickel and Wichura (1971). The application of the latter requires new moment estimates that are of independent interest. Our results stand in sharp contrast with Berry's random wave model, where the required integrability conditions fail and the question of tightness remains open.

math.PR↗

arXiv2025

Universal Cancellations in Uniform Random Waves

A vast literature over the past fifteen years has been devoted to the study of the geometric properties of Gaussian random waves. In this work, we investigate the geometric behavior of \emph{uniform random waves}, a much less studied non-Gaussian model in which the $L^2$ norm is constrained to be exactly equal to one in every realization (a normalization that is natural from the standpoint of quantum mechanics). We show that this norm-constrained formulation has deep consequences for the universality of the so-called \emph{Berry's cancellation phenomenon}, as well as for novel high-frequency asymptotic variance estimates. These effects manifest themselves in both local geometric functionals, such as the Lipschitz--Killing curvatures, and global ones, such as the number of connected components above a fixed threshold. A key byproduct of our analysis is a new explicit relation between Hermite expansions and spherical harmonic decompositions for $0$-homogeneous functionals of Gaussian vectors, which enables a systematic chaos-based analysis of non-Gaussian random waves.

math.PR↗

arXiv2025

Zeros and critical points of Gaussian fields: cumulants asymptotics and limit theorems

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 $(d-k)$-dimensional volume of $f^{-1}(0) \cap R\mathcal{B}$. When $k=1$, we obtain similar asymptotics for the number of critical points of $f$ in $R\mathcal{B}$. Our main hypotheses are some regularity and non-degeneracy of the field, as well as mild integrability conditions on the first derivatives of its covariance kernel. As corollaries of these cumulants estimates, we deduce a strong Law of Large Numbers and a Central Limit Theorem for the nodal volume (resp.~the number of critical points) of a regular and non-degenerate enough field whose covariance decays fast enough at infinity. Our results hold more generally for a one-parameter family $(f_R)$ of Gaussian fields admitting a stationary local scaling limit as $R \to \infty$, for example Kostlan polynomials in the large degree limit. They also hold for the random measures of integration over the vanishing locus of $f_R$ as $R \to +\infty$.

math.PR↗

arXiv2025

Spectral Criteria for the Asymptotics of Local Functionals of Gaussian Fields and Their Application to Nodal Volumes and Critical Points

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 function, without requiring integrability or isotropy. It offers a simple and robust framework for analyzing variance asymptotics in such models. We apply this approach to the study of the nodal volume and the number of critical points of a Gaussian field, proving the positivity of the limiting variance under mild conditions on the covariance function. Additionally, we examine the asymptotics of nodal volume and critical points of Euclidean random waves, deriving the central limit theorem through an analysis of the second and fourth chaotic components. As a byproduct, we unify and generalize many existing results on the volume of intersections of random waves and their critical points, bypassing the need for traditional, intricate variance computations. Our findings shed new light on the second-chaos cancellation phenomenon from a spectral perspective and can be extended to any local, possibly singular, functional of Gaussian fields.

math.PR↗

arXiv2023

The number of critical points of a Gaussian field: finiteness of moments

Let $f$ be a Gaussian random field on $\mathbb{R}^d$ and let $X$ be the number of critical points of $f$ contained in a compact subset. A long-standing conjecture is that, under mild regularity and non-degeneracy conditions on $f$, the random variable $X$ has finite moments. So far, this has been established only for moments of order lower than three. In this paper, we prove the conjecture. Precisely, we show that $X$ has finite moment of order $p$, as soon as, at any given point, the Taylor polynomial of order $p$ of $f$ is non-degenerate. We present a simple and general approach that is not specific to critical points and we provide various applications. In particular, we show the finiteness of moments of the nodal volumes and the number of critical points of a large class of smooth, or holomorphic, Gaussian fields, including the Bargmann-Fock ensemble.

math.PR↗

arXiv2021

Cumulants asymptotics for the zeros counting measure of real Gaussian processes

We compute the exact asymptotics for the cumulants of linear statistics associated with the zeros counting measure of a large class of real Gaussian processes. Precisely, we show that if the underlying covariance function is regular and square integrable, the cumulants of order higher than two of these statistics asymptotically vanish. This result implies in particular that the number of zeros of such processes satisfies a central limit theorem. Our methods refines the recent approach by T. Letendre and M. Ancona and allows us to prove a stronger quantitative asymptotics, under weaker hypotheses on the underlying process. The proof exploits in particular the elegant interplay between the combinatorial structures of cumulants and factorial moments in order to simplify the determination of the asymptotics of nodal observables. The class of processes addressed by our main theorem englobes as motivating examples random Gaussian trigonometric polynomials, random orthogonal polynomials and the universal Gaussian process with sinc kernel on the real line, for which the asymptotics of higher moments of the number of zeros were so far only conjectured.

math.PR↗

arXiv2021

Variance of the number of zeros of dependent Gaussian trigonometric polynomials

We compute the variance asymptotics for the number of real zeros of trigonometric polynomials with random dependent Gaussian coefficients and show that under mild conditions, the asymptotic behavior is the same as in the independent framework. In fact our proof goes beyond this framework and makes explicit the variance asymptotics of various models of random Gaussian polynomials. Though we use the Kac--Rice formula, we do not use the explicit closed formula for the second moment of the number of zeros, but we rather rely on intrinsic properties of the Kac--Rice density.

math.PR↗

arXiv2020

Almost sure asymptotics for Riemannian random waves

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 large band and monochromatic models, the process properly rescaled and evaluated at an independently and uniformly chosen point $X$ on the manifold, converges in distribution under the sole randomness of $X$ towards an universal Gaussian field as the frequency tends to infinity. This result extends the celebrated central limit Theorem of Salem--Zygmund for trigonometric polynomials series to the more general framework of compact Riemannian manifolds. We then deduce from the above convergence the almost-sure asymptotics of the nodal volume associated with the random wave. To the best of our knowledge, in the real Riemannian case, these asymptotics were only known in expectation and not in the almost sure sense due to the lack of sufficiently accurate variance estimates. This in particular addresses a question of S. Zelditch regarding the almost sure equidistribution of nodal volume.

math.PR↗