Random nodal domains


Michele Ancona (Université de Nice)

Title: Metric and spectral aspects of random plane curves

Abstract: A (complex) plane curve is the zero locus in CP2 of a homogeneous complex polynomial in three variables. Any plane curve is endowed with a Riemannian metric induced by the ambient Fubini-Study metric of the complex projective plane. We give probabilistic lower bounds on some metric and spectral quantities (such as the systole or the spectral gap) of the plane curves when these are chosen randomly in the Fubini-Study ensemble. This is a joint work with Damien Gayet.


Raphaël Butez (Université de Lille)

Title: Zeros of random orthogonal polynomials, outliers and open questions

Abstract: In this talk I will discuss the emergence of Bergman processes and Szego random analytic functions for the extremal zeros of random polynomials. In the recent years, the Bergman process appeared in many models to describe the outlier process of eigenvalues of random matrices or zeros of random polynomials. I will present results for zeros of random polynomials obtained in the last years with David Garcia Zelada, Alon Nishry and Aron Wennman and I will present some open questions related to these objects.


Dmitry Belyaev (University of Oxford)

Title: Variance estimates and CLT for the number of level sets

Abstract: The number of nodal domains or, more generally, level/excursion sets has been extensively studied in the last 10-15 years. The first-order asymptotic i.e the law of large numbers has been obtained by Nazarov and Sodin under extremely mild assumptions. In this talk, I will discuss recent progress in the understanding of the variance and central limit theorems for a large class of stationary Gaussian fields. The talk is based on a series of papers by M. McAuley and S. Muirhead and myself.


Valentina Cammarota (Sapienza University of Rome )

Title: Nodal Deficiency of Random Spherical Harmonics in Presence of Boundary

Abstract: We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere satisfying the Dirichlet boundary conditions along the equator. For this model we find a precise asymptotic law for the corresponding zero density functions, in both short range (around the boundary) and long range (far away from the boundary) regimes. As a corollary, we were able to find a logarithmic negative bias for the total nodal length of this ensemble relatively to the rotation invariant model of random spherical harmonics. Join work with Domenico Marinucci and Igor Wigman.


Federico Dalmao (Universidad de la República Uruguay)

Title: On the number of roots of Sturm-Liouville random sums

Abstract: In this talk we consider the number of roots of linear combinations of a system of n orthogonal eigenfunctions of a Sturm-Liouville initial value problem with i.i.d. standard Gaussian coefficients. We prove that its distribution inherits the asymptotic behavior of the number of roots of Quall’s random trigonometric polynomials. This result can be thought of as a robustness result for the central limit theorem for the number of roots of Quall’s random trigonometric polynomials in the sense that small uniform perturbations of sines and cosines do not change the limit distribution.


Louis Gass (Université du Luxembourg)

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

Abstract: In this talk, I will present a proof of the finiteness of the moments of the number of critical points of a smooth non-degenerate Gaussian field. The proof is based on the Kac-Rice formula, that gives an integral expression for the moments of the number of critical points in a domain.

We rely on the observation that the Kac--Rice densities of two smooth non-degenerate Gaussian fields have the same integrability properties. It thus suffices to prove the finiteness of moments for one particular random field. The choice of a random polynomial field allows us to conclude by Bezout's theorem.

In the end, I will discuss possible extensions to the non-Gaussian framework and to some degenerate fields such as the Berry Random waves model. Joint work with Michele Stecconi.


Maxime Ingremeau (Université de Nice)

Title: Semiclassical evolution of Lagrangian states into random waves

Abstract: In 1977, Berry conjectured that eigenfunctions of the Laplacian on manifolds of negative curvature behave, in the high-energy (or semiclassical) limit, as a random superposition of plane waves. This conjecture, central in quantum chaos, is still completely open. In this talk, we will consider a much simpler situation. On a manifold of negative curvature, we will consider a Lagrangian state (the generalization of a plane wave on a manifold). We aim to show that, when evolved during a long time by the Schrödinger equation, these functions do behave, in the semiclassical limit, as a random superposition of plane waves. We will show such a result in two situations: when the phase of the Lagrangian state is generic, and when the Laplace-Beltrami operator is perturbed by a small generic potential. This is based on joint work with Alejandro Rivera, and with Martin Vogel.


Raphaël Lachieze-Rey (Université Paris Cité)

Title: Replication of arithmetic random waves

Abstract: Motivated by the full correlation phenomenon of arithmetic random waves on the torus, unveiled by Benatar, Marinucci, and Wigman, we investigate the smallest pseudo-period of this random Gaussian field, that is the smallest scale to which the random field a.s. replicates almost identically. We identify a pseudo-period slightly above Planck scale, of the order $n^{-1/2}l_n$ where $l_n$'s decay is between the logarithmic and polynomial scales. It turns out that the important feature is that the number of eigenfunctions is logarithmically small with respect to the volume of the torus (on a density one subsequence), and the result extends to a more general model. We also give a heuristic explanation for the existence of an even smaller pseudo period, that is made rigourous for a related toy-model. Joint work with Loïc Thomassey.


Thomas Letendre (Université Paris-Saclay)

Title: On the (non)-singularity of Kac-Rice densities in dimension $1$

Abstract: Let $f$ be a smooth non-degenerate Gaussian field on $\mathbb{R}$. The Kac--Rice formula allows to write the $p$-th moment of the linear statistics associated with the zero set of $f$ in terms of a function $\rho_p:\mathbb{R}^p \setminus \Delta_p \to \mathbb{R}$, where $\Delta_p$ stands for the large diagonal in $\mathbb{R}^p$. This function $\rho_p$ is continuous outside of $\Delta_p$ but its standard expression is singular along $\Delta_p$, and an important step in moments computations is to prove its local integrability along the diagonal. In a joint work with Michele Ancona we proved that $\rho_p$ admits a continuous extension to the whole of $\mathbb{R}^p$. The proof uses divided differences, which might seem like a computational trick. In this talk, I will present a geometric perspective on this matter which yields a more conceptual proof of the continuity of $\rho_p$. If time permits, I will discuss why the situation is much more complicated for fields on $\mathbb{R}^n$ with $n \geq 2$.


Domenico Marinucci (University of Rome "Tor Vergata")

Title: Spin Random Fields: a Challenge for Mathematicians

Abstract: in this talk, we will try to illustrate a number of issues, open problems and conjectures (and some mathematical results) arising in the investigation of the geometry and topology of spin random fields, as motivated by Cosmological applications (Cosmic Microwave Background polarisation). We will discuss alternative approaches, such as "lifting" spin random fields to scalar-valued fields on the group of rotations SO(3) or defining directly excursion sets on sections of the fiber bundles, and we will present some results on Lipschitz-Killing Curvatures and Betti numbers in these two frameworks. We will also show how in the asymptotic regimes where the spin parameter diverges to infinity one can recover in the scaling limit both (some complex form of) standard random spherical harmonics/Berry's random waves and Bargmann-Fock model/Kostlan polynomials, together with an (apparently new) class of random fields which seems to interpolate between the two.

The talk is based also on some recent papers (arXiv 2022-2023) involving Maurizia Rossi, Michele Stecconi and Antonio Lerario (on the mathematical side) and Alessandro Carones and Javier Carron (on the cosmological side).


Lakshmi Priya M. E (Tel-Aviv University)

Title: Almost sharp lower bound for the nodal volume of harmonic functions

Abstract: In this talk, I will discuss the relation between the growth of harmonic functions and their nodal volume. One way to quantify the growth of a harmonic function $u$ in the ball $B(0, 1) \subset \mathbb R^n$ is via the doubling index $N$, defined by $$ \sup_{B(0,1)} |u| = 2^N \sup_{B(0,1/2)} |u|. $$ I will present a result, joint with A. Logunov and A. Sartori, where we prove an almost sharp result, namely: $$H^{n−1}({u=0}∩B(0,2))\gtrsim_{n,\varepsilon} N^{1−\varepsilon}.$$


Stephen Muirhead (University of Melbourne)

Title: Box-crossing estimates for the nodal sets of planar Gaussian fields

Abstract: A planar Gaussian field satisfies the `nodal box-crossing estimates' if the probability that the nodal set `crosses' a box is uniformly bounded away from 0 and 1 in the scale; among other consequences, these give a strong indication that the nodal set possesses a non-degenerate scaling limit. They are expected to hold under quite general conditions, but are yet to be proven for many important examples, such long-range correlated fields and monochromatic waves.

We establish the box-crossing estimates in two cases: (i) for short-range and positively correlated fields; and (ii) for the Cauchy fields with parameter $\alpha > 0$. The first generalises a result from [M. & Vanneuville '20] which required stronger conditions. The second provides the first examples of long-range correlated fields which are known to satisfy the estimates.

Based on the works arXiv:2206.10724 and arXiv:2303.06309.


Hoi Nguyen (Ohio State University)

Title: Concentration of the number of real roots of random polynomials

Abstract: We will discuss a simple framework that helps establish strong concentration of the number of real roots for a broad family of random polynomials. The results work for ensembles with both gaussian and non-gaussian coefficients.


Oanh Nguyen (Brown University)

Title: Hole radii for the Kac polynomials

Abstract: The Kac polynomial is one of the most studied models of random polynomials. It has the form $$f_n(x) = \sum_{i=0}^{n} \xi_i x^i.$$ It is known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. In a beautiful work, Michelen showed that the holes at $\pm 1$ are of order $1/n$. We show that in fact, all the hole radii are of the same order.

Joint work with Hoi Nguyen.


Ivan Nourdin (Université du Luxembourg)

Title: Total variation bound for Hadwiger's functional using Stein's method

Abstract: Let $K$ be a convex body in $\mathbb{R}^d$. Let $X_K$ be a $d$-dimensional random vector distributed according to the Hadwiger-Wills density $\mu_K$ associated with $K$, defined as $\mu_K(x)=ce^{-\pi {\rm dist}^2(x,K)}$, $x\in \mathbb{R}^d$. Finally, let the information content $H_K$ be defined as $H_K={\rm dist}^2(X_K,K)$.

In this talk, we will study the fluctuations of $H_K$ around its expectation as the dimension $d$ go to infinity. Stein's method plays a crucial role in our analysis. This is joint work with Valentin Garino.


Igor Pritsker (Oklahoma State University)

Title: Uniform distribution of roots for random polynomials

Abstract: Zeros of Kac polynomials are asymptotically uniformly distributed near the unit circumference with probablility one, under very mild assumptions on random coefficients. Similar equidistribution results hold for general random polynomials spanned by various deterministic bases. The case of i.i.d. coefficients is the most studied and well understood. We consider dependent coefficients, and give a description of the almost sure weak limits for the zero counting measures of random polynomials. In addition, we quantify this weak convergence via the expected discrepancy between the zero counting measures and the limiting measure.


Maurizia Rossi (University of Milano-Bicocca)

Title: Quasi-critical fluctuations for 2D directed polymers

Abstract: We study the 2d directed polymer in random environment in a novel quasi-critical regime, which interpolates between the much studied sub-critical and critical-regimes. We prove Edwards-Wilkinson fluctuations throughout the quasi-critical regime: the diffusively rescaled partition functions are asymptotically Gaussian, under a rescaling which diverges arbitrarily slowly as criticality is approached. A key challenge is the lack of hypercontractivity, which we overcome deriving new sharp moment estimates. This talk is based on a joint work with Francesco Caravenna (Università di Milano-Bicocca) and Francesca Cottini (Université du Luxembourg & National University of Singapore).


Andrea Sartori (Tel-Aviv University)

Title: On the universality of the Nazarov-Sodin constant

Abstract: In the past decade, a central topic in the study of random functions has been their zero sets and, in particular, their dependence on the underlying distribution. In a significant contribution to the field, Nazarov and Sodin computed the expected number of nodal domains for random Gaussian spherical harmonic. In this talk, I will present new findings showing that this expectation is universal, provided that the underlying distribution has a finite second moment.


Michele Stecconi (Université du Luxembourg)

Title: Regularity of the nodal volume

Abstract: Consider the nodal volume of a non-degenerate (in a sense to be specified) Gaussian random field defined on a compact Riemannian manifold of dimension at least 2. We prove that the law of such random variable has an absolutely continuous component, as a direct consequence of its Fréchet differentiability. Moreover, we give an explicit formula for the Malliavin derivative. The latter property of the law of the nodal volume had already been established by Angst and Poly (2020) for stationary fields on the $d-$torus and on $\mathbb R^d$, in dimension $d$ greater than 3, via Malliavin calculus. In this work, the two dimensional case remained open, in particular, the Malliavin differentiability of the nodal length. We prove that in general the nodal volume admits an $L^2$ Malliavin derivative, for $d$ greater than 3 and that in the two dimensional case, this is false, but the Malliavin derivative still exists in $L^1$. (A joint work with Giovanni Peccati.)


Anna Paola Todino (Sapienza University of Rome)

Title: Spherical Poisson Waves

Abstract: We discuss the universality of Gaussian behaviour for spherical Laplace eigenfunctions introducing a model of Poisson random waves in S^2. We study Quantitative Central Limit Theorems when both the rate of the Poisson process and the frequency of the waves diverge to infinity. We consider finite-dimensional distributions, harmonic coefficients and convergence in law in functional spaces, and we investigate carefully the interplay between the rates of divergence of eigenvalues and Poisson governing measures. Based on a joint work with Solesne Bourguin, Claudio Durastanti and Domenico Marinucci.


Anna Vidotto (University of Naples Federico II)

Title: Functional Convergence of Berry's Nodal Lengths

Abstract: In this talk, we consider Berry’s random planar wave model (1977), and prove spatial functional limit theorems - in the high-energy limit - for discretized and truncated versions of the random field obtained by restricting its nodal length to rectangular domains. We will see that our results are crucially based on a detailed study of the projection of nodal lengths onto the so-called second Wiener chaos, whose high-energy fluctuations are given by a Gaussian total disorder field indexed by polygonal curves. Such an exact characterization is then combined with moment estimates for suprema of stationary Gaussian random fields, and with a tightness criterion by Davydov and Zitikis (2005). The talk is based on a joint work with M. Notarnicola and G. Peccati.


Igor Wigman (King's College London)

Title : Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus.

Abstract: This talk is based on a joint work with Zeev Rudnick. We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. We show that in this same limit, the energy variance for a typical surface is close to the GOE result, a feature called "ergodicity" in the random matrix theory literature. Our analysis rests on an application of the Selberg trace formula, and the convergence of the length spectrum of random hyperbolic surfaces to a Poisson point process, established in essence by Mirzakhani-Petri.


Oren Yakir (Tel Aviv University)

Title: Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere

Abstract: Smale's 7th Problem asks for an efficient algorithm to generate a configuration of n points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltran and Shub considered the roots of the random elliptic polynomial of degree $n$ and computed the expected logarithmic energy. We study fluctuations of the logarithmic energy of this random configuration and prove a central limit theorem. Our analysis shows that the energy is well concentrated around its mean on the scale of $\sqrt{n}$.

Based on a joint work with Marcus Michelen.