Random nodal domains

More titles & abstracts to come...


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.


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.


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.


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.


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.


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.