Conference - Geometry of random nodal domains



Michele Ancona (University of Tel Aviv)

Title: Moments of the number of roots of Kostlan polynomials

Abstract: In this talk, I will explain how to calculate the moments of the number of roots of Kostlan polynomials. This is a joint work with Thomas Letendre. 

Jürgen Angst (Université de Rennes 1)

Title: Almost sure asymptotics for the number of zeros of random trigonometric polynomials.

Abstract: We will explain how variations on the celebrated almost sure CLT by Salem-Zygmund on random trigonometric polynomials allow to deduce the almost sure asymptotics for their number of zeros. This approach is robust enough to be implemented in both independent and dependent frameworks and more generally for random Laplace eigenfunctions on manifolds.  This is joint work with G. Poly and T. Pautrel.


Jean-Marc Azaïs (IMT, Université de Toulouse)

Title: Studying the winding number of a Gaussian process: the real method.

Abstract: We consider the winding number of planar stationary Gaussian processes defined on the line. Our model allows general dependence of the coordinates of the process and non-differentiability of one of them. Furthermore, we consider as an example an approximation to the winding number of a process which coordinates are both non-differentiable. Under mild conditions, we obtain the asymptotic variance and the Central Limit Theorem for the number of winding turns as the underlying time interval grows to $[0,\infty)$. In the asymptotic regime, our discrete approach is equivalent to the continuous one studied previously in the literature and our main result extends the existing ones.

Raphaël Butez (Université de Lille)

Title: Outliers for weakly confining Coulomb gases and zeros of random polynomials

Abstract:   This talk is based on a joint work with David Garcia-Zelada, Alon Nishry and Aron Wennman. We study two models of particle systems: on one hand the zeros of random polynomials with Gaussian coefficients as introduced by Zeitouni and Zelditch, and on the other hand, a determinantal Coulomb gas model which is inspired from zeros of random polynomials. For both models, the behavior of the empirical measures is well understood, and we are interested in the existence of particles outside of the support of the equilibrium measure: the outliers. For both models, we establish that the point process of the outliers in a simply connected domain converge towards the Bergman point process of this domain. The limiting point processes are universal, as they don't depend on the choice of the parameters of the model, and show some conformal equivariance properties, as the limiting objects can be mapped one to another by conformal mappings.

Valentina Cammarota (Sapienza University of Rome)

Title: On the correlation between critical points and the critical values for random spherical harmonics

Abstract: We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval $I \subset \mathbb R$. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random $L^2-$norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics. Joint work with Anna Paola Todino.

Laure Coutin (IMT, Université de Toulouse)

Title: Donsker theorem in Wasserstein 1 distance and rate of convergence for the number of zeros of random trigonometric polynomials

Abstract:  This talk is based on two joint works with L. Decreusefond and L. Peralta. We compute the Wasserstein 1 (or Kolmogorov-Rubinstein) distance between a random walk in $\mathbb R^d$ and the Brownian motion. The proof is based on a new estimate of the  Lipschitz modulus of the solution of the Stein's equation. Then, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process, whose covariance function is given by the $\sin_c$ function.

Federico Dalmao (Universitad de la Republica, Salto)

Title: On the number of roots of random invariant homogeneous polynomials.

Abstract: the number of roots of random polynomials have been intensively studied for a long time. In the case of systems of polynomial equations the first important results can be traced back to the nineties when Kostlan, Shub and Smale computed the expectation of the number of roots of some random polynomial systems with invariant distributions. Nowadays this is a very active field. In this talk we are concerned with the variance and the asymptotic distribution of the number of roots of invariant polynomial systems. Among other examples, we consider Kostlan--Shub--Smale, random spherical harmonics and Real Fubini Study systems.

Yohann De Castro (Ecole Centrale de Lyon)

Title: Maximum of Gaussian fields on Stiefel manifolds

Abstract: TBA

Céline Delmas (INRAE, Toulouse)

Title: Critical points of isotropic Gaussian fields on the sphere

Abstract:  Let $\mathcal{X}=\left\{X(t): t \in \mathcal{S}^{d} \subset \mathbb{R}^{d+1}\right\}$ be a real-valued centered isotropic Gaussian field defined on the unit sphere $\mathcal{S}^{d} \subset \mathbb{R}^{d+1}$ whose covariance function denoted by $r(\langle s, t \rangle):=\mathrm{E}(X(s) X(t))$ is assumed to be $\mathcal{C}^{4}$. We set $\lambda_0:=r(1)$, $\lambda_2:=r'(1)$ and , $\lambda_4=3r''(1)+r(1)$. We then define
$C_u^k(X,\mathcal S^d):=\# \{t \in \mathcal{S}^d : X(t) \leq u, \nabla X(t)=0, i(\nabla^2 X(t))=k\}$ and $C_u(X,\mathcal S^d):=\#\{t\in {\mathcal{S}}^d : X(t)\leq u, \nabla X(t)=0 \}.$
Auffinger, Ben Arous, Cerny (2013) have obtained sharp asymptotics for $\mathrm{E}\left(\mathrm{Crt}_{\mathrm{u}}\left(\mathrm{X}, \mathcal{S}^{\mathrm{d}}\right)\right)$; logarithmic asymptotics for $\mathrm{E}\left(\mathrm{Crt}_{\mathrm{u}}^{\mathrm{k}}\left(\mathrm{X}, \mathcal{S}^{\mathrm{d}}\right)\right)$ and a layered structure for the lowest critical values of $X$ when the covariance function of $X$ is of the form $r(<s, t>)=\lambda_{0}\langle s, t \rangle^{p}$ with $p \geq 2$ ( $p$-spin spherical spin glasses models). This particular case of covariance functions corresponds to the condition $\lambda_{0} \lambda_{4}+2 \lambda_{0} \lambda_{2}-3 \lambda_{2}^{2}=0$ We extend their results to any covariance function such that $\lambda_{0} \lambda_{4}+2 \lambda_{0} \lambda_{2}-3 \lambda_{2}^{2} \geq 0 .$ We remark that there exist some random fields for which we do not have layered structure for the lowest critical values. We extend these results to planar isotropic Gaussian fields.

Vivek Dewan (Université Grenoble-Alpes)

Title: First passage percolation for Gaussian fields

Abstract: Given the recently uncovered similarity between the phase transitions in the discrete model of Bernoulli percolation and the continuous one given by excursion sets of Gaussian fields, it is natural to try and extend these analogies to the random pseudometric model of first passage percolation (FPP). We will define the FPP model in both settings, state Kesten's classical results in the discrete, and the corresponding ones we were able to establish in the continuous model in joint work with Damien Gayet. We will give brief comments on how the techniques differ between the two.

Louis Gass (Université de Rennes 1)

Title: Moments of the number of zeros of non-stationary Gaussian processes.

Abstract: I this talk I will compute the variance asymptotics for the number of real zeros of trigonometric polynomials with random dependent Gaussian coefficients, and show that the asymptotic behavior is the same as in the independent framework. The proof goes beyond this framework and makes explicit the variance asymptotics of various models of random Gaussian polynomials. At last, I will explain how to generalize this proof in order to recover the higher central moments asymptotics (ongoing work).

Damien Gayet (Université Grenoble-Alpes)

Title: Asymptotic topology of random nodal sets

Abstract: Let $f$ be a smooth random Gaussian field over the unit ball of $\mathbb R^n$. It is very natural to imagine that for a high level $u$, $\{f>u\}$ is mainly composed of small components homeomorphic to $n-$balls. I will explain that in average, this intuition is true. After recalling the historical background of this subject, and will present the ideas of the proof, which holds on (deterministic) Morse theory and a control of random critical points of given index.


Maxime Ingremeau (Université de Nice Côte d'Azur)

Title: Spectral asymptotics of large quantum graphs

Abstract: Since Weyl’s work a century ago, a lot of papers have studied the asymptotics of the eigenvalues of the Laplacian (in a domain or on a compact manifold) in the limit where the eigenvalues become large. Recently, an other kind of asymptotics has been of interest: consider a sequence of domains, whose size grow to infinity, and count the asymptotic number of eigenvalues of the Laplacian in these domains in a fixed spectral interval. In this talk, we will deal with this kind of asymptotics in the case of quantum graphs (also known as metric graphs), which are just some compact one-dimensional objects where Laplacians can be defined. If time allows it, I will mention analogous results for the scattering resonances of open quantum graphs. Part of this is joint work with N. Anantharaman, M. Sabri and B. Winn


Zakhar Kabluchko (Muenster Universität)

Title: Dynamics of zeroes under repeated differentiation

Abstract: Consider a random polynomial $P_n$ of degree $n$ whose roots are independent random variables sampled according to some probability distribution $\mu_0$ on the complex plane. It is natural to conjecture that, for a fixed $t\in [0,1)$ and as $n\to\infty$, the zeroes of the $[tn]$-th derivative of $P_n$ are asymptotically distributed according to some measure $\mu_t$ on the complex plane. We shall review known results in this direction, in particular the connection to free probability and PDE's. The talk is partially based on a joint work with Jeremy Hoskins (arXiv:2010.14320).

Raphael Lachieze-Rey (MAP 5, Université de Paris)

Title: Diophantine Gaussian excursions

Abstract: We give a general formula for the excursion variance of stationary Euclidean Gaussian fields, related to the properties of the random walk whose increment measure is the spectral measure. We apply this formula to spectral measures  with finite incommensurable support and give the variance magnitude in function of the diophantine properties of the atoms. It turns out that if the atoms are well approximated by rationals, then the variance is small, and more generally any reasonable asymptotic behaviour of the variance can be achieved with this model, from minimal surface order behaviour, to maximal quadratic order.

Antonio Lerario (SISSA, Trieste)

Title: Differential topology of gaussian random fields

Abstract:  In this seminar I will report on joint work with Michele Stecconi discussing a new approach to smooth gaussian random fields. In this approach, motivated by questions in random topology, we view a smooth gaussian field as a random variable in the space of smooth maps and exploit ideas from differential topology (e.g. Thom's transversality theorem) for the study of its structure.

Thomas Letendre (IMO, Université Paris-Saclay)

Title: How to resolve the singularities of higher order Kac--Rice densities (in dimension $1$)

Abstract: Let $Z$ be a point process obtained as the zero set of a smooth stationary Gaussian field on~$\mathbb{R}$. The distribution of $Z$ is characterized by its $k$-points functions $(\rho_k)_{k \geq 1}$, which also appear as densities in the Kac--Rice formulas for the moments of the number of points of $Z$ in an interval. The function $\rho_k$ is well-defined on the complement of the large diagonal in $\mathbb{R}^k$, but is \textit{a priori} singular along the diagonal. In this talk, we will explain how to resolve this singularity under mild assumptions on the field, using divided differences. This allows to quantify the short range repulsion and long range decorrelation between points of $Z$. This is a joint work with Michele Ancona.

Stephen Muirhead (University of Melbourne)

Title: Decay of subcritical connection probabilities for long-range correlated Gaussian fields

Abstract: It is well-known that the excursion sets of smooth stationary Gaussian fields undergo a phase transition at a critical level $\ell_c$, from a subcritical phase of bounded components to a supercritical phase in which there is (at least) one unbounded component. In this work we consider a quantitative aspect of the phase transition, namely the likelihood of finding large excursion sets in the subcritical phase. For fields with short-range correlations, subcritical excursion sets are known to be exponentially small. We consider fields with long-range regularly-varying correlations $K(x) \sim |x|^{-\alpha}$, $\alpha \in (0,d)$. Intuitively, long-range correlations might be thought to promote large excursion sets, and we establish how the radius of subcritical clusters depends on $\alpha$. If $\alpha>1$ the tail decay is exponential, whereas if $\alpha < 1$ the decay is stretched-exponential with exponent $\alpha$; in the latter case we also identify explicitly the leading-order constant and its dependence on the level $\ell$. The cases $\alpha = 1$ and $\alpha = 0$, which involve logarithmic corrections, are also considered. This result extends recent work on the Gaussian free field (which corresponds roughly to the case $\alpha = d-2$) to a more general class of smooth fields. The result is unconditional in the planar case, whereas in $d \ge 3$ it requires an assumption about the sharpness of the phase transition (believed to be true but not yet verified). Joint work with Franco Severo.

Oahn Nguyen (University of Illinois)

Title: The number of limit cycles bifurcating from a randomly perturbed center

Abstract:  We consider the average number of limit cycles that bifurcate from a randomly perturbed linear center where the perturbation consists of random (bivariate) polynomials with i.i.d. coefficients. We reduce this problem to the number of real roots of the random polynomial $$f(x) = \sum_{k=0}^{n} k^\rho \xi_k x^k$$ where the $\xi_k$ are independent with mean 0 and variance 1 and $\rho \le -1/2$ is a constant. In earlier work, Do, Vu, and myself established this number for $\rho > -1/2$ via the universality method which naturally breaks down for $\rho \le -1/2$. In this talk, we discuss the solution for the $\rho \le -1/2$. Joint work with Erik Lundberg.


Massimo Notarnicola (Université du Luxembourg)

Title: Geometric functionals of multiple Arithmetic Random Waves, Berry’s Cancellation and Wiener Chaos

Abstract: We study the probabilistic fluctuations of local geometric functionals associated with multi-dimensional Gaussian Laplace eigenfunctions on the three-torus as the eigenvalue diverges to infinity, namely (i) the volume of their zero sets and (ii) a generalized notion of the total variation. For the first quantity, we present a non-Central Limit Theorem, whose proof builds on Wiener chaos expansions and a more general cancellation phenomenon, also observed by M. Berry (2002) in the context of planar eigenfunctions. For the total variation, we present a Central Limit Theorem in the high-energy regime: In order to achieve this task, we make use of generalized Hermite polynomials with matrix argument introduced by A. James (1963), which we link to classical Wiener chaos structures.

Thibault Pautrel (Université de Rennes 1)

Title: Zeros of random trigonometric polynomials with dependent Gaussian coefficients

Abstract: We study the large degree asymptotics of the expected number of zeros of random trigonometric polynomials with dependent Gaussian coefficients. Quite surprisingly, we show that the universality of this asymptotics is not related to the rate of decorrelation of the coefficients but rather to the positivity of the absolutely continuous component of the associated spectral measure.

Giovanni Peccati (Université du Luxembourg)

Title: Local fluctuations of nodal volumes via coupling of Gaussian fields

Abstract: I will illustrate a strategy for proving central limit theorems for the nodal length of pullback monochromatic random waves - associated with compact Riemann surfaces - restricted to slowly growing domains. The crucial element of our proof is a novel strategy for coupling non-stationary Gaussian fields, which is based on some eigenvalue decay estimates for covariance kernels, partially extending results by Heinrich and Kühn (1984). Based on joint work with G. Dierickx, I. Nourdin, and M. Rossi.

Hung Viet Pham (Institute of Mathematics, Hanoi)

Title: Conjunction probability of smooth Gaussian fields

Abstract: In this talk, we firstly recall some facts about the distribution of the maximum of Gaussian field such as: Euler characteristic method, Rice method and the asymptotic formula for the case of non-convex index domain. Then we will present a relevant problem, so-called conjunction probability, founded by Worsley and Friston. We provide the asymptotic formula for the conjunction probability and compare it with the heuristic approximation given by the Euler characteristic method.

Ali Pirhadi (Georgia State University)

Title: Real zeros of random trigonometric polynomials with $\ell$-periodic coefficients

Abstract: The large degree asymptotics of the expected number of real zeros of a random trigonometric polynomial $$ T_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) + b_j \sin (j x), \ x \in (0,2\pi), $$ with i.i.d. real-valued standard Gaussian coefficients is known to be $ 2n / \sqrt{3} $. We consider quite a different and extreme setting on the set of the coefficients of $ T_n $ and show that a random trigonometric polynomial of degree $ n $ with $ \ell $-periodic coefficients is expected to have significantly more real zeros compared to the classical case with i.i.d. Gaussian coefficients.

Guillaume Poly (Université de Rennes 1)

Title: Non universality of Fluctuations in Salem-Zygmund CLT

Abstract: We shall study the fluctuations in the CLT theorem of Salem-Zygmund that was previously described by J. Angst and establish an underlying central limit Theorem under mild conditions on the random coefficients. For Gaussian coefficients, a classical and very efficient way consists in expanding over the Hermite basis the studied functional and establish normality of each projection. Here, based on some combinatoric arguments and universality results from noise sensitivity theory, we shall explain how to extend this methodology to the case of general non Gaussian coefficients and precisely force the use of Hermite expansion outside the sole scope of functionals of a Gaussian field. At the end, we will see that fluctuations are not universal and involve the fourth cumulant of the coefficients but also depend on the second Hermite-coefficient of the considered function. Joint work with J. Angst.

Igor Pritsker (Oklahoma State University)

Title: Real zeros of random orthogonal polynomials.

Abstract: We survey recent work on the real zeros of random linear combinations of orthogonal polynomials associated with deterministic measures supported on the real line. These random ensembles generalize classical random trigonometric polynomials. In particular, we discuss local and global asymptotics for the expectation and variance of the number of real zeros, when the random coefficients are standard Gaussian. We shall also mention recent and ongoing work on extending such asymptotic results to certain ensembles with general random coefficients, as well as on establishing the Central Limit Theorems for the number of real zeros.

Lakshmi Priya (Indian Institute of Science)

Title: Overcrowding estimates for the nodal volume of stationary Gaussian processes on $\mathbb R^d$

Abstract: We consider centered stationary Gaussian processes (SGPs) on $\mathbb{R}^{d}$ (for $d \geq 1$ ) and study an aspect of theit nodal set: for $T>0$, we study the nodal volume in $[0, T]^{d}$. In earlier studies, under varying assumptions on the spectral measures of SGPs, the following stat istics were obtained for the nodal volume in $[0, T]^{d}:$ expectation, variance asymptotics, CLT, exponential concentration (only for $d=1$ ), and finiteness of moments. We study the unlikely event of overcrowding of the nodal set in $[0, T]^{d} ;$ this is the event that the volume of the nodal set in $[0, T]^{d}$ is much larger than its expected value. Under some mild assumptions on the spectral measure, we obtain estimates for the overcrowding event's probability. We first get overcrowding estimates for the zero count of SGPs on $\mathbb{R}$. In higher dimensions, we consider Crofton's formula which gives the volume of the nodal set in terms of the number of intersections of the nodal set with all lines in $\mathbb{R}^{d}$. We discretize this formula to get a more workable version of it; this along with the ideas used to obtain overcrowding estimates in one dimension are used to get overcrowding estimates in higher dimensions.

Andrea Sartori (Tel Aviv University)

Title: Asymptotic nodal length and log-integrability of toral eigenfunctions

Abstract: Yau conjectured that the nodal length, volume of the zero set, of Laplace eigenfunction should be comparable to the square root of their eigenvalue. Going further, should we also expect an asymptotic law for the nodal length? In this talk I will focus on Laplace eigenfunctions on the 2d torus and discuss how their “peculiar” spectral properties allowed us to find the asymptotic behavior of their nodal length both at large and small scales.

Anna-Paola Todino (Politecnico di Torino)

Title: Random Spherical Harmonics: Overview and Recent Results

Abstract: In this talk we consider geometric functionals (Lipschitz-Killing curvatures, hereafter LKCs) for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In dimension 2, LKCs correspond to the area, half of the boundary length and the Euler-Poincaré characteristic. We give an overview on the asymptotic behavior of their expected values and variances in the high energy limits focussing in particular on the nodal length of random eigenfunctions both on the whole sphere and on shrinking domains. After discussing Central Limit Theorems we prove Moderate Deviation estimates for these geometric functionals. The proofs are based on a combination of a Moderate Deviation Principle by Schulte and Thale (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.

Hugo Vanneuville (Université Grenoble-Alpes)

Title: An unbounded nodal surface for 3D Bargmann-Fock

Abstract: In a joint work with Hugo Duminil-Copin, Alejandro Rivera and Pierre-François Rodriguez we show that, contrary to the planar case, there exists a.s. an unbounded nodal surface for the Bargmann Fock field in dimension 3 (and that this surface ''visits almost all the space''). This comes from the following result that we also show: the critical level for the percolation model formed by the excursion sets $\{ f < u \}$ (where $f$ is Bargmann-Fock and $u$ is a real number) is strictly less than 0 in dimension 3. In this talk, I will explain the general idea of the proof (which is very different from the proof for Bernoulli percolation; in particular, our main intermediate result uses crucially the continuous nature of the model). Also, I will probably say that for the moment we cannot prove unicity of the unbounded nodal hypersurface!, and I will state a conjecture that I like a lot about the existence of somes subplanes of $\mathbb R^3$ in which Bargmann-Fock behaves very atypically.

Anna Vidotto (U. Chieti-Pescara)

Title: Non-Universal Fluctuations of the Empirical Measure for Isotropic Stationary Fields on $\mathbb{S}^2\times \mathbb{R}$

Abstract: In this talk, we consider isotropic and stationary real Gaussian random fields defined on $\mathbb{S}^2\times \mathbb{R}$ and we investigate the asymptotic behavior, as $T\to\infty$, of the empirical measure (excursion area) in $\mathbb{S}^2\times [0,T]$ at any threshold, covering both cases when the field exhibits short and long memory, i.e. integrable and non-integrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry’s cancellation phenomenon occurs at zero-level, inducing phase transitions for both variance rates and limiting laws. This is a joint work with Domenico Marinucci and Maurizia Rossi.

Igor Wigman (King's College London)

Title: Expected nodal volume for non-Gaussian random band-limited functions

Abstract:  This talk is based on a joint work with Z. Kabluchko and A. Sartori. The asymptotic law for the expected nodal volume of random non-Gaussian monochromatic band-limited functions is determined in vast generality. Our methods combine microlocal analytic techniques and modern probability theory. A particularly challenging obstacle needed to overcome is the possible concentration of nodal volume on a small proportion of the manifold, requiring solutions in both disciplines. As for the fine aspects of the distribution of nodal volume, such as its variance, it is expected that the non-Gaussian monochromatic functions behave qualitatively differently compared to their Gaussian counterpart, with some conjectures been put forward.

Aaron Yeager (College of Coastal Georgia)

Title: Random polynomials and their zeros

Abstract: We investigate the distribution of zeros of random polynomials with independent and identically distributed standard normal coefficients in the complex domain, obtain explicit formulas for the density and mean distribution of the zeros and level-crossings, and inquire into the consequences of their asymptotical evaluations for a variety of orthogonal polynomials. In addition, we bridge a small gap in the method of proof devised by Shepp and Vanderbei. Our approach makes use of the Jacobians of functions of several complex variables and the mean ratio of complex normal random variables. This is a joint work with Christopher Corley and Andrew Ledoan.

Nadav Yesha (Université de Haifa)

Title: The defect distribution of toral Laplace eigenfunctions via Bourgain's de-randomization

Abstract: In this talk, we will discuss the defect (signed area) distribution of deterministic toral Laplace eigenfunctions restricted to balls of radii above the Planck scale. We show that for flat eigenfunctions, the defect variance with respect to the spatial variable vanishes along a full density sequence of energies, so that the eigenfunctions are almost sign-balanced. Our proof uses Bourgain's de-randomization method, which allows us to approximate the eigenfunctions by monochromatic random waves. Joint work with P. Kurlberg and I. Wigman.