Related papers: Random Nodal Lengths and Wiener Chaos
We study of the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. We give upper…
We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our…
We study the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position, in 2 and 3 dimensions. In 2d, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy…
We study the plane (not necessarily monochromatic) gravitational waves at nonlinear quadratic order on a flat background in vacuum. We show that, in the harmonic gauge, the nonlinear waves are unstable. We argue that, at this order, this…
We apply techniques of microlocal analysis to the study of the transverse geometry of Riemannian foliations in order to analyze spectral invariants of the basic Laplacian acting on functions on a Riemannian foliation with a bundle-like…
The objective of the paper is to identify and investigate all possible types of asymptotic behavior for the maximum likelihood estimators of the unknown parameters in the second-order linear stochastic ordinary differential equation driven…
We define a random model for the moments of the new eigenfunctions of a point scat-terer on a 2-dimensional rectangular flat torus. In the deterministic setting,Seba conjectured these moments to be asymptotically Gaussian, in the…
The asymptotic behavior of a nonlinear oscillator subject to a multiplicative Ornstein-Uhlenbeck noise is investigated. When the dynamics is expressed in terms of energy-angle coordinates, it is observed that the angle is a fast variable as…
On the unit tangent bundle of a compact Riemannian surface, we consider a natural sub-Riemannian Laplacian associated with the canonical contact structure. In the large eigenvalue limit, we study the escape of mass at infinity in the…
Black holes and gravitational waves are consequences of the nonlinear character of the Einstein equations. Yet, the remarkable properties of General Relativity point to the existence of other effects. Here we uncover new nonlinear facets of…
Inspired by the recent work [MRW20], we prove that the nodal length of a planar random wave $B_{E}$, i.e. the length of its zero set $B_{E}^{-1}(0)$, is asymptotically equivalent, in the $L^{2}$-sense and in the high-frequency limit…
In this paper, we study the motion of the free surface of a body of fluid over a variable bottom, in a long wave asymptotic regime. We assume that the bottom of the fluid region can be described by a stationary random process $\beta(x,…
We prove the convergence of the solutions of the parabolic wave equation to that of the Gaussian white-noise model widely used in the physical literature. The random medium is isotropic and is assumed to have integrable correlation…
In this paper, we study the asymptotic behavior of the quadratic variation for the class of AR(1) processes driven by white noise in the second Wiener chaos. Using tools from the analysis on Wiener space, we give an upper bound for the…
We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a…
A Lagrangian relativistic approach to the non--linear dynamics of cosmological perturbations of an irrotational collisionless fluid is considered. Solutions are given at second order in perturbation theory for the relevant fluid and…
We consider the canonical ensemble of multilayered constrained Erdos-Renyi networks (CERN) and regular random graphs (RRG), where each layer represents graph vertices painted in a specific color. We study the critical behavior in such…
We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our…
Let $(M,g)$ be a smooth compact Riemannian surface with no boundary. Given a smooth vector field $V$ with finitely many zeroes on $M$, we study the distribution of the number of tangencies to $V$ of the nodal components of random…
We investigate monotonicity properties of eigenvalues of the Dirichlet Laplacian in polyhedral layers of fixed width. We establish that eigenvalues below the essential spectrum threshold monotonically depend on geometric parameters defining…