Concentration for nodal component count of Gaussian Laplace eigenfunctions

Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator

We consider Gaussian random eigenfunctions (Hermite functions) of fixed energy level of the isotropic semi-classical Harmonic Oscillator on ${\bf R}^n$. We calculate the expected density of zeros of a random eigenfunction in the semi-classical limit $h \to 0.$ In the allowed region the density is of order $h^{-1},$ while in the forbidden region the density is of order $h^{-\frac{1}{2}}$. The computer graphics due to E.J. Heller illustrate this difference in "frequency" between the allowed and forbidden nodal sets.

Nodal intersections for random eigenfunctions on the torus

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard two-dimensional flat torus ("arithmetic random waves") with a fixed real-analytic reference curve with nonvanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result prescribes the asymptotic behaviour of the nodal intersections variance for analytic curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lyi…

随机拉普拉斯特征函数的符号平衡

Sign-balance of random Laplace eigenfunctions

本文提出了一种强意义下的符号平衡概念，证明了在特定尺度以上，随机特征函数在概率意义上具有符号平衡性，且该尺度最优至多与能量的对数有关，涵盖了随机球面谐波和更一般的光滑黎曼流形上的带限随机波。

Random waves on $\mathbb{T}^3$: nodal area variance and lattice point correlations

We consider the ensemble of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ (`$3d$ arithmetic random waves'), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or `energy', of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.

Random Nodal Lengths and Wiener Chaos

In this survey we collect some of the recent results on the "nodal geometry" of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic behavior, for high energy levels, of the nodal length of Gaussian Laplace eigenfunctions on the torus (arithmetic random waves) and on the sphere (random spherical harmonics). We give some insight on both Berry's cancellation phenomenon and the nature of nodal length second order fluctuations (non-Gaussian on the torus and Gaussian on the sphere) in terms of chaotic components. Finally we consider the general case of monochromatic random waves…

Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

We study the scaling asymptotics of the eigenspace projection kernels $Π_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 Δ+ |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $Π_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as $\hbar \to 0$. In previous work we proved that the den…

Nodal deficiency of random spherical harmonics in presence of boundary

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. Jean Bourgain's research, and his enthusiastic approach to the nodal geometry of Lap…

Nodal length fluctuations for arithmetic random waves

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is non-universal, and is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.