Related papers: Nodal Replication of Planar Random Waves

We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic"…

We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for…

We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $E>0$, both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy…

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…

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…

Spectral quasi correlations are small sums of lattice points lying on the same circle; we show that, for generic integers representable as the sum of two squares, there are no spectral quasi-correlations. Moreover, we apply our result to…

We are interested in the effect of Dirichlet boundary conditions on the nodal length of Laplace eigenfunctions. We study random Gaussian Laplace eigenfunctions on the two dimensional square and find a two terms asymptotic expansion for the…

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…

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…

We study the asymptotic behaviour of the nodal length of random $2d$-spherical harmonics $f_{\ell}$ of high degree $\ell \rightarrow\infty$, i.e. the length of their zero set $f_{\ell}^{-1}(0)$. It is found that the nodal lengths are…

We introduce two versions of a new sketch for approximately embedding the Gaussian kernel into Euclidean inner product space. These work by truncating infinite expansions of the Gaussian kernel, and carefully invoking the…

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X of R^N of dimension (1-o(1))N such that for every element x in X, |x|_1 and N^{1/2} |x|_2 are equivalent up to a factor of (log N)^{log log log…

Let $X$ be a symmetric, isotropic random vector in $\mathbb{R}^m$ and let $X_1...,X_n$ be independent copies of $X$. We show that under mild assumptions on $\|X\|_2$ (a suitable thin-shell bound) and on the tail-decay of the marginals…

We discuss an approach to compute the first and second moments of the number of eigenvalues $I_N$ that lie in an arbitrary interval of the real line for $N \times N$ Gaussian random matrices. The method combines the standard…

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random…

"Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013)). In this paper we find that their nodal length converges to a non-universal…

Let $d\geq 3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector $v$ of $G$ (with entry sum 0 and normalized to have length…

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3= \mathbb{R}^3/ \mathbb{Z}^3$ ($3$-dimensional 'arithmetic random waves'). We prove that, as the multiplicity of the eigenspace…

In 2017, Benatar and Maffucci arXiv:1708.07015 established an asymptotic law for the variance of the nodal surface of arithmetic random waves on the 3-torus in the high-energy limit. In a subsequent work, Cammarota arXiv:1708.07679 proved a…

It is shown that spatially periodic one-dimensional surface waves in shallow water behave almost linearly, provided large part of the energy is contained in sufficiently high frequencies. The amplitude is not required to be small (apart…