Related papers: On the correlation between critical points and cri…
We study the correlation between the nodal length of random spherical harmonics and the measure of the boundary for excursion sets at any non-zero level. We show that the correlation is asymptotically zero, while the partial correlation…
We prove a Central Limit Theorem for the Critical Points of Random Spherical Harmonics, in the High-Energy Limit. The result is a consequence of a deeper characterizations of the total number of critical points, which are shown to be…
We determine the asymptotic law for the fluctuations of the total number of critical points of random Gaussian spherical harmonics in the high degree limit. Our results have implications on the sophistication degree of an appropriate…
We study here the random fluctuations in the number of critical points with values in an interval $I\subset \mathbb{R}$ for Gaussian spherical eigenfunctions $\left\{f_{\ell }\right\} $, in the high energy regime where $\ell \rightarrow…
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…
The article contains several observations on spherical harmonics and their nodal sets: a construction for harmonics with prescribed zeroes; a kind of canonical representation of this type for harmonics on $\bbS^2$; upper and lower bounds…
We prove two conjectures from [M. R. Douglas, B. Shiffman and S. Zelditch, Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics. J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the expected number of…
We study the two-point correlation $K^m_n(z,w)$ between zeros and critical points of Gaussian random holomorphic sections $s_n$ over K\"ahler manifolds. The critical points are points $\nabla_{h^n} s_n=0$ where $\nabla_{h^n}$ is the smooth…
In this paper we prove that as N goes to infinity, the scaling limit of the correlation between critical points z1 and z2 of random holomorphic sections of the N-th power of a positive line bundle over a compact Riemann surface tends to…
We consider fluctuations in the distribution of critical points - saddle points, minima and maxima - of random gaussian fields. We calculate the asymptotic limits of the two point correlation function for various critical point densities,…
On the two dimensional sphere, we consider axisymmetric critical points of an isoperimetric problem perturbed by a long-range interaction term. When the parameter controlling the nonlocal term is sufficiently large, we prove the existence…
In this note, we consider a fixed vector field $V$ on $S^2$ and study the distribution of points which lie on the nodal set (of a random spherical harmonic) where $V$ is also tangent. We show that the expected value of the corresponding…
We prove the main conjecture from [M. R. Douglas, B. Shiffman and S. Zelditch, Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics. J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the metric dependence…
We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system which has a…
The connection between spherical harmonics and symmetric tensors is explored. For each spherical harmonic, a corresponding traceless symmetric tensor is constructed. These tensors are then extended to include nonzero traces, providing an…
This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves $\phi_\lambda$ of frequency $\lambda$ on a compact, smooth, Riemannian manifold $(M,g)$ as $\lambda \rightarrow \infty$. We prove that…
In the present work we study the two-point correlation function $R(\epsilon)$ of the quantum mechanical spectrum of a classically chaotic system. Recently this quantity has been computed for chaotic and for disordered systems using periodic…
We continue the study, begun in \cite{FA}, of the critical radius of embeddings, via deterministic spherical harmonics, of fixed dimensional spheres into higher dimensional ones, along with the associated problem of the distribution of the…
We investigate the asymptotic behavior of the nodal lines for random spherical harmonics restricted to shrinking domains, in the 2-dimensional case: i.e., the length of the zero set $\mathcal{Z}_{\ell,r_\ell} :=…
We study random spherical harmonics at shrinking scales. We compare the mass assigned to a small spherical cap with its area, and find the smallest possible scale at which, with high probability, the discrepancy between them is small…