相关论文: Critical radius and supremum of random spherical h…
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 study random waves on smooth, compact, Riemannian manifolds under the spherical ensemble. Our first main result shows that there is a positive universal limit for the critical radius of a specific deterministic embedding, defined via the…
It is widely known that the tube method, or equivalently the Euler characteristic heuristic, provides a very accurate approximation for the tail probability that the supremum of a smooth Gaussian random field exceeds a threshold value $c$.…
We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for…
In this note, we investigate the behaviour of suprema for band-limited spherical random fields. We prove upper and lower bound for the expected values of these suprema, by means of metric entropy arguments and discrete approximations; we…
We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy [Spherical dispersion with an application to polygonal approximation of curves, Anz. \"Osterreich. Akad. Wiss.…
Given a closed Riemann surface $\Sigma$ equipped with a volume form $\omega$, we construct a natural probability measure on the space $\mathcal{M}_d(\Sigma)$ of degree $d$ branched coverings from $\Sigma$ to the Riemann sphere…
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 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 Hausdorff limit-set of the Euclidean hypersurfaces with large $\lambda_1$ or small extrinsic radius. The result depends on the $L^p$ norm of the curvature that is assumed to be bounded a priori, with a critical behaviour…
We give an explicit estimate of the distance of a closed, connected, oriented and immersed hypersurface of a space form to a geodesic sphere and show that the spherical closeness can be controlled by a power of an integral norm of the…
We derive fundamental asymptotic results for the expected covering radius $\rho(X_N)$ for $N$ points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For…
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,…
We characterise high-dimensional topology that arises from a random Cech complex constructed on the circle. Expected Euler characteristic curve is computed, where we observe limiting spikes. The spikes correspond to expected Betti numbers…
Consider a Boolean model $\Sigma$ in $\R^d$. The centers are given by a homogeneous Poisson point process with intensity $\lambda$ and the radii of distinct balls are i.i.d.\ with common distribution $\nu$. The critical covered volume is…
The distribution of products of random matrices chosen from fixed spherical classes is determined for classical rank 1 symmetric spaces. It is observed that $n\to\infty$ limit behaves approximately as in the abelian case. A theorem on the…
We consider random $n\times n$ matrices $X$ with independent and centered entries and a general variance profile. We show that the spectral radius of $X$ converges with very high probability to the square root of the spectral radius of the…
We study the asymptotic distribution of critical values of random holomorphic `polynomials' s_n on a Kaehler manifold M as the degree n tends to infinity. By `polynomial' of degree n we mean a holomorphic section of the nth power of a…
We consider random polytopes in the $d$-dimensional Euclidean space that are the convex hulls i.i.d. random points selected according to beta-prime distributions. These distributions are rotationally symmetric, heavy-tailed, and their…
We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a…