Related papers: Number variance of random zeros on complex manifol…

In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain…

Starting with a collection of $n$ oriented polygonal discs, with an even number $N$ of sides in total, we generate a random oriented surface by randomly matching the sides of discs and properly gluing them together. Encoding the surface in…

In this paper we study bounds for the total variation distance between two second degree polynomials in normal random variables provided that they essentially depend on at least three variables.

In this paper we find the asymptotic main term of the variance of the number of roots of Kostlan-Shub-Smale random polynomials and prove a central limit theorem for the number of roots as the degree goes to infinity.

We study divisibility properties of a set $\{f_1(\mathbf{U}_n^{(s)}),\ldots,f_m(\mathbf{U}_n^{(s)})\}$, where $f_1,\ldots,f_m$ are polynomials in $s$ variables over $\mathbb{Z}$ and $\mathbf{U}_n^{(s)}$ is a point picked uniformly at random…

We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of…

In this paper we develop the "local part" of our local/global approach to globally valued fields (GVFs). The "global part", which relies on these results, is developed in a subsequent paper.We study virtual divisors on projective varieties…

Let G be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if G has polynomial growth, then (U^n) is a Folner sequence: that is, the volume of the boundary of U^n divided by U^n goes to zero.…

We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lov\'asz Local Lemma. For metric measure spaces with volume doubling…

We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, non-degenerated and stationary Gaussian field $(f(x), {x \in \mathbb R^d})$. Under mild conditions, we…

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…

We study the number of facets of the convex hull of n independent standard Gaussian points in d-dimensional Euclidean space. In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the…

We investigate the global density of zeros of generalized Hermite orthogonal polynomials, subject to certain truncated conditions on its weight. We shall given explicitly the global density of zeros under some asymptotic conditions on the…

We study zero sets of twisted stationary Gaussian random functions on the complex plane, i.e., Gaussian random functions that are stochastically invariant under the action of the Weyl-Heisenberg group. This model includes translation…

We survey results on the distribution of zeros of random polynomials and of random holomorphic sections of line bundles, especially for large classes of probability measures on the spaces of holomorphic sections. We provide furthermore some…

We prove almost sure strong asymptotic freeness of i.i.d. random unitaries with the following law: sample a Haar unitary matrix of dimension $n$ and then send this unitary into an irreducible representation of $U(n)$. The strong convergence…

We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result,…

We identify the scaling region of a width O(n^{-1}) in the vicinity of the accumulation points $t=\pm 1$ of the real roots of a random Kac-like polynomial of large degree n. We argue that the density of the real roots in this region tends…

We consider random trigonometric polynomials of the form \[ f_n(t):=\frac{1}{\sqrt{n}} \sum_{k=1}^{n}a_k \cos(k t)+b_k \sin(k t), \] where $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent stationary Gaussian processes with the…

Let $(X,g)$ be a closed, connected surface, with variable negative curvature. We consider the distribution of eigenvalues of the Laplacian on random covers $X_n\to X$ of degree $n$. We focus on the ensemble variance of the smoothed number…