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

We investigate the spectrum for partial sums of m position (or gaussian) operators on monotone Fock space based on $\ell^2(\mathbb{N})$. In the basic case of the first consecutive operators, we prove it coincides with the support of the…

Let $n\geq 2$ and $r\in \{1, \cdots, n-1\}$ be integers, $M$ be a compact smooth K\''ahler manifold of complex dimension $n$, $E$ be a holomorphic vector bundle with complex rank $r$ and equipped with an hermitian metric $h_E$, and $L$ be…

We prove a central limit theorem for smooth linear statistics related to the zero divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.

We study (higher order) asymptotic behaviour near branch points of stationary $n$-dimensional two-valued $C^{1, \mu}$ graphs in an open subset of ${\mathbb R}^{n+m}$. Specifically, if $M$ is the graph of a two-valued $C^{1, \mu}$ function…

We study the zero distribution of non-orthogonal polynomials attached to $g(n)=s(n)=n^2$: \begin{equation*} Q_n^g(x)= x \sum_{k=1}^n g(k) \, Q_{n-k}^g(x), \quad Q_0^g(x):=1. \end{equation*} It is known that the case $g=id$ involves…

We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient…

It has been shown recently [10] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random NxN Hermitian matrices. Our main goal is to…

We prove a general multidimensional invariance principle for a family of U-statistics based on freely independent non-commutative random variables of the type $U_n(S)$, where $U_n(x)$ is the $n$-th Chebyshev polynomial and $S$ is a standard…

In this paper we prove that, for every integers $m\geq 2$ and $n\geq 1$, the support of any stationary $m$-dimensional integer rectifiable varifold $V$ in an open set $U\subset \mathbb R^{m+n}$ is $C^\infty$ rectifiable, namely it can be…

We study the regularity of densities of distributions that are polynomial images of the standard Gaussian measure on $\mathbb{R}^n$. We assume that the degree of a polynomial is fixed and that each variable enters to a power bounded by…

In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials.…

In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.

This note is concerned with the scaling limit as N approaches infinity of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic…

For every $m\in\mathbb{N}$, we establish the convergence of the averaged distributions of the zeros of the $m$-th order derivatives $(f^n)^{(m)}$ of the iterated polynomials $f^n$ of a polynomial $f\in\mathbb{C}[z]$ of degree $>1$ towards…

We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree…

In this paper we sharpen significantly several known estimates on the maximal number of zeros of complex harmonic polynomials. We also study the relation between the curvature of critical lemniscates and its impact on geometry of caustics…

An exact map was established by Lacroix-A-Chez-Toine, Majumdar, and Schehr in [44] between the $N$ complex eigenvalues of complex non-Hermitian random matrices from the Ginibre ensemble, and the positions of $N$ non-interacting Fermions in…

Let $ \{\varphi_i\}_{i=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure $ \mu $ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected…

This note concerns the topology of the connected components of the zero sets of monochromatic random waves on compact Riemannian manifolds without boundary. In [SW] it is shown that these are distributed according to a universal measure on…

The Newton polytope $P_f$ of a polynomial $f$ is well known to have a strong impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of simultaneous zeros of $m$ polynomials with given Newton polytopes. In this article,…