Related papers: Hole Probability for Entire Functions represented …

We study the hole probabilities for ${\mathcal X}_{\infty}^{(\alpha)}$ ($\alpha>0$), a determinantal point process in the complex plane with the kernel $\mathbb…

By random complex zeroes we mean the zero set of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This zero set is distribution invariant…

The spacetime singularities in classical general relativity are inevitable, which are also predicated by the celebrated singularity theorems. However, it is general belief that singularities do not exist in the nature and they are the…

Let ($\Sigma$, g) be a closed connected surface equipped with a riemannian metric. Let ($\lambda$ n) n$\in$N and ($\psi$ n) n$\in$N be the increasing sequence of eigenvalues and the sequence of corresponding L 2-normalized eigenfunctions of…

One of the major goals of cosmological observations is to test theories of structure formation. The most straightforward way to carry out such tests is to compute the likelihood function L, the probability of getting the data given the…

Using the Hartle-Hawking no-boundary proposal for the wave function of the universe, we can study the wave function and probability of a single black hole created at the birth of the universe. The black hole originates from a constrained…

In this paper we consider the distribution of the zeros of a real random Bargmann-Fock function of one or more variables. For these random functions we prove estimates for two types of families of events, both of which are large deviations…

We consider random analytic functions given by a Taylor series with independent, centered complex Gaussian coefficients. We give a new sufficient condition for such a function to have bounded mean oscillations. Under a mild regularity…

We consider the range of random analytic functions with finite radius of convergence. We show that any unbounded random Taylor series with rotationally invariant coefficients has dense image in the plane. We moreover show that if in…

We study hyperbolic Gaussian analytic functions in the unit polydisk of $\mathbb C^n$. Following the scheme previously used in the unit ball we first study the asymptotics of fluctuations of linear statistics as the directional intensities…

We consider particle systems (also known as point processes) on the line and in the plane, and are particularly interested in "hole" events, when there are no particles in a large disk (or some other domain). We survey the extensive work on…

We study sampling properties of the zero set of the Gaussian entire function on Fock spaces. Firstly, we relax Seip and Wallst\'en's density and separation conditions for sampling sets on Fock spaces to obtain weighted inequalities for sets…

We investigate the complex Gaussian as well as non-Gaussian distributed random analytical and entire functions (complex entire random field) and calculate their domain of definiteness (radius of convergence) as well as some important…

Context: Two-point correlation functions are used throughout cosmology as a measure for the statistics of random fields. When used in Bayesian parameter estimation, their likelihood function is usually replaced by a Gaussian approximation.…

We study entire functions whose zeros and one-points lie on distinct finite systems of rays. General restrictions on these rays are obtained. Non-trivial examples of entire functions with zeros and one-points on different rays are…

We generalize the maximum likelihood method to non-Gaussian distribution functions by means of the multivariate Edgeworth expansion. We stress the potential interest of this technique in all those cosmological problems in which the…

In this work, we study probability functions associated with Gaussian mixture models. Our primary focus is on extending the use of spherical radial decomposition for multivariate Gaussian random vectors to the context of Gaussian mixture…

Let $f$ be a rational function on an algebraic curve over the complex numbers. For a point $p$ and local parameter $x$ we can consider the Taylor series for $f$ in the variable $x$. In this paper we give an upper bound on the frequency with…

We construct random point processes in the complex plane that are asymptotically close to a given doubling measure. The processes we construct are the zero sets of random entire functions that are constructed through generalised Fock…

We prove a formula for the Taylor series coefficients of a zero of the sum of a complex-exponent polynomial and a base function which is a general holomorphic function with a simple zero. Such a Taylor series is more general than a Puiseux…