Searcharxiv⌕ Search

arXiv subjects

Mikhail Sodin

Publications and source records attributed to Mikhail Sodin.

47 records · Page 3Linked to original sources
arXiv2004

Zeroes of Gaussian analytic functions

Geometrically, zeroes of a Gaussian analytic function are intersection points of an analytic curve in a Hilbert space with a randomly chosen hyperplane. Mathematical physics provides another interpretation as a gas of interacting particles. In the last decade, these interpretations influenced progress in understanding statistical patterns in the zeroes of Gaussian analytic functions, and led to the discovery of canonical models with invariant zero distribution. We shall discuss some of recent results in this area and mention several open questions.

math.CV↗
arXiv2004

Random complex zeroes, I. Asymptotic normality

We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of zeroes.

math.CV↗
arXiv2004

Sign and area in nodal geometry of Laplace eigenfunctions

The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line {f=0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mild local asymmetry may appear. The proof combines methods due to Donnelly-Fefferman and Nadirashvili with a new result on harmonic functions in the unit disc.

math.AP↗
arXiv2003

Random complex zeroes, III. Decay of the hole probability

By a hole we mean a disc that contains no flat chaotic analytic zero points (i.e. zeroes 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!). A given disc of radius r has a probability of being a hole, - the hole probability. We show that for large r the hole probability decays as exp(-cr^4).

math.CV↗
arXiv2003

Lower bounds for quasianalytic functions, II. The Bernstein quasianalytic functions

Let F be a class of functions with the uniqueness property: if a function f in F vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a lower bound for |f| outside a small exceptional set. Such estimates are well-known and useful for polynomials, complex- and real-analytic functions, exponential polynomials. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions. In the first part (this arXiv:math.CA/0208233), we considered quasianalytically smooth functions. Here, we deal with classes of functions characterized by exponentially fast approximation by polynomials whose degrees belong to a given very lacunar sequence. We also prove the polynomial spreading lemma and a comparison lemma which are of a certain interest on their own.

math.CA↗
arXiv2001

Asymptotics of the Fourier and Laplace transforms in weighted spaces of analytic functions

We study the asymptotics near the origin of the Fourier transform in weighted Hardy spaces of analytic functions in the upper half-plane, and of the Laplace transform in weighted spaces of entire functions of zero exponential type. These results are applied to two closely related problems posed by E. Dyn'kin: we find the asymptotics of the depth of zero in non-quasianalytic Denjoy-Carleman classes, and of the exact Levinson-Sjoberg majorant.

math.CV↗
arXiv2001

A question by Alexei Aleksandrov and logarithmic determinants

We answer negatively to the question posed by Aleksandrov concerning analytic functions of Smirnov's class in the unit disk with pure imaginary boundary values. We also find new sufficient conditions for representations of functions of Smirnov's class by the Schwartz and Cauchy integrals. These conditions extend a previous result of Aleksandrov.

math.CV↗
arXiv2000

Zeros of Gaussian Analytic Functions

We prove and discuss three results on zero distribution of gaussian analytic functions: (i) the Edeleman-Kostlan formula for the expectation of the counting measure; (ii) a variation on the theme of Calabi's rigidity theorem; (iii) Offord's estimate of exponential decay of the tail probabilities of an anlytic function having an access or deficiency of zeros in a given region.

math.CV↗