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Ozan Günyüz

Publications and source records attributed to Ozan Günyüz.

10 recordsLinked to original sources
arXiv2026

Equidistribution for Random Polynomials and Systems of Random Holomorphic Sections

This article addresses an equidistribution problem concerning the zeros of systems of random holomorphic sections of positive line bundles on compact Kähler manifolds and random polynomials on $\mathbb{C}^{m}$ in the setting of the weighted pluripotential theory. For random polynomials, we consider non-orthonormal bases and prove an equidistribution result which is more general than the ones acquired before for non-discrete probability measures. More precisely, our result demonstrates that the equidistribution holds true even when the random coefficients in the basis representation are not independent and identically distributed (i.i.d.), and moreover, they are not constrained to any particular probability distribution. For random holomorphic sections, by extending the concept of a sequence of asymptotically Bernstein-Markov measures introduced by Bayraktar, Bloom and Levenberg in their recent paper to the setting of holomorphic line bundles over compact Kahler manifolds, we derive a global equidistribution, variance estimate and expected distribution theorems related to the zeros of systems of random holomorphic sections for large tensor powers of a fixed holomorphic line bundle for any codimension k, generalizing a previous result of Bayraktar in his 2016 paper and giving also a positive answer to a question posed in the same paper, asking whether the equidistribution is true for non-homogeneous manifolds. For both random holomorphic polynomials on $\mathbb{C}^{m}$ and systems of random holomorphic sections, the variance estimation method detailed in another paper of the author with Bojnik is significant.

math.CV↗
arXiv2026

Random Systems of Holomorphic Sections of a Sequence of Line bundles on Compact Kähler Manifolds

This paper primarily establishes an asymptotic variance estimate for smooth linear statistics associated with zero sets of systems of random holomorphic sections in a sequence of positive Hermitian holomorphic line bundles on a compact Kähler manifold $(X, ω)$ in a general non-Gaussian setting. Using this variance estimate and the expected distribution, we derive an equidistribution result for zeros of these random systems, which proves that the smooth positive closed form $ω^{k}$ can be approximated by currents of integration along analytic subsets of $X$ of codimension $k$, $k \in \{1, \ldots, n\}$. The probability measures taken into consideration in this paper are sufficiently general to include a wide range of the measures commonly encountered in the literature, for which we give equidistribution results at the end, such as the standard Gaussian measure, Fubini-Study measure, the area measure of spheres, probability measures whose distributions have bounded densities with logarithmic decaying tails and locally moderate measures among others.

math.CV↗
arXiv2025

A central limit theorem associated with a sequence of positive line bundles

We prove a central limit theorem for smooth linear statistics associated with zero divisors of standard Gaussian holomorphic sections in a sequence of holomorphic line bundles with Hermitian metrics of class $\mathscr{C}^{3}$ over a compact Kähler manifold. In the course of our analysis, we derive first-order asymptotics and upper decay estimates for near and off-diagonal Bergman kernels, respectively. These results are essential for determining the statistical properties of the zeros of random holomorphic sections.

math.CV↗
arXiv2025

Equidistribution of Zeros of Random Polynomials and Random Polynomial mappings on $\mathbb{C}^m$

We study equidistribution problem of zeros in relation to a sequence of $Z$-asymptotically Chebyshev polynomials on $\mathbb{C}^{m}$. We use certain results obtained in a very recent work of Bayraktar, Bloom and Levenberg and have an equidistribution result in a more general probabilistic setting than what the paper of Bayraktar, Bloom and Levenberg considers even though the basis polynomials they use are more general than $Z$-asymptotically Chebyshev polynomials. Our equidistribution result is based on the expected distribution and the variance estimate of random zero currents corresponding to the zero sets (zero divisors) of polynomials. This equidistribution result of general nature shows that equidistribution result turns out to be true without the random coefficients being i.i.d. (independent and identically distributed), which also means that there is no need to use any specific probability distribution function for these random coefficients. In the last section, unlike from the $1$-codimensional case, we study the basis of polynomials orthogonal with respect to the $L^{2}$-inner product defined by the weighted asymptotically Bernstein-Markov measures on a given locally regular compact set, and with a probability distribution studied well by Bayraktar including the (standard) Gaussian and the Fubini-Study probability distributions as special cases, we have an equidistribution result for codimensions bigger than $1$.

math.CV↗
arXiv2024

Core Sets in Kahler Manifolds

The primary objective of this paper is to study core sets in the setting of m-subharmonic functions on the class of (non-compact) Kahler manifolds. Core sets are investigated in different aspects by considering various classes of plurisubharmonic functions. One of the crucial concepts in studying the structure of this kind of sets is the pseudoconcavity. In a more general way, we will have the structure of core defined with respect to the m-subharmonic functions, which we call m-core in our setting, in terms of m-pseudoconcave sets. In the context of m-subharmonic functions, we define m-harmonic functions and show that, in $\mathbb{C}^{n},\,(n \geq 2)$ and more generally in any Kahler manifold of dimension at least 2, m-harmonic functions are pluriharmonic functions for $m \geq 2$.

math.CV↗
arXiv2024

The Internal Polya Inequality for $\mathbb{C}$-convex Domains in $\mathbb{C}^n$

Let $K\subset \mathbb{C}$ be a polynomially convex compact set, $f$ be a function analytic in a domain $\overline{\mathbb{C}}\smallsetminus K$ with Taylor expansion $f\left( z\right) =\sum_{k=0}^{\infty }\frac{a_{k}}{z^{k+1}} $ at $\infty $, and $H_{s}\left( f\right) :=\det \left( a_{k+l}\right) _{k,l=0}^{s}$ related Hankel determinants. The classical Polya theorem \cite% {P} says that \[ \limsup_{s\rightarrow \infty }\left\vert H_{s}\left( f\right) \right\vert ^{1/s^{2}}\leq d\left( K\right) , \]% where $d\left( K\right) $ is the transfinite diameter of $K$. The main result of this paper is multivariate internal analogs of Polya's inequality for $\mathbb{C}$-convex (=strictly linearly convex) domains $D\subset \mathbb{C}^{n}$ and weighted Hankel-type determinants, constructed from the Taylor coefficients of a function $f\in A\left( D\right) $ at a given point $% a\in D$; therewith the weights are generated by $s$-indicatrices of the sequence of analytic functionals biorthogonal to the system of monomials in $% \mathbb{C}^{n}$. It is proved by the reduction to the outer multivariate analog of Polya's inequality (Zakharyuta, Math. USSR Sbornik, \textbf{25 }% (1975)) and is based on the characterization of the strict linear convexity in terms of $s$-indicatrices (S. Znamenskii, Siberian Math. J. \textbf{26 } (1985)).

math.CV↗
arXiv2023

Variance Estimate of Systems of Random Holomorphic Sections in a Sequence of Line Bundles on Compact Kahler Manifolds

This paper primarily concerns the variance estimate of zeros of systems of random holomorphic sections associated with a sequence of smooth Hermitian holomorphic line bundles on a compact Kahler manifold X. The probability measures taken into consideration in this paper satisfy a certain condition which makes it much more general(possibly the most general) than (all) the measures mostly studied in the literature. In particular we also give variance estimates for several known measures such as Gaussian and Fubini-Study measures.

math.CV↗
arXiv2016

Weighted Polya Inequality in Cn

In this article, we obtain a weighted version of Polya inequality in Cn whose unweighted form was established by V. Zakharyuta in USSR Sbornik, 25 (1975), 350-364.

math.CV↗
arXiv2016

On Polya' Theorem in Several Complex Variables

Let $K$ be a compact set in $\mathbb{C}$, $f$ a function analytic in $\overline{\mathbb{C}}\smallsetminus K$ vanishing at $\infty $. Let $% f\left( z\right) =\sum_{k=0}^{\infty }a_{k}\ z^{-k-1}$ be its Taylor expansion at $\infty $, and $H_{s}\left( f\right) =\det \left( a_{k+l}\right) _{k,l=0}^{s}$ the sequence of Hankel determinants. The classical Polya inequality says that \[ \limsup\limits_{s\rightarrow \infty }\left\vert H_{s}\left( f\right) \right\vert ^{1/s^{2}}\leq d\left( K\right) , \]% where $d\left( K\right) $ is the transfinite diameter of $K$. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Polya's inequality, considered by the second author in Math. USSR Sbornik, 25 (1975), 350-364.

math.CV↗