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Turgay Bayraktar

Publications and source records attributed to Turgay Bayraktar.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotic expansion of induced Grassmannian Chern forms and distribution of random degeneracy sets

For the Grassmannian embeddings defined by the spaces $H^0(X,L^p\otimes E)$, where $L$ is a positive line bundle and $E$ is a holomorphic vector bundle over a compact complex manifold, we prove a complete asymptotic expansion of the induced Grassmannian Chern forms and compute the first coefficients explicitly. As an application of the first-order asymptotics and of the theory of meromorphic transforms by Dinh and Sibony, we prove that on a compact Kähler manifold, the normalized currents of integration over the loci where several random sections become linearly dependent converge almost surely to the corresponding power of the curvature form of the positive line bundle, with a quantitative estimate for the speed of convergence. Moreover, in the determinant case, we additionally present an alternative method based on the Wishart distribution, together with variance estimates.

math.CV↗
arXiv2026

The Expected Depth of Random Real Algebraic Plane Curves

In this note we study asymptotic isotopy of random real algebraic plane curves. More precisely, we obtain a Kac-Rice type formula that gives the expected number of two-sided components (i.e.\ ovals) of a random real algebraic plane curve winding around a given point. In particular, we show that expected number of such ovals for an even degree Kostlan polynomial is $\frac{\sqrt{d}}{2}$ and independent of the given point.

math.AG↗
arXiv2026

On Nests and Large Components of Random Real Algebraic Curves

We develop a variant of the barrier method in order to address questions about topology of Kostlan random real algebraic plane curves. In particular we prove that the expected number of connected components of the curve of length at least $\displaystyle{O\left(\sqrt{d^{-1}\log \log d}\right)}$ grows to infinity with $d$, and likewise, the expected number of nests of the curve of depth at least $\displaystyle{O\left(\log\log d\right)}$ grows to infinity with $d$. In another direction, we adapt an $L^{\infty}$-norm bound result of Shifmann and Zelditch to subspaces and employ it to obtain a lower bound for the probability that a finite number of points remain all in different components of the complement of a large degree random curve.

math.AG↗
arXiv2026

Zeros of random $P$-polynomials in $\mathbb{C}^d$ with exponential profiles

We study random multivariate $P$-polynomials in $\mathbb{C}^d$ with monomial supports constrained to $nP\cap\mathbb{Z}_+^d$ for a convex body $P\subset\mathbb{R}_+^d$, and deterministic coefficients admitting a uniform exponential profile $f$ on $P$. Assuming the tail condition $\mathbb{P}(\log(1+|ξ_0|)>t)=o(t^{-d})$ on the i.i.d. complex coefficients, we prove that the normalized potentials $\frac1n\log|\mathbf{P}_n|$ converge in probability in $L^1_{\mathrm{loc}}(\mathbb{C}^d)$ to a deterministic toric plurisubharmonic function $Φ_{P,f}$, and consequently the normalized zero currents $\frac1n[Z_{\mathbf{P}_n}]$ converge weakly to the closed positive $(1,1)$-current $dd^cΦ_{P,f}$. Under the stronger logarithmic moment assumption $\mathbb{E}[(\log(1+|ξ_0|))^d]<\infty$, we prove almost sure weak convergence of the zero currents along the full sequence for $d>2$, and along sparse subsequences for $d \le 2$. On $(\mathbb{C}^*)^d$, the limiting potential is given by $Φ_{P,f}(z)=I_{P,f}(\operatorname{Log} z)$, where $I_{P,f}$ is the Legendre-Fenchel transform of the profile over $P$ and $\operatorname{Log} (z)=(\log|z_1|,\dots,\log|z_d|)$. These results extend the exponential-profile mechanism of Kabluchko and Zaporozhets from one complex variable to the genuinely multivariate $P$-polynomial setting under relaxed probabilistic assumptions, directly connecting random zero hypersurfaces with convex-analytic data determined by $(P,f)$.

math.CV↗
arXiv2026

A Dynamical Fekete-Szegő Theorem

Let $E\subset\Bbb{C}$ be a compact set symmetric with respect to the real axis. A classical theorem of Fekete-Szegő asserts that such a compact set is of logarithmic capacity at least one if and only if it admits approximation by algebraic integers whose Galois conjugates lie arbitrarily close to $E$. In this note we prove a dynamical analogue of this phenomenon. When $\mathrm{cap}(E)=1$, we also show that the algebraic polynomials arising from the Fekete-Szegő theorem generate filled Julia sets $K_{P_n}$ which converge to the polynomially convex hull $Pc(E)$ in the Klimek topology, while their Brolin measures converge to the equilibrium measure $μ_E$. In particular, when $E\subset\Bbb{R}$, this provides a genuine approximation of $E$ by algebraic filled Julia sets. As an arithmetic application, we prove that the Rumely height associated to $E$ arises as a limit of canonical dynamical heights in the sense of Call and Silverman, giving a dynamical counterpart to the equidistribution theorems of Bilu and Rumely.

math.DS↗
arXiv2025

Asymptotic Mass Distribution of Random Holomorphic Sections

In this note, we prove a central limit theorem for the mass distribution of random holomorphic sections associated with a sequence of positive line bundles endowed with $\mathscr{C}^3$ Hermitian metrics over a compact Kähler manifold. In addition, we show that almost every sequence of such random holomorphic sections exhibits quantum ergodicity in the sense of Zelditch.

math.CV↗
arXiv2025

Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections

Let $(X,ω)$ be a compact Kähler manifold, $(L,h^L)$ be a positive line bundle, and $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$. We prove that the pullback by the Kodaira embedding associated to $L^p\otimes E$ of the $k$-th Chern class of the dual of the universal bundle over the Grassmannian converges as $p\to\infty$ to the $k$-th power of the Chern form $c_1(L,h^L)$, for $0\leq k\leq r$. If $c_1(L,h^L)=ω$ we also determine the second term in the semiclassical expansion, which involves $c_1(E,h^E)$. As a consequence we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers $L^p\otimes E$ is $c_1(L,h^L)^r$. Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections.

math.CV↗
arXiv2025

Widom factors in $\mathbb C^n$

We generalize the theory of Widom factors to the $\mathbb C^n$ setting. We define Widom factors of compact subsets $K\subset \mathbb C^n$ associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets $K=K_1\times\cdots\times K_n$ of $\mathbb C^n$, where each $K_j$ is a non-polar compact subset of $\mathbb C$, these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each $K_j$ is a subset of the real line, we provide improved lower bounds for Widom factors for some weight functions $w$; in particular, for the case $w\equiv 1$. Finally, we define the Mahler measure of a multivariate polynomial relative to $K\subset \mathbb C^n$ and obtain lower bounds for this quantity on product sets.

math.CV↗
arXiv2024

Zeros of random holomorphic sections of big line bundles with continuous metrics

Let $X$ be a compact normal complex space, $L$ be a big holomorphic line bundle on $X$ and $h$ be a continuous Hermitian metric on $L$. We consider the spaces of holomorphic sections $H^0(X, L^{\otimes p})$ endowed with the inner product induced by $h^{\otimes p}$ and a volume form on $X$, and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current $c_1(L,h_{\mathrm{eq}})$ of the equilibrium metric $h_{\mathrm{eq}}$ associated to $h$. We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to $c_1(L,h_{\mathrm{eq}})$, for very general classes of probability measures on $H^0(X, L^{\otimes p})$.

math.CV↗
arXiv2023

Expected Multivolumes of Random Amoebas

We compute the expected multivolume of the amoeba of a random half dimensional complete intersection in $\mathbb{CP}^{2n}$. We also give a relative generalization of our result to the toric case.

math.AG↗
arXiv2023

Zero Distribution of Random Bernoulli Polynomial Mappings

In this note, we study asymptotic zero distribution of multivariable full system of random polynomials with independent Bernoulli coefficients. We prove that with overwhelming probability their simultaneous zeros sets are discrete and the associated normalized empirical measure of zeros asymptotic to the Haar measure on the unit torus.

math.CV↗
arXiv2023

On Dynamics of Asymptotically Minimal Polynomials

We study dynamical properties of asymptotically extremal polynomials associated with a non-polar planar compact set E. In particular, we prove that if the zeros of such polynomials are uniformly bounded then their Brolin measures converge weakly to the equilibrium measure of E. In addition, if E is regular and the zeros of such polynomials are sufficiently close to E then we prove that the filled Julia sets converge to polynomial convex hull of E in the Klimek topology.

math.CV↗
arXiv2022

Random Polynomials in Several Complex Variables

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$ with i.i.d. complex random variable coefficients $\{a_j\}$ where $\{p_j\}$ form an orthonormal basis for a Bernstein-Markov measure on a compact set $K\subset {\bf C}^d$. Here $m_n$ is the dimension of $\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in ${\bf C}^d$. We consider more general bases $\{p_j\}$, which include, e.g., higher-dimensional generalizations of Fekete polynomials. Moreover we allow $H_n(z):=\sum_{j=1}^{m_n} a_{nj}p_{nj}(z)$; i.e., we have an array of basis polynomials $\{p_{nj}\}$ and random coefficients $\{a_{nj}\}$. This always occurs in a weighted situation. We prove results on convergence in probability and on almost sure convergence of $\frac{1}{n}\log |H_n|$ in $L^1_{loc}({\bf C}^d)$ to the (weighted) extremal plurisubharmonic function for $K$. We aim for weakest possible sufficient conditions on the random coefficients to guarantee convergence.

math.CV↗
arXiv2020

A survey on zeros of random holomorphic sections

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 new examples of measures supported in totally real subsets of the complex probability space.

math.CV↗
arXiv2020

Mass equidistribution for random polynomials

The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contains a wide range of random variables including standard Gaussian and all bounded random variables. We prove that for almost every sequence of random polynomials their normalized zero currents become equidistributed with respect to a deterministic extremal current. The main ingredients of the proof are Bergman kernel asymptotics, mass equidistribution of random polynomials and concentration inequalities for subgaussian quadratic forms.

math.CV↗
arXiv2020

Universality results for zeros of random holomorphic sections

In this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact Kähler complex space $X$. Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections. In the case when $X$ is a compact Kähler manifold, we also prove an off-diagonal exponential decay estimate for the Bergman kernels of a sequence of positive line bundles on $X$.

math.CV↗
arXiv2020

Sampling Property of Fekete Points for Random Holomorphic Sections

In this note, we obtain the growth order of Lebesgue constants for Fekete points associated with tensor powers of a positive line bundle. Moreover, by endowing the space of global holomorphic sections with a natural Gaussian probability measure we prove that Fekete points are sampling for random holomorphic sections.

math.CV↗