Zeros Of Random Holomorphic Sections Research Articles

Universality results for zeros of random holomorphic sections

In this work we prove a 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 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 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 X .

Equidistribution for sequences of line bundles on normal Kähler spaces

We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.

Equidistribution and convergence speed for zeros of holomorphic sections of singular Hermitian line bundles

We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.

Equidistribution of zeros of random holomorphic sections for moderate measures

We establish an equidistribution theorem for the zeros of random holomorphic sections of high powers of a positive holomorphic line bundle. The equidistribution is associated with a family of singular moderate measures. We also give a convergence speed for the equidistribution.

Equidistribution of zeros of random holomorphic sections

We study asymptotic distribution of zeros of random holomorphic sections of high powers of positive line bundles defined over projective homogenous manifolds. We work with a wide class of distributions that includes real and complex Gaussians. As a special case, we obtain asymptotic zero distribution of multivariate complex polynomials given by linear combinations of orthogonal polynomials with i.i.d. random coefficients. Namely, we prove that normalized zero measures of m i.i.d random polynomials, orthonormalized on a regular compact set $K\subset \Bbb{C}^m,$ are almost surely asymptotic to the equilibrium measure of $K$.

CONVERGENCE OF FUBINI–STUDY CURRENTS FOR ORBIFOLD LINE BUNDLES

We discuss positive closed currents and Fubini–Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini–Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvature is strictly positive, generalizing a well-known theorem of Tian. We include applications to the asymptotic distribution of zeros of random holomorphic sections.

Large Deviations of Empirical Measures of Zeros on Riemann Surfaces

This is a continuation of a project on large deviations for the empirical measures of zeros of random holomorphic sections of random line bundles over a Riemann surface X. In a previous article with O. Zeitouni (arXiv:0904.4271), we proved an LDP for random polynomials in the genus zero case. In higher genus, there is a Picard variety of line bundles and so the line bundle L is a random variable as well as the section s. The space of pairs (L, s) is known as the vortex moduli space. The zeros of (L, s) fill out the configuration space $X^{(N)}$ of $N$ points of $X$. The LDP shows that the configurations concentrate at one equilibrium measure exponentially fast. The new features of the proof involve Abel-Jacobi theory, the prime form and bosonization.

Number Variance of Random Zeros on Complex Manifolds

We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set \(U \subset {\mathbb{C}}^m\) with smooth boundary is asymptotic to \(N^{{m-1}/2} \nu_{mm} {\rm Vol}(\partial U)\), where \(\nu_{mm}\) is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on \({\mathbb{C}}^{m}\). Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact Kahler manifold.

Energy of zeros of random sections on Riemann surfaces

The purpose of this paper is to determine the asymptotic of the average energy of a configuration of N zeros of a system of random polynomials of degree N as N?8 and more generally the zeros of random holomorphic sections of a line bundle L?M over any Riemann surface M . We also compare our results to the well-known minimum of energies.