5,971 research outputs found
Correlations and Pairing Between Zeros and Critical Points of Gaussian Random Polynomials
We study the asymptotics of correlations and nearest neighbor spacings
between zeros and holomorphic critical points of , a degree N Hermitian
Gaussian random polynomial in the sense of Shiffman and Zeldtich, as N goes to
infinity. By holomorphic critical point we mean a solution to the equation
Our principal result is an explicit asymptotic formula
for the local scaling limit of \E{Z_{p_N}\wedge C_{p_N}}, the expected joint
intensity of zeros and critical points, around any point on the Riemann sphere.
Here and are the currents of integration (i.e. counting
measures) over the zeros and critical points of , respectively. We prove
that correlations between zeros and critical points are short range, decaying
like e^{-N\abs{z-w}^2}. With \abs{z-w} on the order of however,
\E{Z_{p_N}\wedge C_{p_N}}(z,w) is sharply peaked near causing zeros
and critical points to appear in rigid pairs. We compute tight bounds on the
expected distance and angular dependence between a critical point and its
paired zero.Comment: 35 pages, 3 figures. Some typos corrected and Introduction revise
Asymptotic expansion of the off-diagonal Bergman kernel on compact K\"ahler manifolds
We compute the first four coefficients of the asymptotic off-diagonal
expansion of the Bergman kernel for the N-th power of a positive line bundle on
a compact Kaehler manifold, and we show that the coefficient b_1 of the
N^{-1/2} term vanishes when we use a K-frame. We also show that all the
coefficients of the expansion are polynomials in the K-coordinates and the
covariant derivatives of the curvature and are homogeneous with respect to the
weight w.Comment: Added references to a paper and a new preprint of X. Ma and G.
Marinescu. Added an exampl
Distribution of zeros of random and quantum chaotic sections of positive line bundles
We study the limit distribution of zeros of certain sequences of holomorphic
sections of high powers of a positive holomorphic Hermitian line bundle
over a compact complex manifold . Our first result concerns `random'
sequences of sections. Using the natural probability measure on the space of
sequences of orthonormal bases of , we show that for
almost every sequence , the associated sequence of zero currents
tends to the curvature form of . Thus, the zeros of
a sequence of sections chosen independently and at random
become uniformly distributed. Our second result concerns the zeros of quantum
ergodic eigenfunctions, where the relevant orthonormal bases of
consist of eigensections of a quantum ergodic map. We show that
also in this case the zeros become uniformly distributed
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