2,936 research outputs found
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
Random zeros on complex manifolds: conditional expectations
We study the conditional distribution of zeros of a Gaussian system of random
polynomials (and more generally, holomorphic sections), given that the
polynomials or sections vanish at a point p (or a fixed finite set of points).
The conditional distribution is analogous to the pair correlation function of
zeros, but we show that it has quite a different small distance behavior. In
particular, the conditional distribution does not exhibit repulsion of zeros in
dimension one. To prove this, we give universal scaling asymptotics for the
conditional zero distribution around p. The key tool is the conditional Szego
kernel and its scaling asymptotics.Comment: 27 page
Universality and scaling of correlations between zeros on complex manifolds
We study the limit as of the correlations between simultaneous
zeros of random sections of the powers of a positive holomorphic line
bundle over a compact complex manifold , when distances are rescaled so
that the average density of zeros is independent of . We show that the limit
correlation is independent of the line bundle and depends only on the dimension
of and the codimension of the zero sets. We also provide some explicit
formulas for pair correlations. In particular, we provide an alternate
derivation of Hannay's limit pair correlation function for SU(2) polynomials,
and we show that this correlation function holds for all compact Riemann
surfaces.Comment: 3 figure
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