Number Variance of Random Zeros on Complex Manifolds

By Bernard Shiffman, Steve Zelditch. 2006

113 citations recorded by the source corpus.

Abstract.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 Kähler manifold.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e42b6533f9dddc5ac94f101d4610ea14e381f498

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Real and complex zeros of Riemannian random waves (2008, methodology): “In the K¨ ahler case, asymptotic formulae for the variance of smooth linear statistics are given in [ SZ2 , SZ3], but the method does not apply in the real case.”
Equidistribution of zeros of random holomorphic sections (2013, unknown): “If a (n) j are standard complex Gaussian then it is classical that (see [26, 27, 7]) the identity (3.”
Fluctuations of eigenvalues of random normal matrices (2008, methodology): “The method has also been used in the parallel work on linear statistics of zeros of Gaussian analytic functions initiated by Sodin–Tsirelson [34] and generalized by Shiffman–Zelditch [33].”
EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS (2013, background): “A similar probabilistic study of complex projective manifolds has been performed by Shiffman and Zelditch (see [2, 30, 33] for example, and also [3, 36]).”