Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n
By Robert J. Berman. 2007
78 citations recorded by the source corpus.
Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is differentiable and all of its first partial derivatives are locally Lipshitz continuous. The convergence is studied in the large k limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the Monge-Ampere measure of the weight function itself on a certain set. A setting of polynomials associated to a given Newton polytope, scaled by k, is also considered. These results apply directly to the study of the distribution of zeroes of random polynomials and of the eigenvalues of random normal matrices.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/446e73bad18cedb4ef246bc9dcfa07a4a9e9526a
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In Applied Mathematics
- Y. Ameur
- Rémi Bardenet
- Adrien Hardy
- H. Hedenmalm
- Nikolai G. Makarov
- J. A. Pel'aez
- Alexei Poltoratski
- Mishko Mitkovski
In Geometry and Topology
Citations in the corpus
- Fluctuations of eigenvalues of random normal matrices (2008, methodology): “For this purpose, we will use a result, due to Berman [7], concerning the near-diagonal bulk asymptotics of the correlation kernel.”
- Monte Carlo with determinantal point processes (2016, background): “Although DPPs on higher-dimensional supports have attracted attention in complex geometry [Berman, 2009a,b, 2013, 2014], in statistics [Lavancier et al., 2015, Møller et al., 2015], and in physics [Torquato et al., 2008, Scardicchio et al., 2009], it seems no CLT has been established yet when d > 3.”