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Random polynomials of high degree and Lévy concentration of measure. (English) Zbl 1060.32012

In this paper on several complex variables, complex geometry, and probability the authors continue their long ongoing studies of random polynomials, random holomorphic sections of high powers of positive line bundles over Kähler manifolds, and random almost holomorphic sections of high powers of positive line bundles over symplectic almost complex manifolds. The authors are among the main contributors to the above topics.
The framework for the paper’s study is that we can regard the sections \(s_N\) of \(L^N\) as \(N\)-th degree homogeneous complex valued functions on the unit circle bundle of the dual line bundle \(L^*\). In this way sections of \(L^N\) for various \(N\) live over the same space, and thus are comparable within the same setup. The authors obtain estimates of the \({\mathcal L}^p\) and \({\mathcal C}^k\) norms of \({\mathcal L}^2\) normalized random sections \(s_N\) of the \(N\)-th power \(L^N\) of a line bundle \(L\) of the type \(\| s_N\| _\infty=O(\sqrt{\log N})\), \(\| s_N\| _p=O(1)\) for \(2\leq p<\infty\), and \(\| \nabla^ks_N\| _\infty=O(\sqrt{N^k\log N})\) in the holomorphic case almost surely as \(N\to\infty\), and in addition in the symplectic almost complex case they find that \(\| \overline\partial s_N\| _\infty=O(\sqrt{\log N})\) and \(\| \nabla^k\overline\partial s_N\| _\infty=O(\sqrt{N^k\log N})\).
The proofs depend on analyzing local Heisenberg models, asymptotics of the relevant Bergman Szegő type kernels, some classical integral inequalities such as the Schur-Young inequality, the Schwarz inequality, and probabilistic tools such as the Chebyshev inequality, and the Lévy concentration of measures phenomenon for Lipschitz functions over spheres around their medians. One of the main novelties of the paper is to bring into play the Lévy concentration – whose relevance could be guessed also from the fact that the \(s_N\) are random points of large finite dimensional unit spheres in the space of sections of \(L^N\).
The paper is very clearly written, informative, and pleasant to read.

MSC:

32Q15 Kähler manifolds
53D35 Global theory of symplectic and contact manifolds
60D05 Geometric probability and stochastic geometry
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