Random Polynomials and Pluripotential-Theoretic Extremal Functions

By Thomas F. Bloom, Norman Levenberg. 2013

76 citations recorded by the source corpus.

There is a natural pluripotential-theoretic extremal function VK,Q associated to a closed subset K of ℂm$\mathbb {C}^{m}$ and a real-valued, continuous function Q on K. We define random polynomials Hn whose coefficients with respect to a related orthonormal basis are independent, identically distributed complex-valued random variables having a very general distribution (which includes both normalized complex and real Gaussian distributions) and we prove results on a.s. convergence of a sequence {1nlog|Hn|}$\{\frac {1}{n}\log |H_{n}|\}$ pointwise and in Lloc1(ℂm)$L^{1}_{loc}(\mathbb {C}^{m})$ to VK,Q. In addition we obtain results on a.s. convergence of a sequence of normalized zero currents {ddc1nlog|Hn|}$\{dd^{c}\left (\frac {1}{n}\log |H_{n}|\right )\}$ to ddcVK,Q as well as asymptotics of expectations of these currents. All these results extend to random polynomial mappings and to a more general setting of positive holomorphic line bundles over a compact Kähler manifold.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c3ab4dd2ec0c2649ae7397e8fd03b497ec301ed6

Citations in the corpus

• Equidistribution of zeros of random holomorphic sections (2013, unknown): “We remark that the scalar L 2 -product used in [12] is different than the one used in [7] and this paper.”