Random polynomials and pluripotential-theoretic extremal functions

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There is a natural pluripotential-theoretic extremal function Vₖ,Q associated to a closed subset K of Cᵐ 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 1/n log |Hₙ| pointwise and in L¹loc(Cᵐ) to Vₖ,Q. In addition we obtain results on a.s. convergence of a sequence of normalized zero currents ddᶜ [1/n log |Hₙ|] to ddᶜ Vₖ,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 Kahler manifold.