Paper detail

Random polynomials and pluripotential-theoretic extremal functions

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n 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 |H_n| pointwise and in L^1_{loc}(C^m) to V_{K,Q}. In addition we obtain results on a.s. convergence of a sequence of normalized zero currents dd^c [1/n log |H_n|] to dd^c V_{K,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.