Random polynomials and Green functions

Abstract

Let K be a regular (in the sense of pluripotential theory) compact set in ℂn and let VK(z) denote its pluricomplex Green function with a logarithmic singularity at ∞. Then, with probability 1, a sequence of random polynomials {fα} (linear combination of monomials of lexicographic order ≤ α) gives the pluricomplex Green function via the formula (lim¯α(1/deg(fα))log|fα(z)|)*=VK(z) for all z ∈ ℂn. In the one-dimensional case, this result may be used to generalize a result of Shiffman-Zelditch on the limiting normalized distribution of zeroes of random polynomials.