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RANDOM POLYNOMIALS IN SEVERAL COMPLEX VARIABLES

2023    Bayraktar, Turgay; Bloom, Thomas; Levenberg, Norm

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials H-n(z) := Sigma(mn)(j=1) a(j)p(j)(z) that are linear combinations of basis polynomials {p(j)} with i.i.d. complex random variable coefficients {a(j)} where {p(j)} form an orthonormal basis for a Bernstein-Markov measure on a compact set K subset of C-d. Here mn is the dimension of P-n, the holomorphic polynomials of degree at most n in C-d. We consider more general bases {p(j)}, which include, e.g., higher-dimensional generalizations of Fekete polynomials. Moreover we allow H-n(z) := Sigma(mn)(j=1) a(nj)p(nj)(z), i.e., we have an array of basis polynomials {p(nj)} and random coefficients {a(nj)}. This always occurs in a weighted situation. We prove results on convergence in probability and on almost sure convergence of 1/n log vertical bar H-n vertical bar in L-loc(1)(C-d) to the (weighted) extremal plurisubharmonic function for K. We aim for weakest possible sufficient conditions on the random coefficients to guarantee convergence.