Random Sums of Weighted Orthogonal Polynomials in \mathbb C^d

2024/12/16 by Thomas Bloom, Duncan Dauvergne, Bloom, T. +3
Mathematics · Computer Science · #Mathematical functions and polynomials #Analytic Number Theory Research #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2412.11969

Abstract

We consider random polynomials of the form G_n(z):= ∑_{|α|≤ n} ξ⁽ⁿ⁾_αp_n,α(z) where \ξ⁽ⁿ⁾_α\_{|α|≤ n} are i.i.d. (complex) random variables and \p_n,α\_{|α|≤ n} form a basis for \mathcal P_n, the holomorphic polynomials of degree at most n in \mathbb C^d. In particular, this includes the setting where \p_n,α\ are orthonormal in a space L²(e^{-2n Q} τ), where τ is a compactly supported Bernstein-Markov measure and Q is a continuous weight function. Under an optimal moment condition on the random variables \ξ⁽ⁿ⁾_α\, in dimension d=1 we prove convergence in probability of the zero measure to the weighted equilibrium measure, and in dimension d ≥ 2 we prove convergence of zero currents.

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