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Mass Equidistribution for Random Polynomials

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Abstract

The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contains a wide range of random variables including standard Gaussian and all bounded random variables. We prove that for almost every sequence of random polynomials their normalized zero currents become equidistributed with respect to a deterministic extremal current. The main ingredients of the proof are Bergman kernel asymptotics, mass equidistribution of random polynomials and concentration inequalities for subgaussian quadratic forms.

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Authors and Affiliations

  1. Faculty of Engineering and Natural Sciences, Sabanci University, İstanbul, Turkey

    Turgay Bayraktar

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  1. Turgay Bayraktar
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Correspondence to Turgay Bayraktar.

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T. Bayraktar is partially supported by TÜ BİTAK grants BİDEB-2232/118C006, ARDEB-3501/118F049 and Science Academy BAGEP grant.

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Bayraktar, T. Mass Equidistribution for Random Polynomials. Potential Anal 53, 1403–1421 (2020). https://doi.org/10.1007/s11118-019-09811-w

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