Equidistribution of zeros of random polynomials

Igor Pritsker; Koushik Ramachandran

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arXiv · 1607.02855

Equidistribution of zeros of random polynomials

Igor Pritsker·Koushik Ramachandran

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Abstract

We study the asymptotic distribution of zeros for the random polynomials $P_n(z) = \sum_{k=0}^n A_k B_k(z)$, where $\{A_k\}_{k=0}^{\infty}$ are non-trivial i.i.d. complex random variables. Polynomials $\{B_k\}_{k=0}^{\infty}$ are deterministic, and are selected from a standard basis such as Szeg\H{o}, Bergman, or Faber polynomials associated with a Jordan domain $G$ bounded by an analytic curve. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$ if and only if $\mathbb{E}[\log^+|A_0|]<\infty$.

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BibTeX RIS

Igor Pritsker, Koushik Ramachandran. 2016-07-11. Equidistribution of zeros of random polynomials. https://arxiv.org/abs/1607.02855

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