## Abstract

Consider random polynomials of the form G_{n}=∑_{i=0}^{n}ξ_{i}p_{i}, where the ξ_{i} are i.i.d. non-degenerate complex random variables, and {p_{i}} is a sequence of orthonormal polynomials with respect to a regular measure τ supported on a compact set K. We show that the zero measure of G_{n} converges weakly almost surely to the equilibrium measure of K if and only if Elog(1+|ξ_{0}|)<∞. This generalizes the corresponding result of Ibragimov and Zaporozhets in the case when p_{i}(z)=z^{i}. We also show that the zero measure of G_{n} converges weakly in probability to the equilibrium measure of K if and only if P(|ξ_{0}|>e^{n})=o(n^{−1}). Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in L^{p}(τ), where p∈(0,∞]. In particular, sequences of L^{p}-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.

Original language | English (US) |
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Article number | 107691 |

Journal | Advances in Mathematics |

Volume | 384 |

DOIs | |

State | Published - Jun 25 2021 |

## All Science Journal Classification (ASJC) codes

- General Mathematics

## Keywords

- General orthogonal polynomials
- Orthogonal polynomials
- Potential theory
- Random polynomials
- Universality
- Zeros of random polynomials