# Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains

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## Abstract

We consider the zero distribution of random polynomials of the form \(P_n(z) = \sum _{k=0}^n a_k B_k(z)\), where \(\{a_k\}_{k=0}^{\infty }\) are non-trivial i.i.d. complex random variables with mean 0 and finite variance. Polynomials \(\{B_k\}_{k=0}^{\infty }\) are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain *G* whose boundary is \(C^{2, \alpha }\) smooth. We show that the zero counting measures of \(P_n\) converge almost surely to the equilibrium measure on the boundary of *G*. We also show that if \(\{a_k\}_{k=0}^{\infty }\) are i.i.d. random variables, and the domain *G* has analytic boundary, then for a random series of the form \(f(z) =\sum _{k=0}^{\infty }a_k B_k(z),\)\(\partial {G}\) is almost surely the natural boundary for *f*(*z*).

## Keywords

Random polynomials Orthogonal polynomials Zero distribution Natural boundary## Mathematics Subject Classification

MSC 60F05 31A15 30B20 30B30## Notes

### Acknowledgements

Research of the first author was partially supported by the National Security Agency (Grant H98230-15-1-0229) and by the American Institute of Mathematics. We are grateful to the referee for pointing out a correction in the statement of Theorem 1.4, and for other remarks and suggestions which helped in improving the exposition.

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