Computational Methods and Function Theory

, Volume 19, Issue 3, pp 401–410 | Cite as

Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains

  • Igor Pritsker
  • Koushik RamachandranEmail author


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).


Random polynomials Orthogonal polynomials Zero distribution Natural boundary 

Mathematics Subject Classification

MSC 60F05 31A15 30B20 30B30 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsOklahoma State UniversityStillwaterUSA
  2. 2.Centre for Applicable MathematicsTata Institute of Fundamental ResearchBengaluruIndia

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