Equidistribution of zeros of random polynomials

2016/07/11 by Pritsker, Igor, Ramachandran, Koushik
#30 #31 #60 #Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1607.02855

Abstract

We study the asymptotic distribution of zeros for the random polynomials P_n(z) = ∑_k=0ⁿ A_k B_k(z), where \A_k\_k=0^∞ are non-trivial i.i.d. complex random variables. Polynomials \B_k\_k=0^∞ are deterministic, and are selected from a standard basis such as Szegő, 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 𝔼[log⁺|A₀|]

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