Equidistribution of zeros of random polynomials

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• 作者：Igor Pritsker ; ^{igor@math.okstate.edu} ; Koushik Ramachandran ^{koushik.math@gmail.com}
• 关键词：Random polynomials ; Orthogonal polynomials ; Zero distribution
• 刊名：Journal of Approximation Theory
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：215
• 期：Complete
• 页码：106-117
• 全文大小：260 K
• 卷排序：215

文摘

We study the asymptotic distribution of zeros for the random polynomials Pn(z)=∑k=0nAkBk(z), where {Ak}k=0∞ are non-trivial i.i.d. complex random variables. Polynomials {Bk}k=0∞ are deterministic, and are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain GG bounded by an analytic curve. We show that the zero counting measures of PnPn converge almost surely to the equilibrium measure on the boundary of GG if and only if E[log+|A0|]<∞E[log+|A0|]<∞.