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Theme for TIFR Centre For Applicable Mathematics, Bangalore

Abstract: We study the asymptotic distribution of zeros for the random polynomials $$Pn(z)$$ = $$\sum^n_{k=0}\xi_kB_k(z),$$ where $$\{\xi_k\}^\infty_{k=0}$$ are non-trivial i.i.d. complex random variables. Polynomials $$\{B_k\}^\infty_{k=0}$$ are deterministic, and are selected from a standard basis such as Bergman or Szeg˝o 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 $$E[log^+|\xi_0|] < \infty$$. This talk is based on joint work with Igor Pritsker.