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Seminar on Complex Analysis (Gonchar Seminar)
May 20, 2013 18:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)

Asymptotic behaviour of zeros of random polynomials and analytic functions

D. N. Zaporozhets

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: For any analytic function $G$ denote by $\mu_{G}$ a measure counting the complex zeros of $G$ according to their multiplicities. Let $\xi_0,\xi_1,\ldots$ be non-degenerate independent and identically distributed random variables. Consider a random polynomial
$$G_n(z)=\sum_{k=0}^n\xi_kz^k.$$
The first question we are interested in is an asymptotic behaviour of the average number of real zeros of $G_n$ as $n\to\infty$ under different assumptions on the distribution of $\xi_0$. Afterwards we consider all complex zeros of $G_n$ and study the asymptotic behaviour of random empirical measure $\mu_{G_n}$.
$$G_n(z)=\sum_{k=0}^{\infty} \xi_k f_{k,n}z^k.$$