

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

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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 nondegenerate 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 $\mu_{G_n}$.
Finally, we consider the generalization of the previous problem to a random analytic function of the following form:
$$
G_n(z)=\sum_{k=0}^{\infty} \xi_k f_{k,n}z^k.
$$

