M.Sc Student | Lulav Ari |
---|---|

Subject | The Empirical Measure of Roots of Random Polynomials |

Department | Department of Electrical Engineering |

Supervisor | Professor Ofer Zeitouni |

We study the asymptotic
distribution of the complex roots of the random polynomial _{} with independent standard
normal coefficients. Based on an earlier work by Erdös and Turan, Arnold showed weak convergence with probability 1 of the empirical measure of the roots of
_{} to
the uniform measure on the unit circle. Ibragimov and Zeitouni showed, using
the Kac - Rice formula, in what manner the roots of _{} tend to approach the
unit circle as they obtained an exact expression for _{} where _{}, _{} fixed and _{} is the density
on Â of the expected empirical measure of roots of _{} in its polar
form.

Here, using the Kac -
Rice formula for the second moment and the result of Ibragimov and Zeitouni, we
extend Arnold's result for the case where the coefficients are i.i.d standard
normal r.v's. We introduce an empirical measure of normalized roots which gives
information not only about the asymptotic angular distribution of the roots,
but also about the way the modulus of the roots tend to 1. We show weak convergence
in probability of that empirical measure and give the limiting law. We show
that the (normalized) roots of _{} have asymptotically independent arguments
and moduli and evaluate their limiting densities.