טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentLulav Ari
SubjectThe Empirical Measure of Roots of Random Polynomials
DepartmentDepartment of Electrical Engineering
Supervisor Professor Ofer Zeitouni


Abstract

   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.