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Zeros of random orthogonal polynomials on the unit circle

Citation

Stoiciu, Mihai (2005) Zeros of random orthogonal polynomials on the unit circle. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-05272005-110242

Abstract

We study the statistical distribution of the zeros of some classes of random orthogonal polynomials on the unit circle. For each n we take the random Verblunsky coefficients alpha_0, alpha_1,...,alpha_{n-2} to be independent identically distributed random variables uniformly distributed in a disk of radius r < 1 and alpha_{n-1} to be another random variable independent of the previous ones and distributed uniformly on the unit circle. These coefficients define a sequence of random paraorthogonal polynomials Phi_n. For any n, the zeros of Phi_n are n random points on the unit circle.

We prove that, for any point p on the unit circle, the distribution of the zeros of Phi_n in intervals of size O(1/n) near p is the same as the distribution of n independent random points uniformly distributed on the unit circle (i.e., Poisson). Therefore, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials.

The same result holds when we take alpha_0, alpha_1,...,alpha_{n-2} to be independent identically distributed random variables uniformly distributed in a circle of radius r < 1 and alpha_{n-1} to be another random variable independent of the previous ones and distributed uniformly on the unit circle.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:distribution of eigenvalues; orthogonal polynomials; random unitary matrices; random Verblunsky coefficients
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Simon, Barry M.
Thesis Committee:
  • Simon, Barry M. (chair)
  • Damanik, David
  • Lorden, Gary A.
  • Killip, Rowan
Defense Date:9 May 2005
Author Email:mihai (AT) caltech.edu
Record Number:CaltechETD:etd-05272005-110242
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-05272005-110242
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2145
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:27 May 2005
Last Modified:26 Dec 2012 02:47

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