Mihai Stoiciu
The Statistical Distribution of the zeros of Random Paraorthogonal Polynomials on the Unit Circle
(460K, pdf)

ABSTRACT.  We consider polynomials on the 
unit circle defined by the recurrence relation 
\[ 
\Phi_{k+1}(z) = z \Phi_{k} (z) - \overline{\alpha}_{k} 
\Phi_k^{*}(z) \qquad k \geq 0, \quad \Phi_0=1 
\] 
For each $n$ we take $\alpha_0, \alpha_1, \ldots ,\alpha_{n-2}$ 
i.i.d. random variables distributed uniformly in a disk of radius $r 
< 1$ and $\alpha_{n-1}$ another random variable independent of the 
previous ones and distributed uniformly on the unit circle. The 
previous recurrence relation gives a sequence of random 
paraorthogonal polynomials $\{\Phi_n\}_{n \geq 0}$. For any $n$, the 
zeros of $\Phi_n$ are $n$ random points on the unit circle. 
We prove that for any $e^{i \theta} \in \partial \bbD$ the 
distribution of the zeros of $\Phi_n$ in intervals of size 
$O(\frac{1}{n})$ near $e^{i \theta}$ is the same as the distribution 
of $n$ independent random points uniformly distributed on the unit 
circle (i.e., Poisson). This means that, for large $n$, there is no 
local correlation between the zeros of the considered random 
paraorthogonal polynomials.