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.