 04405 Mihai Stoiciu
 The Statistical Distribution of the zeros of Random Paraorthogonal Polynomials on the Unit Circle
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Dec 8, 04

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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 $\alpha_0, \alpha_1, \ldots ,\alpha_{n2}$
i.i.d. random variables distributed uniformly in a disk of radius $r
< 1$ and $\alpha_{n1}$ 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.
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