04-405 Mihai Stoiciu

The Statistical Distribution of the zeros of Random Paraorthogonal Polynomials on the Unit Circle (460K, pdf) 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 $\alpha_0, \alpha_1, \ldots ,\alpha_{n-2}$ 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.

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