The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle

• Authors: Mihai Stoiciu
• Venue: Journal of Approximation Theory
• Year: 2006

The orthogonal polynomials on the unit circle are defined by the recurrence relation Φk+1(z) = zΦk(z) - α-kΦk*(z), k ≥ 0, Φ0 = 1, where αk ∈ D for any k ≥ 0. If we consider n complex numbers α0, α1,...,αn-2 ∈ D and αn-1 ∈ ∂D, we can use the previous recurrence relation to define the monic polynomials Φ0, Φ1,...,Φi. The polynomial Φn(z) = Φn (z; α0,...,αn-2, αn-1) obtained in this way is called the paraorthogonal polynomial associated to the coefficients α0, α1,...,αn-1.We take α0, α1,...,αn-2 i.i.d, random variables distributed uniformly in a disk of radius r n-1 another random variable independent of the previous ones and distributed uniformly on the unit circle. For any n we will consider the random paraorthogonal polynomial Φn(z) = Φn(z; α0,...,αn-2, αn-1). The zeros of Φn are n random points on the unit circle.We prove that for any eiθ; ∈ ∂D the distribution of the zeros of Φn in intervals of size O(1/n) near eiθ; 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.