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

Abstract

The orthogonal polynomials on the unit circle are defined by the recurrence relation Φ k + 1 ( z ) = z Φ k ( z ) - α ¯ k Φ k * ( z ) , k ⩾ 0 , Φ 0 = 1 , where α k ∈ D for any k ⩾ 0 . If we consider n complex numbers α 0 , α 1 , … , α n - 2 ∈ D and α n - 1 ∈ ∂ D , we can use the previous recurrence relation to define the monic polynomials Φ 0 , Φ 1 , … , Φ n . The polynomial Φ n ( z ) = Φ n ( z ; α 0 , … , α n - 2 , α n - 1 ) obtained in this way is called the paraorthogonal polynomial associated to the coefficients α 0 , α 1 , … , α n - 1 . We take α 0 , α 1 , … , α n - 2 i.i.d. random variables distributed uniformly in a disk of radius r < 1 and α 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 Φ n ( z ) = Φ n ( z ; α 0 , … , α n - 2 , α n - 1 ) . The zeros of Φ n are n random points on the unit circle. We prove that for any e i θ ∈ ∂ D the distribution of the zeros of Φ n in intervals of size O ( 1 n ) near e i θ 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.