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

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∈Dαk∈D for any k⩾0k⩾0. If we consider nn complex numbers α0,α1,…,αn-2∈Dα0,α1,…,αn-2∈D and αn-1∈∂Dαn-1∈∂D, we can use the previous recurrence relation to define the monic polynomials Φ0,Φ1,…,ΦnΦ0,Φ1,…,Φn. The polynomial Φn(z)=Φn(z;α0,…,αn-2,αn-1)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-1α0,α1,…,αn-1.We take α0,α1,…,αn-2α0,α1,…,αn-2 i.i.d. random variables distributed uniformly in a disk of radius r<1r<1 and αn-1αn-1 another random variable independent of the previous ones and distributed uniformly on the unit circle. For any nn we will consider the random paraorthogonal polynomial Φn(z)=Φn(z;α0,…,αn-2,αn-1)Φn(z)=Φn(z;α0,…,αn-2,αn-1). The zeros of ΦnΦn are nn random points on the unit circle.We prove that for any eiθ∈∂Deiθ∈∂D the distribution of the zeros of ΦnΦn in intervals of size O(1n) near eiθeiθ is the same as the distribution of nn independent random points uniformly distributed on the unit circle (i.e., Poisson). This means that, for large nn, there is no local correlation between the zeros of the considered random paraorthogonal polynomials.