The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle |
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Authors: | Mihai Stoiciu |
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Institution: | Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA |
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Abstract: | The orthogonal polynomials on the unit circle are defined by the recurrence relation where for any k 0. If we consider n complex numbers and , we can use the previous recurrence relation to define the monic polynomials Φ0,Φ1,…,Φn. 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<1 and αn-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 the distribution of the zeros of Φn in intervals of size 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. |
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Keywords: | Zeros of orthogonal polynomials Random verblunsky coefficients Random CMV matrices |
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