Real zeros of random polynomials: Scaling and universality

Xiaojun Di, Purdue University

Abstract

The objective of the present study is to investigate the asymptotic properties and behaviors of the distributions and correlations between real zeros of random polynomials. We first obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial, we obtain similar formula for the limit 2-point correlation function as Hannay did, and obtain more explicit formula for the limit m-point correlation functions than Hannay did [1]. We further show that for Kac random polynomial the zeros in the interval (−1,1) are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Secondly, we study the correlation functions for SO(2) random polynomial. We calculate the exact analytical limit 2-point correlation function and obtain analytical formula for limit m-point correlation functions between the scaled zeros of the SO(2) random polynomial. In addition, we calculate the variance of the number of real zeros for a SO(2) random polynomial. Thirdly, we study the existence and universality of the scaling limit of the distributions and correlations between real zeros of a general class of random polynomials. For a rather general class of random polynomials, generalized from SO(2) random polynomial, we prove that away from the origin such scaling limits exist and are universal so that they do not depend on the distribution of the coefficients. Near to the origin, we prove that the scaling limits are not universal, and we find a crossover from the nonuniversal asymptotics of the density of the probability distribution of zeros at the origin to the universal one away from the origin. We further show that the average number of zeros of a random polynomial from this class is asymptotically equal to [special characters omitted]. Finally, we simulate our results through numeric computations and the comparison between analytical results and simulated results show great consistency.

Degree

Ph.D.

Advisors

Bleher, Purdue University.

Subject Area

Mathematics

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