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## Zeros of Random Orthogonal Polynomials

Yeager, Aaron Michael

Yeager, Aaron Michael

##### Abstract

Let $\{f_j\}$ be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line (OPRL) or on the unit circle (OPUC). We study zero distribution of random linear combinations of the form

$$P_n(z)=\sum_{j=0}^n\eta_jf_j(z),$$

where $\{\eta_j\}$ are random variables. We give quantitative estimates on the zeros accumulating on the unit circle for a wide class of random polynomials $P_n$. When the coefficients $\{\eta_j\}$ are independent identically distributed (i.i.d.) real-valued standard Gaussian, we give asymptotics for the expected number of zeros of various classes of random sums $P_n$ spanned by OPUC. For the case when the coefficients $\{\eta_j\}$ are i.i.d.~complex-valued standard Gaussian coefficients, we derive a formula for the expected number of zeros of $P_n$. The formula is then applied to give asymptotics of the expected number of zeros of $P_n$ when $\{f_j\}$ are from the Nevai class. We also compute the limiting value as $n\rightarrow \infty$ of the variance of the number of zeros of $P_n$ in annuli that do not contain the unit circle for the case when $\{\eta_j\}$ are i.i.d.~complex-valued standard Gaussian random variables, and $\{f_j\}$ are OPUC from the Nevai class. In the case of annuli that contain the unit circle, for a wide class of random variables $\{\eta_j\}$ and $\{f_j\}$ that are OPUC, we give quantitative results that show the variance of the number of zeros of $P_n$ scaled by $n^2$ tends to zero as $n$ tends to infinity. The work is concluded by providing formulas for the variance of the number of zeros of a random orthogonal power series, specifically when $\sum_{j=0}^{\infty}\eta_j f_j(z)$, with $\{\eta_j\}$ being i.i.d.~complex-valued standard Gaussian, and $\{f_j\}$ are OPUC from the Szeg\H{o} class.

$$P_n(z)=\sum_{j=0}^n\eta_jf_j(z),$$

where $\{\eta_j\}$ are random variables. We give quantitative estimates on the zeros accumulating on the unit circle for a wide class of random polynomials $P_n$. When the coefficients $\{\eta_j\}$ are independent identically distributed (i.i.d.) real-valued standard Gaussian, we give asymptotics for the expected number of zeros of various classes of random sums $P_n$ spanned by OPUC. For the case when the coefficients $\{\eta_j\}$ are i.i.d.~complex-valued standard Gaussian coefficients, we derive a formula for the expected number of zeros of $P_n$. The formula is then applied to give asymptotics of the expected number of zeros of $P_n$ when $\{f_j\}$ are from the Nevai class. We also compute the limiting value as $n\rightarrow \infty$ of the variance of the number of zeros of $P_n$ in annuli that do not contain the unit circle for the case when $\{\eta_j\}$ are i.i.d.~complex-valued standard Gaussian random variables, and $\{f_j\}$ are OPUC from the Nevai class. In the case of annuli that contain the unit circle, for a wide class of random variables $\{\eta_j\}$ and $\{f_j\}$ that are OPUC, we give quantitative results that show the variance of the number of zeros of $P_n$ scaled by $n^2$ tends to zero as $n$ tends to infinity. The work is concluded by providing formulas for the variance of the number of zeros of a random orthogonal power series, specifically when $\sum_{j=0}^{\infty}\eta_j f_j(z)$, with $\{\eta_j\}$ being i.i.d.~complex-valued standard Gaussian, and $\{f_j\}$ are OPUC from the Szeg\H{o} class.

##### Date

2019-05-01