arXiv subjects

Aaron Yeager

Publications and source records attributed to Aaron Yeager.

6 recordsLinked to original sources

arXiv2021

The complex zeros of random orthogonal polynomials

We utilize Cauchy's argument principle in combination with the Jacobian of a holomorphic function in several complex variables and the first moment of a ratio of two correlated complex normal random variables to prove explicit formulas for the density and the mean distribution of complex zeros of random polynomials spanned by orthogonal polynomials on the unit circle and on the unit disk. We then inquire into the consequences of their asymptotical evaluations.

math.PR↗

arXiv2020

Zeros of Complex Random Polynomials Spanned by Bergman Polynomials

We study the expected number of zeros of $$P_n(z)=\sum_{k=0}^n\eta_kp_k(z),$$ where $\{\eta_k\}$ are complex-valued i.i.d standard Gaussian random variables, and $\{p_k(z)\}$ are polynomials orthogonal on the unit disk. When $p_k(z)=\sqrt{(k+1)/\pi} z^k$, $k\in \{0,1,\dots, n\}$, we give an explicit formula for the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin. From our formula we establish the limiting value of the expected number of zeros, the expected number of zeros in a radially expanding disk, and show that the expected number of zeros in the unit disk is $2n/3$. Generalizing our basis functions $\{p_k(z)\}$ to be regular in the sense of Ullman--Stahl--Totik, and that the measure of orthogonality associated to polynomials is absolutely continuous with respect to planar Lebesgue measure, we give the limiting value of the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is $2n/3$.

math.CA↗

arXiv2017

Zeros of random orthogonal polynomials with complex Gaussian coefficients

Let $\{f_j\}_{j=0}^n$ be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the form $$P_n(z)=\sum_{j=0}^n\eta_jf_j(z),$$ where $\eta_0,\dots,\eta_n$ are complex-valued i.i.d.~standard Gaussian random variables. Using the Christoffel-Darboux formula, the density function for the expected number of zeros of $P_n$ in these cases takes a very simple shape. From these expressions, under the mere assumption that the orthogonal polynomials are from the Nevai class, we give the limiting value of the density function away from their respective sets where the orthogonality holds. In the case when $\{f_j\}$ are orthogonal polynomials on the unit circle, the density function shows that the expected number of zeros of $P_n$ are clustering near the unit circle. To quantify this phenomenon, we give a result that estimates the expected number of complex zeros of $P_n$ in shrinking neighborhoods of compact subsets of the unit circle.

math.CA↗

arXiv2017

Zeros of real random polynomials spanned by OPUC

Let $ \{\varphi_i\}_{i=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure $ \mu $. We study zero distribution of random linear combinations of the form \[ P_n(z)=\sum_{i=0}^{n-1}\eta_i\varphi_i(z), \] where $ \eta_0,\dots,\eta_{n-1} $ are i.i.d. standard Gaussian variables. We use the Christoffel-Darboux formula to simplify the density functions provided by Vanderbei for the expected number real and complex of zeros of $ P_n $. From these expressions, under the assumption that $ \mu $ is in the Nevai class, we deduce the limiting value of these density functions away from the unit circle. Under the mere assumption that $ \mu $ is doubling on subarcs of $ \T $ centered at $ 1 $ and $ -1 $, we show that the expected number of real zeros of $ P_n $ is at most \[ (2/\pi) \log n +O(1), \] and that the asymptotic equality holds when the corresponding recurrence coefficients decay no slower than $ n^{-(3+\epsilon)/2} $, $ \epsilon>0 $. We conclude with providing results that estimate the expected number of complex zeros of $ P_n $ in shrinking neighborhoods of compact subsets of $ \T $.

math.CA↗

arXiv2016

Zeros of random linear combinations of entire functions with complex Gaussian coefficients

We study zero distribution of random linear combinations of the form $$P_n(z)=\sum_{j=0}^nη_jf_j(z),$$ in any Jordan region $Ω\subset \mathbb C$. The basis functions $f_j$ are entire functions that are real-valued on the real line, and $η_0,\dots,η_n$ are complex-valued iid Gaussian random variables. We derive an explicit intensity function for the number of zeros of $P_n$ in $Ω$ for each fixed $n$. Our main application is to polynomials orthogonal on the real line. Using the Christoffel-Darboux formula the intensity function takes a very simple shape. Moreover, we give the limiting value of the intensity function when the orthogonal polynomials are associated to Szegő weights.

math.PR↗

arXiv2010

The Radio Number of $C_n \square C_n$

Radio labeling is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph $G$ subject to certain constraints involving the distances between the vertices. Specifically, a radio labeling of a connected graph $G$ is a function $c:V(G) \rightarrow \mathbb Z_+$ such that $$d(u,v)+|c(u)-c(v)|\geq 1+\text{diam}(G)$$ for every two distinct vertices $u$ and $v$ of $G$ (where $d(u,v)$ is the distance between $u$ and $v$). The span of a radio labeling is the maximum integer assigned to a vertex. The radio number of a graph $G$ is the minimum span, taken over all radio labelings of $G$. This paper establishes the radio number of the Cartesian product of a cycle graph with itself (i.e., of $C_n\square C_n$.)

math.CO↗