On the distribution patterns of zeros for random polynomials with regularly varying coefficients

Abstract

This paper investigates asymptotic distribution of complex zeros of random polynomials $P_n(z):={\sum }_{k=0}^{n}b(k)\hat{I}{¾}_k{z}^k$, as $n\to \infty$, where $b$ is a regularly varying function at infinity with index $\hat{I}\pm \in \mathbb{R}$ and $(\hat{I}{¾}_k{)}_{k\ge 0}$ is a sequence of independent copies of a complex-valued random variable $\hat{I}¾$. The limiting distribution of zeros both inside and outside the unit disk is determined assuming $\mathbb{E}[{\log }^{+}|\hat{I}¾|]<\infty$. Under the additional assumptions $\mathbb{E}[\hat{I}¾]=0$ and $\mathbb{E}[|\hat{I}¾{|}^2]<\infty$, local universality results for zeros near the boundary of the unit disk are established. Notably, it is shown that the point process of zeros undergoes a transition from liquid-like to crystalline phases as $\hat{I}\pm$ crosses the critical value $\hat{I}{\pm }_c=-1/2$ from right to left. In the liquid phase ($\hat{I}\pm >\hat{I}{\pm }_c$), the limiting point process of zeros is universal. In the crystalline phase, it is universal if and only if $\hat{I}\pm =\hat{I}{\pm }_c$ and ${\sum }_k{b}^2(k)=+\infty$ (the weak crystalline phase), and non-universal when ${\sum }_k{b}^2(k)<+\infty$ (the strong crystalline phase). The zeros of the so-called random self-inversive polynomials on the unit circle exhibit a similar phase transition.

44 pages, 3 figures, 1 table