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

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