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 $n\to\infty$, where $b$ is a regularly varying function at infinity with index $\alpha\in \mathbb{R}$ and $(\xi_k)_{k\geq 0}$ 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 $\mathbb{E}[\log^{+}|\xi|]<\infty$. Under the additional assumptions $\mathbb{E}[\xi]=0$ and $\mathbb{E}[|\xi|^2]<\infty$, 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 $\alpha$ crosses the critical value $\alpha_c = -1/2$ 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 $\sum_k b^2(k) = +\infty$ (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.