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

TL;DR

The paper addresses the asymptotic distribution of complex zeros for random polynomials with regularly varying variance profiles, establishing a trichotomy of boundary behavior depending on the index $\alpha$. The authors develop local convergence results: in the liquid phase ($\alpha>-\tfrac12$) zeros near the unit circle converge to zeros of a Gaussian analytic function with universal statistics, while in the crystalline phases ($\alpha\le -\tfrac12$) zeros concentrate on a near-unit circle lattice, with a strong crystalline regime (finite $\sum b^2(k)$) yielding non-universal limits and a weak crystalline regime (infinite $\sum b^2(k)$) yielding universal Gaussian limits. A precise crossover result as $\alpha\to -\tfrac12^+$ links the liquid and weak crystalline limits, and the framework is extended to self-inversive polynomials, where similar phase transitions govern unit-circle zeros. Concrete examples include hyperbolic polynomials and Kac-type polynomials, and the isotropic Gaussian case highlights universal zero statistics via Edelman–Kostlan formulas and, outside the unit disk, determinantal structures. These results illuminate phase transitions in zero patterns and connect to broader phenomena in random analytic functions and quantum-chaos-inspired models.

Abstract

This paper investigates asymptotic distribution of complex zeros of random polynomials $P_n(z):=\sum_{k=0}^{n}b(k)ξ_k z^k$, as $n\to\infty$, where $b$ is a regularly varying function at infinity with index $α\in \mathbb{R}$ and $(ξ_k)_{k\geq 0}$ is a sequence of independent copies of a complex-valued random variable $ξ$. The limiting distribution of zeros both inside and outside the unit disk is determined assuming $\mathbb{E}[\log^{+}|ξ|]<\infty$. Under the additional assumptions $\mathbb{E}[ξ]=0$ and $\mathbb{E}[|ξ|^2]<\infty$, 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 $α$ crosses the critical value $α_c = -1/2$ from right to left. In the liquid phase ($α> α_c$), the limiting point process of zeros is universal. In the crystalline phase, it is universal if and only if $α= α_c$ 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.

Figures (3)

• Figure 1: Zeros of the random polynomial $P_n(z)= \sum_{k=0}^{n} (k+1)^{\alpha} \xi_k z^k$ of degree $n=100$ with $\alpha = -2$ (left, crystalline phase) and $\alpha = 0$ (right, liquid phase). The random variables $\xi_k$ are i.i.d. standard complex normals. The zeros are represented by dots and the unit circle is shown as the solid line.
• Figure 2: Each row shows zeros of $5$ independent realizations of the random polynomial $P_n(z) = \sum_{k=0}^{n} (k+1)^{\alpha} \xi_k z^k$ with $\alpha = 0$ (top row, liquid phase) and $\alpha = -3$ (bottom row, crystalline phase). The random variables $\xi_k$ are i.i.d. complex standard normal. The degree is $n=1000$. The unit circle is shown in blue. In the bottom row, the circle of radius $r_n$ is shown in red. Only zeros contained in the window $[0.9,1.1] \times [-0.2,0.2]$ are shown.
• Figure 3: Graphs of the intensity $\rho_1 (\alpha, \cdot )$ of zeros of $G_{\psi}$ restricted to the real axis for $\alpha = -0.1$ (dashed line), $\alpha= -0.5 +10^{-4}$ (dotted line) and $\alpha=- 0.5 +10^{-9}$ (dash-dotted line). The solid line is the graph of the intensity of zeros of the Gaussian analytic function $u\mapsto \hat{N}_1+{\rm e}^{u}N_1$ restricted to the real axis and is the limiting form of $\rho_1 (\alpha, \cdot )$, as $\alpha\to-1/2+0$, after an appropriate shift. The intensity $\rho_1 (\alpha, \cdot )$ is defined in Eqs. \ref{['eq:rho']}--\ref{['eq:Phi_def']} and $G_{\psi}$ is assumed to have isotropic Gaussian marginals.

Theorems & Definitions (57)

• Proposition 2.1: Convergence inside the unit disk: polynomials
• Corollary 2.2: Convergence inside the unit disk: zeros
• Theorem 2.3: Convergence outside the unit disk: polynomials
• proof
• Corollary 2.4: Convergence outside the unit disk: zeros
• Example 2.5
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• proof