Asymptotic Distribution of Complex Zeros

Updated 23 November 2025

The topic is defined as the study of how complex zeros of polynomial sequences or analytic functions converge to unique equilibrium measures in the complex plane.
Researchers use potential theory, explicit limiting measures, and Legendre-Fenchel transforms to characterize zero distributions in both random and deterministic ensembles.
Key insights include universality in random ensembles, phase transition phenomena, and extensions to multivariate settings under pluripotential theory.

The asymptotic distribution of complex zeros is a central topic in random polynomial theory, spectral geometry, and potential theory. It describes how the zeros of sequences of polynomials (often random or with large parameters) organize in the complex plane as the degree or other scaling parameter diverges. This phenomenon reveals connections to equilibrium measures, log-potential theory, determinantal point processes, universality, and phase transitions in random systems. The modern approach, as distilled from recent work, emphasizes explicit limiting measures, potential-theoretic variational principles, and precise phase-space characterizations for both deterministic and random ensembles.

1. Potential-Theoretic Framework and Equilibrium Measures

The limiting (asymptotic) distribution of complex zeros for sequences of polynomials or analytic functions typically emerges as the equilibrium measure on a compact set $K \subset \mathbb{C}$ with respect to the logarithmic energy functional

$I[\mu] = \iint_{K \times K} \log |z - w|^{-1} \, d\mu(z) \, d\mu(w)$

which is minimized by a unique probability measure $\mu_K$ supported on $K$ (Bloom et al., 2018). The Siciak extremal (Green) function,

$V_K(z) = \sup \{ \frac{1}{\deg p} \log |p(z)| : p \text{ polynomial}, \, \|p\|_K \leq 1 \}$

has Laplacian $\frac{1}{2\pi} \Delta V_K = \mu_K$ , with boundary conditions $V_K|_K = 0$ , and logarithmic growth at infinity.

The zeros of orthogonal or random polynomials, suitably normalized, converge (in probability or almost surely) to $\mu_K$ , provided either the Bernstein-Markov property for the underlying measure and suitable tail decay for coefficients (random case), or large parameters in the deterministic (e.g. oscillatory or hypergeometric) case (Bloom et al., 2018, Bayraktar, 2013).

2. Universality and Random Ensembles

For random polynomials with i.i.d. coefficients $\{\xi_j\}$ , the normalized zero counting measures

$\nu_{H_n} = \frac{1}{n} \sum_{k=1}^n \delta_{\zeta_k}$

(where the $\zeta_k$ are zeros of $H_n(z) = \sum_{j=0}^n \xi_j q_j(z)$ , and $\{q_j\}$ are orthonormal polynomials on $K$ ), converge to $\mu_K$ if and only if the coefficients satisfy a logarithmic tail condition $\mathbb{P}(|\xi_0| > e^t) = o(t^{-1})$ . This condition is sharp; for example, in the classical Kac ensemble ( $G_n(z) = \sum \xi_j z^j$ ), if $\mathbb{E} \log(1 + |\xi_0|) = \infty$ , zeros fail to concentrate near the unit circle (Ibragimov et al., 2011, Bloom et al., 2018, Pritsker et al., 2016).

General random analytic functions

$G_n(z) = \sum_{k=0}^{\infty} \xi_k f_{k,n} z^k$

with suitable deterministic coefficients $f_{k,n}$ and moment assumption $\mathbb{E} \log(1+|\xi_0|) < \infty$ yield limiting zero measures of the form

$\mu = \frac{1}{2\pi}\Delta[I(\log|z|)]$

where $I$ is the Legendre-Fenchel transform of the large-index coefficient growth rate $u(t) = -\log f(t)$ (Kabluchko et al., 2014, Eremenko, 2022).

3. Deterministic Polynomials: Orthogonal, Hypergeometric, and Special Functions

In high-degree deterministic settings, zeros of polynomials orthogonal with respect to complex or oscillatory weights, or of generalized hypergeometric type, often condense on analytic arcs, curves, or skeletons in $\mathbb{C}$ linked to critical trajectories of quadratic differentials, level curves of Abelian (complex potential) integrals, or solutions of nonlinear equilibrium problems (Deano et al., 2010, Nazarov et al., 2010, Blaschke et al., 2019, Abathun et al., 2013).

For example, orthogonal polynomials with exponential weights on complex contours result in zeros aligning on an S-curve $\gamma$ defined as a critical trajectory of a quadratic differential. The limiting zero density is expressible as

$d\mu(s) = \frac{1}{\pi i} Q_+^{1/2}(s) ds$

along $\gamma$ , where $Q$ is constructed from the weight (Deano et al., 2010). In general, exceptional or generalized Hermite polynomials have a fixed number of non-real zeros converging (at rate $O(n^{-1/2})$ ) to the simple zeros of an auxiliary polynomial as the degree grows, rendering the complex-zero part of the asymptotic measure discrete (Kuijlaars et al., 2014).

Hypergeometric polynomial zeros cluster along level sets given by

$\operatorname{Re}[\phi_i(z) - \phi_j(z)] = \text{constant}$

for branches of Abelian integrals arising from the ODE, and in certain limit cases yield explicit lemniscates or more intricate skeletons (Abathun et al., 2013).

4. Multivariate Extensions and Pluripotential Theory

Random holomorphic sections of high tensor powers of positive line bundles on compact projective manifolds ( $X$ ) produce zero currents $Z_{s_n} = \frac{1}{n}[Z_{s_n}]$ whose distributions converge, in the weak topology of currents, to the pluripotential equilibrium current $T_{K,q}=ω+dd^c V_{K,q}$ associated to a non-pluripolar compact $K$ and weight $q$ (Bayraktar, 2013). For random multivariate polynomials in $\mathbb{C}^d$ , or sections over homogeneous spaces, the zero measure converges almost surely to the normalized Monge–Ampère measure $(dd^c V_{K,q})^d$ , analogously to the one-dimensional case.

5. Fine Structure, Fluctuations, and Correlation Phenomena

Beyond the macroscopic distribution, the local fluctuations and fine-scale statistics of zeros exhibit universality and determinantal structure in many cases. For the Gaussian Entire Function (GEF)

$F(z) = \sum_{k=0}^\infty a_k z^k / \sqrt{k!}$

(with $a_k$ i.i.d. complex Gaussians), zeros form a translation and rotation-invariant determinantal point process of intensity $1/\pi$ . Linear statistics of zeros satisfy central limit theorems for sufficiently smooth test functions, and the two-point correlation functions can be described via explicit kernels (Nazarov et al., 2010). In random matrix models (e.g., products of Ginibre matrices plus truncations), the large- $n$ limit of the zero counting measure weakly converges to Raney distributions, with densities computable through explicit trigonometric parametrizations, extending Fuss–Catalan universality (Neuschel et al., 2014, Neuschel et al., 2015).

6. Non-Universal and Phase Transition Phenomena

When random coefficients exhibit regularly varying behavior with index $\alpha$ , the zero process near the boundary ( $|z| = 1$ ) can undergo a transition: for $\alpha > -1/2$ (liquid phase), the local process is universal (GAF), but for $\alpha \leq -1/2$ (crystalline phase), the zeros arrange on arithmetic lines with nontrivial global shifts depending on the law of the coefficients, breaking universality. At the critical index $\alpha_c = -1/2$ , weak and strong crystalline phases are distinguished by the summability of the squares of the deterministic weights (Kabluchko et al., 15 Nov 2025).

7. Extensions, Limitations, and Open Problems

Generalizations include:

Asymptotics for zeros of specific special function polynomials (parabolic cylinder, Bessel, Lommel), with rigorous Airy-type expansions and explicit analytic curves as support (Dunster et al., 2024, Dunster et al., 16 Oct 2025, Deaño et al., 2014, Blaschke et al., 2019).
Relations between zero distributions and critical point distributions, governed by subharmonic potential theory and Legendre transform techniques (Eremenko, 2022).
Invariance and scaling limits on compact Riemann surfaces, or other geometries of constant curvature (Kabluchko et al., 2014).

Limitations of the global approach include lack of direct control over local repulsion, hole probabilities, or multipoint statistics, which typically require more refined probabilistic/determinantal techniques not covered by pure potential theory.

Open directions include explicit characterizations for non-self-adjoint Jacobi-type matrix models, understanding the universality in strong crystalline regimes, and precise asymptotic in multi-variable contexts.

References (arXiv IDs):