Random Self-Inversive Polynomials

• Random self-inversive polynomials are random polynomials with symmetry constraints that force zeros to appear in reciprocal-conjugate pairs, often aligning on the unit circle.
• They exhibit distinct liquid-like and crystalline phases, with the transition governed by the growth rate of the weight sequence and characterized by universal Gaussian statistics.
• Rigorous local limit theorems detail scaling laws and phase transitions in zero distributions, providing critical insights for advancing spectral analysis in complex systems.

Random self-inversive polynomials are a class of random polynomials defined by specific symmetry constraints on their coefficients, which impose a strong algebraic structure on their zero sets. The canonical form considered is

$$K_m(z) = 1 + \sum_{k=1}^m b(k)\,\xi_k\,z^k + \sum_{k=1}^m b(k)\,\overline{\xi_k}\, z^{2m-k+1} + z^{2m+1},$$

where $b(k)$ is a weight sequence, $\xi_k$ are i.i.d. complex-valued random variables, and $m$ determines the degree $2m+1$. The crucial algebraic property is self-inversiveness: $K_m(z) = z^{2m+1} \overline{K_m(1/\overline{z})}$, enforcing that zeros off the unit circle occur in reciprocal-conjugate pairs and a significant fraction of zeros are located on $|z|=1$. Recent advances rigorously establish the precise statistical behavior of their zeros in the limit of large degree, revealing a transition from “liquid-like” to “crystalline” regimes determined by the growth rate of the weights $b(k)$ (Kabluchko et al., 15 Nov 2025).

1. Model Definition and Symmetry Structure

Random self-inversive polynomials of degree $2m+1$ are defined by combining i.i.d. random coefficients $\{\xi_k\}$ with weights $b(k)$ for $k=1,\dots,m$ as above. The weights are given by $b(k)=k^\alpha \ell(k)$, where $\alpha\in\mathbb{R}$ and $\ell$ is a positive slowly varying function. The law of the coefficients is such that $\mathbb{E}[\xi]=0$, $\mathbb{E}[|\xi|^2]=\sigma^2\in(0,\infty)$, and $\mathbb{E}[\xi^2]=0$ (the latter imposes no real restriction, as it can be absorbed by a global phase).

The self-inversive property yields that if $z_0$ is a zero, then so is $1/\bar{z}_0$, enforcing spectral symmetry with respect to the unit circle. As a result, many zeros are exactly or approximately located on $|z|=1$.

2. Zeros: Local Scaling Windows and Regimes

Analysis focuses on the local statistics of zeros of $K_m(z)$ in the large-degree limit, specifically in microscopic neighborhoods of points $e^{i\psi}$ on the unit circle. The local regime is set by a scaling $z = e^{i(\psi + u/m)}$ (liquid phase) or, in crystalline regimes, on circles of radius $r_m > 1$ parametrized logarithmically.

The behavior falls into three distinct phases determined by the index $\alpha$ and the divergence or convergence of $\sum b(k)^2$:

• Liquid phase ($\alpha > -\frac{1}{2}$): Weights are large enough for $\sum b(k)^2 = \infty$.
• Weak crystalline phase ($\alpha = -\frac{1}{2}$, $\sum b(k)^2 = \infty$): Edge case with slowly diverging weights.
• Strong crystalline phase ($\alpha < -\frac{1}{2}$, or $\alpha=-\frac{1}{2}$ with $\sum b(k)^2 < \infty$): Weights decay quickly; series becomes square-summable.

Within each regime, the zeros in a local window are described by a limiting point process.

3. Liquid and Crystalline Phase Descriptions

The phase transition is characterized as follows (Kabluchko et al., 15 Nov 2025):

Phase	$\alpha$ condition	Zeros
Liquid	$\alpha > -\frac{1}{2}$	2D “liquid-like”
Weak crystalline	$\alpha = -\frac{1}{2}$, $\sum b^2 = \infty$	1D “weak crystal”
Strong crystalline	$\alpha < -\frac{1}{2}$;	1D “strong crystal”

Liquid phase

Microscopically, zeros in an arc $z = e^{i(\psi + u/m)}$ and normalized by $c_m = b(m)\sqrt{m}$, obey

$$\{ K_m(e^{i(\psi + u/m)}) / c_m \}_{u\in\mathbb{C}} \to \text{G}_\psi(u),$$

where $\text{G}_\psi$ is a centered Gaussian entire process with covariance derived from the weights and variance $\sigma^2$:

$$\mathbb{E}[\text{G}_\psi(u_1)\overline{\text{G}_\psi(u_2)}] = \sigma^2 \int_0^1 x^{2\alpha} e^{(u_1 + \overline{u_2})x} dx.$$

Zeros in this regime are distributed genuinely two-dimensionally (“liquid-like”) in the arc at scale $1/m$, with intensity expressed explicitly by

$$\rho_1(\alpha;u) = \frac{1}{\pi} \left( \frac{\Phi_{2\alpha+2}(u+\bar{u})}{\Phi_{2\alpha}(u+\bar{u})} - \left(\frac{\Phi_{2\alpha+1}(u+\bar{u})}{\Phi_{2\alpha}(u+\bar{u})}\right)^2 \right),$$

where $\Phi_\beta(w) = \int_0^1 x^\beta e^{w x} dx$.

Strong crystalline phase

Weights are summable; zeros concentrate on a circle of radius $r_m>1$. In local logarithmic coordinates

$z_m(\psi,u) = r_m e^{i\psi} e^{u/m},$

$\{K_m(z_m(\psi,u))\}_{u\in\mathbb{C}}$ converges in law to $u\mapsto P_\infty(e^{i\psi})+e^u N$, with $$P_\infty(e^{i\psi}) = \sum_{k\geq 0} b(k) \xi_k e^{i\psi k}$$ convergent, and $N$ an independent complex normal. The zeros form a rigid 1D “lattice” in these coordinates at

$$u = \log\big[-N^{-1} P_\infty(e^{i\psi})\big] + 2\pi i\,\mathbb{Z}.$$

Weak crystalline phase

Weights satisfy $b(k)^2$ just diverging; zeros pile on $|z|=r_m$, and after normalization,

$$\{K_m(\hat{z}_m(\psi,u))/\sqrt{L(m)}\} \to u\mapsto \hat{H} + e^u N,$$

where $L(m) = \sum_{k=0}^m b(k)^2 \to \infty$, $\hat{H}$ and $N$ are independent centered complex normals with covariances matching the liquid phase. Again, zeros form a 1D lattice

$u = \log[-N^{-1} \hat{H}] + 2\pi i\,\mathbb{Z}.$

4. Universality, Non-Universality, and Phase Transition

The “liquid $\leftrightarrow$ crystal” transition is continuous as $\alpha \downarrow -1/2$: the Gaussian process in the liquid regime rescales and converges to the process governing the weak crystal lattice. In the liquid phase, the local zero process is universal: it depends only on $\alpha$ and the second moments $\sigma^2$, $\sigma_1^2 - \sigma_2^2$, not on the higher moments or detailed tail of $\xi$. In the weak crystalline phase, the 1D lattice statistics are again universal—any sufficiently regular law for $\xi$, with $\mathbb{E}[|\xi|^2]<\infty$, produces the same point process. Non-universality emerges in the strong crystalline phase: the deterministic lattice shift $\log[-N^{-1}P_\infty(e^{i\psi})]$ depends on the entire sequence of coefficients, and hence on the detailed law of $\xi$.

5. Zeros on the Unit Circle

Analyzing zeros exactly on $|z|=1$, $K_m(e^{it})=e^{i(m+1/2)t}\,\widetilde{K}_m(t)$ reduces the problem to studying real roots of $\widetilde{K}_m(t) \in \mathbb{R}$. In the liquid phase,

$$\{\widetilde{K}_m(t/m) / (b(m)\sqrt{m})\}_{t\in\mathbb{R}} \to t \mapsto 2\,\text{Re}\left[e^{-t} G_0(t)\right],$$

with $G_0$ as above, so the corresponding real zeros converge to a Gaussian “random trigonometric” zero set (with two-point correlations determined by classical Kac–Rice expressions). In crystalline regimes, unit-circle roots become equispaced (angles shifted by a random offset), again corresponding to random lattice statistics in the relevant process.

6. Joint Local Statistics and Rigorous Theorems

The local limit theorems (as formalized in Theorems 6.1, 6.3, and 6.5 of (Kabluchko et al., 15 Nov 2025)) hold jointly for any finite set of points $e^{i\psi_1},\dots,e^{i\psi_m}$ on the unit circle: the behavior at these points is governed by a collection of independent local limits described above. The results rigorously verify and sharpen the conjectural liquid–crystal transition proposed heuristically in the physics literature by Bogomolny, Bohigas, and Lebœuf.

The regimes and their characteristics are summarized as follows:

Regime	Universal?	Description
$\alpha > -1/2$ (liquid)	Yes	2D universal, Gaussian zeros
$\alpha = -1/2$, $\sum b^2=\infty$ (weak crystal)	Yes	1D universal lattice
$\alpha \leq -1/2$, $\sum b^2<\infty$ (strong crystal)	No	1D non-universal lattice

The explicit analytic formulas for covariance kernels and first-intensity of zeros allow for detailed predictions of statistical correlations and their scaling in the large-degree limit (Kabluchko et al., 15 Nov 2025).