Three topological phases of the elliptic Ginibre ensembles with a point charge

TL;DR

The paper addresses the limiting spectral distribution of conditioned elliptic Ginibre ensembles with a deterministic eigenvalue, revealing a three-phase topological classification of the droplet: doubly connected, simply connected, or two disjoint simply connected components. The authors develop a potential-theoretic approach based on quadrature domains and a conformal-mapping framework to explicitly characterize the droplets and their electrostatic energies across Regimes I–III, depending on $(p,c,\tau)$. They provide exact droplet descriptions for the doubly connected and simply connected cases, derive the associated energies, and connect these results to the asymptotics of moments of characteristic polynomials in the elliptic Ginibre setting. The work extends known results for $\tau=0$ and $p=0$ by incorporating the non-Hermiticity parameter and detailing the phase transitions, with implications for free-energy expansions and potential universality near critical points.

Abstract

We consider the complex and symplectic elliptic Ginibre matrices of size $(c+1)N \times (c+1)N$, conditioned to have a deterministic eigenvalue at $ p \in \mathbb{R} $ with multiplicity $ c N $. We show that their limiting spectrum is either simply connected, doubly connected, or composed of two disjoint simply connected components. Moreover, denoting by $τ\in [0,1]$ the non-Hermiticity parameter, we explicitly characterise the regions in the parameter space $ (p, c, τ) $ where each topological type emerges. For cases where the droplet is either simply or doubly connected, we provide an explicit description of the limiting spectrum and the corresponding electrostatic energies. As an application, we derive the asymptotic behaviour of the moments of the characteristic polynomial for elliptic Ginibre matrices in the exponentially varying regime.

Figures (6)

• Figure 1: The plot illustrates Regimes I, II, and III in Definition \ref{['Def_regimes of p and c']}, across various values of $\tau$. Notably, for $\tau = 0$, only Regimes I and II are present, while for $p = 0$, only Regimes I and III persist, which is consistent with discussions in Remark \ref{['Remark_regimes extremal']}.
• Figure 2: The plots illustrate the configurations of Fekete points associated with the Hamiltonian \ref{['Ham complex']}, with parameters $c = 0.4$, $\tau = 0.5$, and $N = 500$. Regime I corresponds to $p < \sqrt{2/5} \approx 0.63$, while Regime III approximately corresponds to $p > 1.12$. In (A) and (C), the black solid lines represent the boundaries of the droplets as determined in Theorem \ref{['Thm_droplet and energy']}.
• Figure 3: The plots display the graphs $p \mapsto I_Q(\mu_Q)$, given by \ref{['weighted energy doubly connected']} and \ref{['weighted energy simply connected']}, for different values of $c$ with $\tau = 0.3$. The vertical dotted line indicates the critical value between Regimes I and II.
• Figure 4: The plots illustrate various critical phases with $\tau = 1/3$. In this case, the triple points \ref{['def of triple pts']} are given by $c_{\rm tri} = 1/7$ and $p_{\rm tri} = 2\sqrt{14}/7 \approx 1.07$. Plot (A) represents the case $c = 1/14 < c_{\rm tri}$, where the intersection of Regimes I and II occurs at $p = 2\sqrt{7}(\sqrt{30} - 1)/21 \approx 1.13$. Plot (B) corresponds to the case $c = 3/7 > c_{\rm tri}$, where $p = 4/\sqrt{21} \approx 0.88$. Plot (C) shows the case when $c = c_{\rm tri}$ and $p = p_{\rm tri}$, where the droplet exhibits identical curvatures for the ellipse and circle at the singularity.
• Figure 5: The plot illustrates the ranges of $\kappa$ for which different geometric properties of $f(\partial \mathbb{D})$ arise.

Theorems & Definitions (48)

• Definition 1: Regimes of the parameters $p$, $c$ and $\tau$
• Theorem 1.1: Topological characterisation of the droplet
• Remark 1.1: Phases in extremal cases
• Theorem 1.2: Description of the droplet and electrostatic energies
• Remark 1.2: Fekete points and numerics
• Remark 1.3: The extremal $\tau=0$ case
• Remark 1.4: Further phases in the general case
• Remark 1.5: Critical phases at intersections of different regimes
• Remark 1.6: Multi-component ensembles in Regime III
• Remark 1.7: Phases of the motherbody