Remarks on the one-point density of Hele-Shaw $β$-ensembles

TL;DR

This work analyzes the microscopic structure of planar Coulomb gases with Hele-Shaw type potentials via the one-point density $R_n^\beta$. It proves equicontinuity of $R_n^\beta$ for $\beta\ge 1/2$, giving Lipschitz subsequential limits after microscopic rescaling and clarifying the role of the effective potential $Q_{\mathrm{eff}}=Q-\check Q$. The paper develops detailed overcrowding bounds, establishes global and local upper bounds, and proves equicontinuity, thereby enabling a rigorous microscopic (Ward-rescaled) analysis and connections to the thermal equilibrium measure. It also discusses almost circular ensembles, Ward identities, and the limitations of using thermal equilibrium as an approximation to the microscopic density, highlighting open questions about uniqueness of bulk scaling limits for general $\beta$. Overall, the results provide a quantitative, rigorous foundation for understanding microscopic density fluctuations and crystallization phenomena in non-Hermitian random matrix ensembles with Hele-Shaw-type confinement.

Abstract

In this note we prove equicontinuity for the family of one-point densities with respect to a two-dimensional Coulomb gas at an inverse temperature $β\ge 1/2$ confined by an external potential of Hele-Shaw (or quasi-harmonic) type. As a consequence, subsequential limiting Lipschitz continuous densities are defined on the microscopic scale. There are several additional results, for example comparing the one-point density with the thermal equilibrium density.

Figures (1)

• Figure 1: A sample from an almost circular Ginibre ensemble at $\beta=1$.

Theorems & Definitions (32)

• Theorem 1.1: Upper bound
• Remark
• Theorem 1.2: Equicontinuity
• Corollary 1.3
• Remark
• Remark
• Remark
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3