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arxiv: 2605.29594 · v1 · pith:QWUCPGAFnew · submitted 2026-05-28 · 🧮 math-ph · math.CV· math.MP· math.PR

Free energy expansion of determinantal Coulomb gases in the quadratic fields with a point charge

Sung-Soo Byun , Meng Yang , Eui Yoo This is my paper

Pith reviewed 2026-06-29 00:51 UTC · model grok-4.3

classification 🧮 math-ph math.CVmath.MPmath.PR
keywords determinantal Coulomb gasfree energy expansionLiouville actionGinibre ensembleorthogonal polynomialsHele-Shaw potentialsquadratic potentialpoint charge
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The pith

The free energy of determinantal Coulomb gases in quadratic fields with a point charge expands up to the constant term identified with the Liouville action on the droplet.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an explicit expansion of the free energy for determinantal Coulomb gases with an anisotropic quadratic potential that includes a point charge. The expansion is carried out to the constant term when the equilibrium droplet is either simply or doubly connected. All coefficients in the expansion are given explicitly, extending the isotropic case. The constant term is identified as the Liouville action associated with the droplet. The proof relies on deforming the system in both the charge position and the anisotropy parameter while tracking the free energy through asymptotics of the orthogonal polynomials.

Core claim

In the regimes where the associated droplet is simply or doubly connected, the free energy expansion up to and including the constant term is derived with all coefficients computed explicitly. This extends recent results in the isotropic case. In particular, the constant term is identified with the Liouville action associated with the droplet. The result also admits an interpretation in terms of asymptotic expansions of moments of characteristic polynomials for the elliptic Ginibre ensemble.

What carries the argument

A deformation framework that varies both the point charge location and the anisotropy parameter to relate changes in the free energy to refined asymptotics of planar orthogonal polynomials.

If this is right

  • The expansion provides asymptotic formulas for moments of characteristic polynomials in the elliptic Ginibre ensemble.
  • The approach indicates a broader connection between free energy expansions, asymptotics of planar orthogonal polynomials, and conformally invariant geometric functionals such as the Liouville action.
  • Intermediate results apply to general algebraic Hele-Shaw potentials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This framework could extend to free energy expansions for other external potentials that lead to algebraic droplets.
  • The link to the Liouville action suggests possible connections to quantities in two-dimensional conformal field theory.
  • Direct computation of the partition function for moderate system sizes might allow numerical checks of the constant term.

Load-bearing premise

Refined asymptotics of the planar orthogonal polynomials are available that suffice to control how the free energy varies under simultaneous deformation of the point charge location and the anisotropy parameter.

What would settle it

A numerical calculation of the free energy for chosen values of the anisotropy parameter, the charge strength, and the charge location, compared to the predicted explicit expansion formula.

Figures

Figures reproduced from arXiv: 2605.29594 by Eui Yoo, Meng Yang, Sung-Soo Byun.

Figure 1
Figure 1. Figure 1: The plot depicts the phase diagram of three regimes—Regime I (doubly con￾nected), Regime II (simply connected), and Regime III (multi-component)—in the (a, τ )-plane for c = 1/7. The arrows indicate the deformation paths used to compute the variation of the free energy from the generic points (A) and (D): in the doubly connected regime, (A)→(B):=(0, τ ) corresponds to variation in a and (B)→(C):=(0, 0) to … view at source ↗
Figure 2
Figure 2. Figure 2: Schematic diagram of the proof strategy for Theorem 2.2. To be more precise, we relate the variations of the free energy with respect to the parameters to the leading￾order coefficients of Pn,N (z). To this end, we denote by An,N and Bn,N the coefficients of z n−1 and z n−2 , respectively, in Pn,N (z), i.e. (2.25) Pn,N (z) = z n + An,N z n−1 + Bn,N z n−2 + O(z n−3 ), z → ∞. The variations of the free energ… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration of the deformation procedures. The labels (A)–(E) are con￾sistent with those in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

We study a determinantal Coulomb gas in the complex plane associated with the external potential $$ Q(z)=\frac{1}{1-\tau^2}\big(|z|^2-\tau \text{Re } z^2\big)-2c\log|z-a|, $$ where $\tau\in[0,1)$, $c\ge0$, and $a\ge0$. In the regimes where the associated droplet is simply or doubly connected, we derive the free energy expansion up to and including the constant term, with all coefficients computed explicitly, thereby extending recent results in the isotropic case $\tau=0$. In particular, we identify the constant term with the Liouville action associated with the droplet. Our result admits a natural interpretation in terms of asymptotic expansions of moments of characteristic polynomials for the elliptic Ginibre ensemble. The proof is based on a deformation framework involving both the singularity location $a$ and the anisotropy parameter $\tau$, relating variations of the free energy to refined asymptotics of planar orthogonal polynomials. The asymptotic analysis relies on the foliation flow method of Hedenmalm and Wennman, providing an alternative to the Riemann--Hilbert approach used in the isotropic setting. The present work suggests a general framework connecting free energy expansions, refined asymptotics of planar orthogonal polynomials, and conformally invariant geometric functionals, with several intermediate results already formulated for general algebraic Hele-Shaw potentials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript derives the free energy expansion, up to and including the constant term with all coefficients explicit, for the determinantal Coulomb gas with external field Q(z) = 1/(1-τ²)(|z|² - τ Re z²) - 2c log|z-a| (τ ∈ [0,1), c ≥ 0, a ≥ 0). The expansion holds in both simply-connected and doubly-connected droplet regimes, extends the isotropic (τ=0) case, and identifies the constant term with the Liouville action of the droplet. The proof proceeds via a joint deformation in the parameters a and τ, relating free-energy variations to refined asymptotics of the associated planar orthogonal polynomials; these asymptotics are obtained from the foliation-flow construction of Hedenmalm–Wennman rather than Riemann–Hilbert analysis.

Significance. If the derivation is correct, the work supplies the first explicit free-energy expansion (including constant) for an anisotropic quadratic potential with an added point charge, thereby linking the constant term to a conformally invariant geometric quantity (Liouville action). It furnishes a deformation framework that connects free-energy expansions, planar-orthogonal-polynomial asymptotics, and algebraic Hele-Shaw potentials, and it offers an alternative analytic route (foliation flow) to the isotropic results. These features would constitute a concrete advance in the asymptotic analysis of elliptic Ginibre ensembles and related determinantal processes.

major comments (1)
  1. [proof strategy / deformation framework paragraph] The central claim that explicit coefficients (including the constant identified with the Liouville action) are obtained in both connectivity regimes rests on the assertion that the Hedenmalm–Wennman foliation-flow construction supplies the required higher-order asymptotics of the planar orthogonal polynomials under the simultaneous deformation in a and τ. The manuscript must therefore contain an explicit verification that the flow hypotheses remain satisfied when the external field includes both the anisotropic quadratic term (τ Re z²) and the logarithmic singularity (-2c log|z-a|), particularly that the error estimates controlling the free-energy variation are preserved for τ > 0 and c > 0. This verification is load-bearing; without it the passage from the deformation identity to the explicit expansion is not justified.
minor comments (2)
  1. [abstract] The abstract states that the result 'admits a natural interpretation in terms of asymptotic expansions of moments of characteristic polynomials for the elliptic Ginibre ensemble'; a brief explicit statement of this correspondence (e.g., which moment corresponds to which term in the free-energy expansion) would improve readability.
  2. [introduction] Notation for the droplet connectivity regimes (simply vs. doubly connected) should be introduced once at the beginning of the main text and used consistently thereafter.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the significance of the work, and the recommendation for major revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The central claim that explicit coefficients (including the constant identified with the Liouville action) are obtained in both connectivity regimes rests on the assertion that the Hedenmalm–Wennman foliation-flow construction supplies the required higher-order asymptotics of the planar orthogonal polynomials under the simultaneous deformation in a and τ. The manuscript must therefore contain an explicit verification that the flow hypotheses remain satisfied when the external field includes both the anisotropic quadratic term (τ Re z²) and the logarithmic singularity (-2c log|z-a|), particularly that the error estimates controlling the free-energy variation are preserved for τ > 0 and c > 0. This verification is load-bearing; without it the passage from the deformation identity to the explicit expansion is not justified.

    Authors: We agree that the manuscript requires an explicit verification that the foliation-flow hypotheses of Hedenmalm–Wennman continue to hold under the joint deformation in a and τ when the external field contains both the anisotropic quadratic term and the point-charge singularity. Although the cited construction applies to general algebraic Hele-Shaw potentials (which formally include our Q(z)), the error estimates controlling the free-energy variation are not re-checked in detail for τ > 0 and c > 0. In the revised version we will insert a dedicated subsection (or appendix) that verifies the relevant hypotheses on the potential, confirms that the foliation remains admissible in both the simply- and doubly-connected regimes, and establishes that the error bounds remain uniform for τ ∈ [0,1) and c ≥ 0. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external foliation flow method

full rationale

The paper derives the free energy expansion explicitly via a deformation framework that relates variations to refined planar orthogonal polynomial asymptotics supplied by the foliation flow method of Hedenmalm and Wennman (external citation, no author overlap). This replaces Riemann-Hilbert analysis for the τ=0 case and is used to control joint (a,τ) deformations in both simply and doubly connected regimes. The identification of the constant term with the Liouville action is presented as a derived output, not an input. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear; the central claim remains independent of the paper's own fitted values or definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; ledger populated from stated dependencies. The work assumes the foliation flow method yields the necessary orthogonal-polynomial asymptotics under joint deformation.

axioms (1)
  • domain assumption Foliation flow method of Hedenmalm and Wennman supplies refined asymptotics of planar orthogonal polynomials under deformation in singularity location and anisotropy parameter
    Paper states the asymptotic analysis relies on this method to relate free-energy variations to orthogonal-polynomial asymptotics.

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