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Partition function of 2D Coulomb gases with radially symmetric potentials and a hard wall

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Pith's one-line read The paper proves that a hard wall strictly inside a two-dimensional Coulomb droplet changes the $\log N$ coefficient of the partition function to $-\frac{1}{4}$ for an annulus and $-\frac{1}{3}$ for a disk, independent of the radial…

desk verdict The annulus expansion is solid, careful work; the disk -1/3 log N result is a conjecture resting on an unproven and dimensionally sloppy relation (5.1). read the letter →

arxiv 2506.14738 v1 pith:64S344AJ submitted 2025-06-17 math.PR math-phmath.MP

Matthias Allard , Peter J. Forrester , Sampad Lahiry , Bojian Shen This is my paper
classification math.PRmath-phmath.MP MSC 60B2082D0541A6060G55
keywords normalmatrixmodeltwo-dimensionalCoulombgashardwallconstraintpartitionfunctionasymptoticslogNcoefficientradiallysymmetricpotentialLaplacemethodEuler-Maclaurinsummation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes the large-$N$ asymptotic expansion of the partition function for a two-dimensional Coulomb gas (equivalently, the normal matrix model at $\beta=2$) when a hard wall confines the particles to the unit disk and the external potential is radial. The coefficient of $\log N$ in the expansion is shown to be universal, independent of the potential, but it changes when the hard wall lies strictly inside the droplet rather than at its boundary. For an annular droplet with the wall strictly inside ($0<\eta<1$), the coefficient is $-\frac{1}{4}$; when the wall coincides with the droplet's outer boundary ($\eta=0$), the $\log N$ term is absent. For a disk droplet with the wall strictly inside, the paper derives $-\frac{1}{3}\log N$. These results sharpen the topological prediction $\chi/12$ for the free-energy expansion by showing how an interior constraint modifies it.

What carries the argument

The load-bearing object is the factorization $\log(Z_N^h/(2\pi)^N)=\sum_{j=0}^{N-1}\log u_j$, with $u_j=\int_0^1 r^{2j+1}e^{-Nq(r)}dr$, which radial symmetry produces from the $N$-particle integral. Each $u_j$ is analyzed by Laplace's method with the critical point $r_\tau$ fixed by $rq'(r)=2\tau$, $\tau=j/N$, and the hard wall enters through $\tau_0$ with $r_{\tau_0}=1$ and $\eta=1-q'(1)/2$. The delicate part is the boundary layer $|\tau-\tau_0|$ of width $N^{-1/2}$, where the $u_j$ expansion is expressed through complementary error functions; those expressions are converted into sums by Euler-Maclaurin summation, producing the universal constants $\gamma_{\rm in},\gamma_{\rm out},\alpha_{\rm in},\alpha_{\rm out},\beta_{\rm in},\beta_{\rm out}$ as definite integrals.

What would settle it

For a potential with a disk droplet, such as $q(r)=a^2r^2$ with $0<a<1$, every $u_j$ is an incomplete gamma function, so $\log Z_N^h$ can be computed numerically at large $N$; comparing the $-\frac{1}{3}\log N$ coefficient and the constant term $\zeta'(-1)+F_D[Q]-F_{S\cap D}[Q]$ with Theorem 2.3 would settle the disk claim. Independently, evaluating the left and right sides of equation (5.1) at finite $N$ for a family of radial potentials and checking whether their difference tends to zero would test the transfer relation directly.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.1: in the in/out annulus case $0<r_0<1\le r_1$ with $\eta=1-q'(1)/2\in(0,1)$, one has $\log(Z_N^h/(2\pi)^N)= -N^2[I_{S\cap D}[\mu_Q]+\eta q(1)] -\frac{(1+\eta)}2 N\log N - N(\cdots) -\sqrt N(\gamma_{\rm in}+\gamma_{\rm out})\sqrt{\Delta Q(1)} -\frac{1}{4}\log N + F_{S\cap D}[Q] -(\alpha_{\rm in}+\alpha_{\rm out})+(\beta_{\rm in}+\beta_{\rm out})\frac{\partial_r\Delta Q(1)}{\Delta Q(1)} + \frac{\Delta Q(1)}{4\eta} -\frac{1}{2}\log\eta+\frac{1}{4}\log(2\pi\Delta Q(1))+o(1)$. The $-\frac{1}{4}\log N$ coefficient is independent of $q(r)$; when $r_1=1$ ($\eta=0$) no $\log N$ term appears. Theorem 2.3 transfers the annulus result to the disk, giving $-\frac{1}{3}\log N$ for $1>\eta>0$, together with a constant-term shift $\zeta'(-1)+F_D[Q]-F_{S\cap D}[Q]$. The expansions also identify universal constants at order $\sqrt N$ and at $O(1)$ as definite integrals involving the complementary error function.

Load-bearing premise

The disk result rests on equation (5.1), an unproved equality stating that replacing an annular droplet by a disk changes the partition function by the same amount whether or not a hard wall is present; if that relation fails, the $-\frac{1}{3}\log N$ coefficient for the disk does not follow.

Editorial extensions

If this is right

  • In an annulus with a hard wall strictly inside the droplet, the $\log N$ coefficient is $-\frac{1}{4}$ for every admissible radial potential, whereas it vanishes when the wall sits exactly at the droplet boundary.
  • In a disk with the wall strictly inside, the $\log N$ coefficient is $-\frac{1}{3}$, and the disk-annulus difference at order $1$ is $\zeta'(-1)+F_D[Q]-F_{S\cap D}[Q]$, independent of the wall position.
  • At order $\sqrt N$, the coefficient is universal up to the factor $\sqrt{\Delta Q(1)}$: it is $(\gamma_{\rm in}+\gamma_{\rm out})\sqrt{\Delta Q(1)}$ when the wall is inside, and $\gamma_{\rm out}\sqrt{\Delta Q(1)}$ when the wall is at the outer boundary, with the same constants appearing in known gap-probability expansions.
  • At order $1$, the constants $\alpha_{\rm in},\alpha_{\rm out},\beta_{\rm in},\beta_{\rm out}$ coincide with the constants in Mittag-Leffler-type large-gap asymptotics through the relations $\beta_{\rm in}=-\frac{1}{2}\alpha_{\rm in}+\frac{1}{2}\log 2$ and $\beta_{\rm out}=-\frac{1}{2}\alpha_{\rm out}+\frac{1}{4}\log\pi$.
  • In the out-annulus case with the wall inside the inner radius ($\eta>1$), the expansion has no $\sqrt N$ or $\log N$ term, while at $\eta=1$ (wall exactly at $r_0=1$) the coefficient $-\frac{1}{4}\log N$ reappears with a $\sqrt N$ term.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The pattern $\{0,-\frac{1}{4},-\frac{1}{3}\}$ for the $\log N$ coefficient looks like one plus the number of hard edges that intersect the droplet; the authors leave open how to anticipate it, and a testable extension would be to compute the same coefficient for $\beta\neq2$ radial hard-wall gases to see whether the fractions become $\beta$-dependent while remaining potential-independent.
  • Equation (5.1) suggests a transfer principle stronger than the paper's radial setting: the disk-annulus difference in the partition function may be independent of the confining wall for general, not necessarily radial, potentials. If so, the $-\frac{1}{3}\log N$ coefficient would extend to non-radial disk droplets with an interior hard wall.
  • The universal constants $\gamma_{\rm in},\gamma_{\rm out}$ are defined by erfc integrals that also encode surface tension; one might look for closed-form evaluations or relations connecting them to $\alpha$ and $\beta$, which would make the constant term fully explicit without numerical integration.
  • The breakdown of the $\eta\to0^+$ limit at order $\sqrt N$ noted in the paper implies that small changes in wall position have non-perturbative effects in $N$; a natural check is to examine the crossover regime where $\eta$ is taken to zero simultaneously with $N$, for instance $\eta\sim N^{-1/2}$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the large-N asymptotics of the partition function of the beta=2 two-dimensional Coulomb gas in the unit disk with a radially symmetric potential and a hard wall. Using the exact factorization log(Z_N^h/(2 pi)^N) = sum_{j=0}^{N-1} log u_j, the authors apply Laplace's method and Euler-Maclaurin summation to obtain expansions in several configurations. Theorem 2.1 gives the annulus case with hard wall inside the droplet (eta in (0,1)) or at the outer edge (eta=0), including log N coefficients -1/4 or 0 and universal constants alpha, beta, gamma. Theorem 2.3 asserts the disk case follows from the annulus case by adding -1/12 log N + zeta'(-1) + F_D - F_{S cap D}; this transfer rests on the unproven identity (5.1). Theorem 2.5 treats an annulus whose inner edge lies outside the hard wall.

Significance. If fully correct, the paper would give the first systematic hard-wall corrections to the free energy expansion for non-quadratic radial potentials, confirming and extending predicted universal log N and sqrt N terms. The annulus analysis is detailed and self-contained, with explicit universal constants (2.3)-(2.4), incomplete-gamma cross-checks in Examples 3.3, 3.5, 3.10, and agreement with the known quadratic example (1.13). However, the advertised disk result, in particular the -1/3 log N coefficient, is not established because Eq. (5.1) is unproven and appears dimensionally inconsistent. Thus the paper's central novelty for the disk is conditional. The strengths of the annulus part, and the clarity of the universal constants, make this a worthwhile contribution once the disk bridge is proved or replaced.

major comments (2)
  1. [Section 5, Eq. (5.1)] The disk result Theorem 2.3 rests entirely on the unproven identity Z_s^disk - Z_s^annulus = Z_h^disk - Z_h^annulus. As written this identity is dimensionally inconsistent: the left and right sides are partition functions, while the following sentence reads off a logarithmic-expansion difference from [11]. Even after replacing Z by log Z, the assertion that the disk/annulus difference is independent of the hard wall is not justified: tau0 determines the upper endpoint floor(N tau0) of the j-sum, and the small-tau block j = O(N^{1-epsilon}) governed by Proposition 3.11 changes when r0 changes from 0 to >0; its contribution to the Euler-Maclaurin summation must be computed explicitly to decide whether it cancels the annulus terms. Since no such computation is supplied, the advertised -1/3 log N coefficient, and the constant terms zeta'(-1) + F_D - F_{S cap D}, are conditional. The quadratic check in Example 2.4 is one special case and cannot establish the universal claim for all radial potentials satisfying (1.16).
  2. [Section 4, Propositions 4.1-4.2] The proofs of Theorem 2.1 combine asymptotic formulas for u_j with Euler-Maclaurin summation in regimes separated by the cutoffs Delta_N and delta_N. The final results are stated with o(1) remainders after cancellation of terms depending on Delta_N, xi_N, xi*_N and {N tau0}. Uniform error bounds are not supplied, and the o(1) errors from Corollaries 3.2, 3.4, 3.7 and Proposition 3.9 are not tracked through summations over O(N) terms. The incomplete-gamma examples are reassuring, but the theorem as stated needs either explicit remainder control or a precise citation of where such control is established.
minor comments (4)
  1. [Section 2.4] In the discussion after Eq. (2.25), 'eliminate alpha_in, alpha_in' should read 'eliminate alpha_in, alpha_out'.
  2. [Examples 2.2, 2.4, 3.5] The numerical verification statements are not accompanied by any displayed data; adding a table of computed versus predicted values would let the reader assess the order of the remainder and the claimed agreement.
  3. [Proposition 3.1, Eq. (3.14)] The displayed formula (3.14) is extremely difficult to read in its current typesetting; restructuring it with named auxiliary quantities would improve usability.
  4. [Theorem 2.3] The notation Z_s^annulus and Z_h^annulus in (2.17) is confusing because the annulus partition function in Theorem 2.1 has r0>0 and r1>=1, while the disk setting has r0=0; the relation should specify exactly which parameters are held fixed in the comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hard-wall annulus expansion is derived from first principles; the disk case relies on an unproved identity but that is a correctness gap, not a circular reduction.

full rationale

The annulus results (Theorems 2.1 and 2.5) are derived directly from the exact factorization (1.18), Laplace-method asymptotics for u_j in Section 3, and Euler-Maclaurin summation in Section 4 and Appendix A. The universal constants alpha_in, alpha_out, beta_in, beta_out, gamma_in, gamma_out are defined as q-independent definite integrals in (2.3)-(2.4) and arise from the sums rather than being fitted to the final expansions. The citations to [11] and [19] are to external works used for specific lemmas, such as Proposition 3.11 and identities for constants, and are not self-citations; the only author-overlapping citation is [25], which is mentioned historically in Section 2.4 and is not load-bearing for the proofs. The one step that is not fully derived is Theorem 2.3, which rests on the asserted identity (5.1), with the justification 'Since this common alteration is independent of tau0'. That identity is unproved and is a genuine correctness risk, but it is not a circular reduction: it is neither equivalent to the theorem by construction nor a fitted parameter renamed as a prediction. Therefore no significant circularity is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fit; the constants alpha, beta, gamma are universal definite integrals evaluated numerically. No invented entities are introduced. The main input axioms are the standard tools and the cited results from [11].

assumptions (6)
  • domain assumption Strict subharmonicity of the potential (1.16) and growth conditions on q(r)
    Guarantees a unique critical point r_tau for V_tau(r) and applies Laplace's method; stated in Section 1.2 and used throughout.
  • standard math Equilibrium measure formula (1.17) from Saff-Totik [38]: d mu_Q = (1/4) Delta Q 1_S dA
    Used to define the droplet, the parameter eta in (2.1), and the functionals (2.5)-(2.8).
  • standard math Laplace's method and Euler-Maclaurin summation (Proposition A.1) are valid for the integrals and sums encountered
    The proofs of Propositions 3.1, 3.6, 4.1, 4.2 rely on these standard tools.
  • standard math Results of Byun-Kang-Seo [11] (Lemmas 3.1, 3.2, 2.2-2.4) are correct and can be adapted by replacing r1 by 1 and accounting for the prefactor
    Used in Propositions 3.9, 3.11, and in the computation of S_in in Proposition 4.2; these are published results, not re-derived here.
  • standard math Factorization (1.18) of the hard-wall partition function into a product of one-dimensional integrals
    Stated as a standard calculation from [27, Exercises 15.3 q.1(ii)]; the starting point of the analysis.
  • ad hoc to paper The disk case difference (5.1): Z_s^disk - Z_s^annulus = Z_h^disk - Z_h^annulus
    Asserted without proof in Section 5; load-bearing for Theorem 2.3. This is the weakest assumption.

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Pith. "Pith review of Partition function of 2D Coulomb gases with radially symmetric potentials and a hard wall." pith.science (2026). https://pith.science/paper/64S344AJ

@misc{pith2026250614738,
  author       = {Pith},
  title        = {Pith review of: Partition function of 2D Coulomb gases with radially symmetric potentials and a hard wall},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64S344AJ}},
  note         = {Machine review of arXiv:2506.14738}
}
abstract

The large $N$ asymptotic expansion of the partition function for the normal matrix model is predicted to have special features inherited from its interpretation as a two-dimensional Coulomb gas. However for the latter, it is most natural to include a hard wall at the boundary of the droplet. We probe how this affects the asymptotic expansion in the solvable case that the potential is radially symmetric and the droplet is a disk or an annulus. We allow too for the hard wall to be strictly inside the boundary of the droplet. It is observed the term of order $\log N$, has then a different rational number prefactor to that when the hard wall is at the droplet boundary. Also found are certain universal (potential independent) numerical constants given by definite integrals, both at order $\sqrt{N}$, and in the constant term.

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Cited by 1 Pith paper

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    math.PR 2025-07 conditional novelty 7.0 of 10

    For a general potential, the smallest gaps of the 2D Coulomb gas at beta=2 are of order n^{-3/4} and converge to a Poisson point process with intensity determined by the equilibrium density.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.