Pith. sign in

REVIEW 1 major objections 5 minor 43 references

Smallest gaps of the two-dimensional Coulomb gas

T0 review · 1 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For a two-dimensional Coulomb gas, the smallest inter-particle gaps are of order $n^{-3/4}$ and the rescaled gap process is Poisson with explicit intensity.

desk verdict Genuine extension of Shi-Jiang to general regular potentials; likely true, but the proof hinges on an edge-kernel expansion whose interior/exterior validity needs refereeing. read the letter →

arxiv 2507.23502 v3 pith:HZDTWCBE submitted 2025-07-31 math.PR math-phmath.MP

Christophe Charlier This is my paper · ORCID
classification math.PRmath-phmath.MP MSC 62G3260G5531A99
keywords CoulombgasesrandomnormalmatricesextremevaluestatisticssmallestgapsPoissonpointprocessdeterminantalprocessesorthogonalpolynomialasymptoticsedgeuniversality
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 proves a law for the tiniest spacings in a two-dimensional Coulomb gas, equivalently the eigenvalues of random normal matrices. For any regular, real-analytic, strictly subharmonic potential whose droplet boundary is a smooth Jordan curve, the smallest gaps between particles are of order $n^{-3/4}$, and after this rescaling the gap locations and sizes form a Poisson point process with an explicit intensity. The intensity is $\pi^2$ times the integral of the cube of the equilibrium density over the droplet, times the area of the rescaled gap disc, so both higher particle density and larger gap radii increase the expected count. A consequence is a closed-form limiting density for the $k$-th smallest rescaled gap, proportional to $x^{4k-1}e^{-\frac{\mathcal{J}}{4}x^4}$. The proof extends the quadratic-potential result to a general class of potentials by adding a new uniform edge expansion of the correlation kernel to existing bulk estimates.

What carries the argument

The argument uses the determinantal structure of the gas: all $k$-point correlation functions are determinants of a single correlation kernel $K_n$ built from orthonormal polynomials. In the bulk, known estimates give $K_n$ to leading order with error $O(1)$. The paper's new engine is Theorem 2.4, a uniform subleading expansion of $K_n$ at the soft edge: for $z_0\in\partial S$ and $|\xi|,|\eta|\le M\sqrt{\log n}$, $$c_n(\xi)c_n(\eta)K_n\left(z_0+\frac{n\sqrt{2}\xi}{\sqrt{\$\Delta$ Q(z_0)n/4}},z_0+\frac{n\sqrt{2}\eta}{\sqrt{\$\Delta$ Q(z_0)n/4}}\right) = \frac{\$\Delta$ Q(z_0)n}{4\pi}k(\xi,\eta)+\frac{k_2(\xi,\eta;z_0)}{\sqrt{n}}+O(1+|\xi|^6+|\eta|^6),$$ with an explicit polynomial-exponential $k_2$. This edge control, proved by Riemann-sum approximation from all-order planar orthogonal polynomial asymptotics, is what keeps the relevant two-point integrals $O(1)$ uniformly even when the reference point approaches $\partial S$.

What would settle it

Take the elliptic Ginibre ensemble, where the correlation kernel is explicit, and compute the edge-region integral $\int_{w+B_n}\rho_2(w,x)\,d^2x$ for $w$ at distance at most $M\sqrt{\log n}/\sqrt{n}$ from $\partial S$. Theorem 2.4 predicts this integral stays $O(1)$ uniformly; an evaluation, exact or numerical, showing growth like $(\log n)^{3/2}$ would contradict Theorem 2.4 and therefore Theorem 1.1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for potentials $Q\in C^2(\mathbb{C})$ satisfying the growth condition, real-analytic and strictly subharmonic near the droplet $S$, regular, and with $\partial S$ a smooth Jordan curve, the point process $\chi^{(n)} = \sum_{i=1}^{n-1}\delta_{(n^{3/4}|z_{i^*}-z_i|, z_i)}$ converges weakly to a Poisson point process on $\mathbb{R}_+\times\mathbb{C}$ with intensity $\mathbb{E}[\chi(A\times\Omega)] = (\pi^2\int_{\Omega\cap S}\rho(v)^3\,d^2v)(\int_A r^3\,dr)$, where $\rho$ is the equilibrium density. Consequently the $k$-th smallest rescaled gap has limiting density $\frac{\mathcal{J}^k}{4^{k-1}\Gamma(k)}x^{4k-1}e^{-\frac{\mathcal{J}}{4}x^4}$ with $\mathcal{J}=\pi^2\int_S\rho^3$. This is the same law previously known for the quadratic potential, now established for a general class of potentials.

Load-bearing premise

The proof hinges on the new uniform edge expansion of the correlation kernel, Theorem 2.4, which must hold with its stated small error all the way to $|\xi|\le M\sqrt{\log n}$; if that expansion failed, gaps forming near the droplet boundary could not be controlled and the Poisson limit would not follow.

Editorial extensions

If this is right

  • The $k$-th smallest rescaled gap has limiting density $\frac{\mathcal{J}^k}{4^{k-1}\Gamma(k)}x^{4k-1}e^{-\frac{\mathcal{J}}{4}x^4}$, with $\mathcal{J} = \pi^2\int_S \rho^3$.
  • The joint law of the $k$ smallest rescaled gaps has the explicit form given in Corollary 1.3, so gaps in disjoint size and location windows are asymptotically independent.
  • Gap locations and sizes decouple asymptotically: the Poisson intensity factorizes as $(\pi^2\int_{\Omega\cap S}\rho^3)(\int_A r^3\,dr)$, so the spatial density of small-gap events is proportional to $\rho^3$.
  • The theorem upgrades the quadratic-potential result to all real-analytic regular potentials with smooth droplet boundary, and numerical evidence in the paper suggests the conclusion survives for annuli and for potentials with singularities outside the droplet.
  • For models where the smallest gaps typically occur at the droplet edge, the convergence is slower, consistent with the need for the new edge estimate.

Reading between the lines

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

  • The uniform edge expansion Theorem 2.4 is likely reusable for other extremal statistics near the droplet boundary, such as large-gap probabilities or fluctuations of counting functions in shrinking windows, since it supplies subleading control on a $\sqrt{\log n}/\sqrt{n}$ scale.
  • If the same Poisson intensity $\pi^2\int\rho^3 \int r^3\,dr$ appears in other planar determinantal processes with a bulk scaling limit, it would point to a universal density-cubed-times-gap-area law for minimal spacings in two dimensions.
  • A quantitative next step is to track finite-$n$ errors in Corollary 1.3; the edge-dominant models in the paper suggest convergence rates degrade when $\rho$ is maximized on $\partial S$, and a refined version of Theorem 2.4 could make this precise.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper studies the two-dimensional Coulomb gas at inverse temperature beta=2, equivalently the eigenvalues of random normal matrices, for a general potential Q. Under assumptions that Q is C^2, satisfies the growth condition (1.2), is real analytic and strictly subharmonic near the droplet S, is regular, and has a smooth Jordan boundary, it proves that the point process of rescaled forward-gaps n^{3/4}|z_{i*}-z_i| together with their locations converges to a Poisson point process on R_+ x C with intensity (pi^2 int_{Omega cap S} rho^3)(int_A r^3 dr). Corollary 1.3 then gives the limiting joint distribution of the k smallest gaps and the explicit density x^{4k-1}e^{-Jx^4/4} for the k-th smallest gap, with J = pi^2 int_S rho^3. The proof adapts the determinantal strategy of Shi and Jiang: thinned factorial moments, inclusion-exclusion for correlation functions, and a three-region bulk/edge/exterior analysis. The edge control rests on a new uniform subleading edge-kernel expansion, Theorem 2.4, proved in Appendix A from all-order planar orthogonal polynomial asymptotics of Hedenmalm and Wennman. Numerical simulations for four ensembles and two consistency checks of Theorem 2.4 are included.

Significance. If correct, this is a substantial generalization of the Ginibre smallest-gap result of Shi and Jiang, with an explicit, parameter-free intensity constant. The proof is carefully structured, and the paper contains useful consistency checks against the elliptic Ginibre kernel and rotationally invariant potentials, as well as numerical confirmations. Theorem 2.4, a uniform subleading expansion of the correlation kernel in a sqrt(log n)/sqrt n neighborhood of the edge, is potentially of independent interest. The main risk is that the entire theorem rests on the validity of this edge expansion, whose proof in Appendix A is the least standard step and depends on precise hypotheses of [25] that are not fully stated in the manuscript.

major comments (1)
  1. [Appendix A, Eq. (A.3)] The application of [25, Theorem 1.3] in (A.3) is not justified for the points that are actually needed. For z = z0 + n c0 xi / sqrt n with Re xi < 0, the point z lies on the interior side of S. The asymptotic (A.3) is written as an exterior asymptotic relative to the shifted droplet S_{1-j/n}, whose boundary has moved inward by O(j/n). For j <= C sqrt n, which is precisely the range that contributes the leading Riemann-sum integral in (A.11)-(A.12), z lies strictly inside S_{1-j/n} unless [25, Theorem 1.3] is known to be valid on both sides of the boundary. The manuscript does not state the exact validity domain of [25, Theorem 1.3], nor does it prove that the formula applies uniformly for all |xi| <= M sqrt(log n), all j <= floor(epsilon_n n), and all z0 in partial S with the O(n^{-1}) error claimed in (A.3). Since Theorem 2.4, and in particular the uniform O((log n)^3 / n) entrywise error used in (3.27) of Lemma 3.5, depends on this step, this is a load-bearing gap for Theorem 1.1. The authors should either quote the precise theorem from [25] and verify that its hypotheses cover Re xi < 0, or provide an independent interior-side asymptotic for p_{n-j,n} in the range j <= C sqrt n.
minor comments (5)
  1. [Theorem 2.4, Eq. (2.8)] The scaling in the displayed argument of the kernel is not consistent with the scaling used in Appendix A and with the announced sqrt(log n)/sqrt n neighborhood of partial S. The theorem should use the same normalization as z = z0 + n c0 xi / sqrt n with c0 = sqrt(2)/sqrt(Delta Q(z0)/4); the current notation appears to contain a typo involving the normal vector n and the dimension n.
  2. [Lemmas 3.2 and 3.5] The bulk cases in Lemmas 3.2 and 3.5 apply Theorem 2.1 to points with dist(w, partial S) >= epsilon_n, but Theorem 2.1 is stated for points outside the 2 epsilon_n-neighborhood of partial S. This is presumably fixable by rescaling M, but the text should be adjusted for consistency.
  3. [Lemma 3.2, Eq. (3.11)] The displayed estimate in the exterior contribution contains a factor (1 + |z|^{-epsilon n/100}); if Omega is genuinely unbounded as stated, the term 1 is not integrable over the unbounded exterior. The argument needs a split into a bounded piece and a far-away tail where the |z|-decay is used, or the statement should be restricted to bounded Omega.
  4. [Corollary 1.3] The proof of Corollary 1.3 applies the convergence statement with Omega = C, whereas Theorem 1.1 is stated only for bounded Borel sets Omega. A short tightness argument using the exterior decay estimates would justify the passage from bounded Omega to Omega = C; as written this step is omitted.
  5. [Section 2, Eq. (2.1)] There is a typo: 'measurably function' should read 'measurable function'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central proof derives the Poisson limit from independent kernel asymptotics, with the intensity constant obtained from the equilibrium measure rather than fitted.

full rationale

The paper's main theorem is not circular. The intensity J = pi^2 integral rho^3 is computed from the equilibrium measure density rho, which is determined by the potential Q through dmu = (Delta Q / 4pi) 1_S, not fitted to gap data. The Poisson limit follows from factorial moment estimates (Lemmas 3.4-3.6) that reduce to known correlation kernel asymptotics: the bulk estimate [1, Theorem 2.1], the decay estimate [5, Theorem 8.1], and the new edge expansion Theorem 2.4. Theorem 2.4 itself is proved in Appendix A from the published all-order planar orthogonal polynomial asymptotics of Hedenmalm-Wennman [25, Theorems 1.3 and 1.5], which are external to this paper and do not presuppose the smallest-gap result. The only place where a known leading-order result is invoked inside the proof of Theorem 2.4 is the identification of the coefficients a8 and a9: the paper states 'we know from [25, Corollary 1.7] that the leading term of Kn(z,w) is equal to Delta Q(z0) n / 4pi k(xi,eta) (up to unimodular factors). To reconcile [25, Corollary 1.7] with (A.4) and (A.12), it is easy to check that we must have a8 = 0 and a9 = -1.' This is a consistency match against an established external result, not a fitting of the target quantity: the new content of Theorem 2.4 is the subleading term k2, which is obtained from the all-order expansion and Euler-Maclaurin summation, not from the leading-order match. The numerical simulations in Section 1 are presented as confirmations, not as evidence used in the proof. The self-citations [3] and [14] are not load-bearing: [3] is used only in an appendix consistency check for rotation-invariant potentials, and [14] is cited in the introduction as related work. There is no step in which a parameter is fitted to data and then renamed a prediction, and no uniqueness claim is imported from the authors' own prior work to force the choice. The skeptic's concern that [25, Theorem 1.3] may be an exterior asymptotic while some points in the edge neighborhood lie on the interior side of the shifted droplet is a correctness risk about uniformity of an external theorem's applicability, not a circularity: it does not show that any claimed prediction is equivalent by construction to an input. Under the stated standards, the derivation chain is self-contained outside of the present paper's own fitted values, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters were fitted: J is a deterministic functional of Q. No new physical entities are introduced. The proof imports standard kernel estimates and the new edge expansion is derived, not postulated.

assumptions (7)
  • standard math Equilibrium measure for (1.1) exists, is unique, has support S, and dmu = Delta Q/(4 pi) 1_S d^2z (Saff-Totik).
    Invoked in the introduction and used throughout to define rho and J; cited as [36].
  • standard math The beta=2 Coulomb gas (1.1) is a determinantal point process with kernel K_n from orthonormal polynomials (2.4)-(2.5).
    Foundation of every correlation function computation in Section 3; cited to Soshnikov [37].
  • standard math Bulk correlation kernel asymptotics: K_n(z,w) = n b0/pi e^{n psi - ...} + O(1) uniformly in S away from partial derivative S ([1, Theorem 2.1]).
    Used in Lemmas 3.2 and 3.4 to evaluate bulk contributions and the pointwise limit of rho_k^A.
  • standard math Fast decay and pointwise bounds for K_n: Theorem 2.2 ([5, Theorem 8.1]) and Proposition 2.3 ([5, Proposition 3.6]) together with obstacle function inequality (2.7).
    Central to controlling exterior and clustering events in Lemmas 3.2, 3.4, 3.5.
  • standard math All-order planar orthogonal polynomial asymptotics near the boundary, including smooth dependence of phi_tau, Q_tau, B_tau on tau near 1 ([25, Theorems 1.3, 1.5, Proposition 2.3]).
    The backbone of the proof of the new edge kernel expansion, Theorem 2.4, in Appendix A.
  • standard math Criteria for Poisson convergence from factorial moments and convergence of point processes ([29, Propositions 2.8 and 2.9]) and Soshnikov's thinning formula for correlation functions ([38, (4.5)]).
    Used to reduce the main theorem to factorial moment limits; stated in Section 3.
  • domain assumption Q is C2, satisfies growth (1.2), is real analytic and strictly subharmonic near S, regular, and partial derivative S is a smooth Jordan curve.
    Explicit hypotheses of Theorem 1.1; they are not derived and the paper notes in Remark 1.2 that some are likely relaxable.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Smallest gaps of the two-dimensional Coulomb gas." pith.science (2026). https://pith.science/paper/HZDTWCBE

@misc{pith2026250723502,
  author       = {Pith},
  title        = {Pith review of: Smallest gaps of the two-dimensional Coulomb gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZDTWCBE}},
  note         = {Machine review of arXiv:2507.23502}
}
abstract

We consider the two-dimensional Coulomb gas with a general potential at the determinantal temperature, or equivalently, the eigenvalues of random normal matrices. We prove that the smallest gaps between particles are typically of order $n^{-3/4}$, and that the associated joint point process of gap locations and gap sizes, after rescaling the gaps by $n^{3/4}$, converges to a Poisson point process. As a consequence, we show that the $k$-th smallest rescaled gap has a limiting density proportional to $x^{4k-1}e^{-\frac{\mathcal{J}}{4}x^{4}}$, where $\mathcal{J}=\pi^{2}\int \rho(z)^{3}d^{2}z$ and $\rho$ is the density of the equilibrium measure. This generalizes a result of Shi and Jiang beyond the quadratic potential.

Figures

Figures reproduced from arXiv: 2507.23502 by the authors.

Figure 1
Figure 1. Each plot shows a sample of (1.1) with n = 1000 and a different potential Q: from left to right, Q(z) = 1 1−τ 2 (|z| 2−τRe z 2 ) with τ = 1 2 , Q(z) = |z| 2−2a log |z| with a = 1 2 , Q(z) = a log(1+|z| 2 ) with a = 1.25, and Q as in (1.9) with a = 0.25. The blue curves represent ∂S. Theorem 1.1. Suppose Q : C → R is C 2 -smooth on C, satisfies the growth condition (1.2), and is real analytic and strictly subharmonic… view at source ↗
Figure 2
Figure 2. Numerical confirmations of Corollary 1.3 for k = 1 (top) and k = 2 (bottom), and for the potentials Q(z) = 1 1−τ 2 (|z| 2 − τRe z 2 ) with τ = 1 2 (left), Q(z) = |z| 2 − 2a log |z| with a = 1 2 (middle), and Q(z) = a log(1 + |z| 2 ) with a = 1.25 (right). In each diagram, the blue curve corresponds to the density (1.8) with either k = 1 (top) or k = 2 (bottom). The histograms were produced as described in the text. … view at source ↗
Figure 3
Figure 3. Numerical confirmations of Corollary 1.3 for k = 1 (top) and k = 2 (bottom), for the potential (1.9) with a = 0.25. The plots show histograms for increasing values of n; as n grows, the empirical distributions seem to converge toward the theoretical prediction of Corollary 1.3. the histograms are already in good agreement with Corollary 1.3 when n = 400. In contrast, for the TUE, the convergence is slower: even at n… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 40 canonical work pages

  1. [25]

    Hedenmalm and A

    H. Hedenmalm and A. Wennman, Planar orthogonal polynomials and boundary universality in the random normal matrix model, Acta Math. 227 (2021), no. 2, 309–406

    work page 2021

  2. [1]

    Ameur, Near-boundary asymptotics for correlation kernels, J

    Y. Ameur, Near-boundary asymptotics for correlation kernels, J. Geom. Anal. 23 (2013), no. 1, 73–95

    work page 2013

  3. [2]

    Ameur, Repulsion in low temperature β-ensembles, Comm

    Y. Ameur, Repulsion in low temperature β-ensembles, Comm. Math. Phys. 359 (2018), no. 3, 1079–1089

    work page 2018

  4. [3]

    The two-dimensional Coulomb gas: fluctuations through a spectral gap

    Y. Ameur, C. Charlier and J. Cronvall, The two-dimensional Coulomb gas: fluctuations through a spectral gap, arXiv:2210.13959. (to appear in Arch. Ration. Mech. Anal.)

    work page Pith review arXiv

  5. [4]

    Ameur and J

    Y. Ameur and J. Cronvall, Szeg¨ o type asymptotics for the reproducing kernel in spaces of full-plane weighted polynomials, Comm. Math. Phys. 398 (2023), no. 3, 1291–1348

    work page 2023

  6. [5]

    Ameur, H

    Y. Ameur, H. Hedenmalm and N. Makarov, Berezin transform in polynomial Bergman spaces, Comm. Pure Appl. Math. 63 (2010), no. 12, 1533–1584

    work page 2010

  7. [6]

    Ameur, H

    Y. Ameur, H. Hedenmalm and N. Makarov, Fluctuations of eigenvalues of random normal matrices, Duke Math. J. 159 (2011), no. 1, 31–81

    work page 2011

  8. [7]

    Gaussian beta ensembles: the perfect freezing transition and its characterization in terms of Beurling-Landau densities

    Y. Ameur, F. Marceca and J.L. Romero, Gaussian beta ensembles: the perfect freezing transition and its char- acterization in terms of Beurling-Landau densities, arXiv:2205.15054. (to appear in Ann. Inst. Henri Poincar´ e Probab. Stat.)

    work page Pith review arXiv

Show all 43 references
  1. [8]

    Allard, P.J

    M. Allard, P.J. Forrester, S. Lahiry and B. Shen, Partition function of 2D Coulomb gases with radially symmetric potentials and a hard wall, arXiv:2506.14738

    arXiv

  2. [9]

    Ben Arous and P

    G. Ben Arous and P. Bourgade, Extreme gaps between eigenvalues of random matrices, Ann. Probab. 41 (2013), no. 4, 2648–2681

    2013

  3. [10]

    Bourgade, Extreme gaps between eigenvalues of Wigner matrices, J

    P. Bourgade, Extreme gaps between eigenvalues of Wigner matrices, J. Eur. Math. Soc. 24 (2022), no. 8, 2823– 2873

    2022

  4. [11]

    Byun and M

    S.-S. Byun and M. Ebke, Universal scaling limits of the symplectic elliptic Ginibre ensemble, Random Matrices Theory Appl. 12 (2023), no. 1, Paper No. 2250047, 33 pp

    2023

  5. [12]

    Byun and P.J

    S.-S. Byun and P.J. Forrester, Progress on the study of the Ginibre ensembles, KIAS Springer Series in Mathe- matics, Vol. 3, Springer, Singapore, 2025

    2025

  6. [13]

    Byun and S

    S.-S. Byun and S. Park, Large gap probabilities of complex and symplectic spherical ensembles with point charges, arXiv:2405.00386

    arXiv

  7. [14]

    Charlier, Hole probabilities and balayage of measures for planar Coulomb gases, arXiv:2311.15285

    C. Charlier, Hole probabilities and balayage of measures for planar Coulomb gases, arXiv:2311.15285

    arXiv

  8. [15]

    Elbau and G

    P. Elbau and G. Felder, Density of eigenvalues of random normal matrices, Comm. Math. Phys. 259 (2005), no. 2, 433–450

    2005

  9. [16]

    R. Feng, F. G¨ otze and D. Yao, Smallest gaps between zeros of stationary Gaussian processes, J. Funct. Anal. 287 (2024), no. 4, Paper No. 110493, 14 pp

    2024

  10. [17]

    R. Feng, J. Li and D. Yao, Small gaps of GSE, arXiv:2409.03324

    arXiv

  11. [18]

    R. Feng, G. Tian and D. Wei, Small gaps of GOE, Geom. Funct. Anal. 29 (2019), no. 6, 1794–1827

    2019

  12. [19]

    Feng and D

    R. Feng and D. Wei, Small gaps of circular β-ensemble, Ann. Probab. 49 (2021), no. 2, 997–1032

    2021

  13. [20]

    Feng and D

    R. Feng and D. Wei, Large gaps of CUE and GUE, arXiv:1807.02149 (to appear in Ann. Probab.)

    arXiv

  14. [21]

    Feng and G

    R. Feng and G. Yao, Smallest distances between zeros of Gaussian analytic functions, DOI: 10.13140/RG.2.2.26674.62409

  15. [22]

    Forrester, Some statistical properties of the eigenvalues of complex random matrices, Phys

    P.J. Forrester, Some statistical properties of the eigenvalues of complex random matrices, Phys. Lett. A 169 (1992), no. 1-2, 21–24

    1992

  16. [23]

    Forrester, Log-gases and random matrices, Princeton University Press, Princeton, NJ, 2010

    P.J. Forrester, Log-gases and random matrices, Princeton University Press, Princeton, NJ, 2010

    2010

  17. [24]

    Hedenmalm, S

    H. Hedenmalm, S. Shimorin, Hele-Shaw flow on hyperbolic surfaces, J. Math. Pures Appl. (9) 81 (2002), no. 3, 187–222

    2002

  18. [26]

    Hedenmalm and A

    H. Hedenmalm and A. Wennman, A global asymptotic expansion of the polynomial Bergman density, preprint

  19. [27]

    Landon, P

    B. Landon, P. Lopatto and J. Marcinek, Comparison theorem for some extremal eigenvalue statistics, Ann. Probab. 48 (2020), no. 6, 2894–2919

    2020

  20. [28]

    Lee and R

    S.-Y. Lee and R. Riser, Fine asymptotic behavior for eigenvalues of random normal matrices: ellipse case, J. Math. Phys. 57 (2016), no. 2, 023302, 29 pp

    2016

  21. [29]

    Lopatto and M

    P. Lopatto and M. Meeker, Smallest gaps between eigenvalues of real Gaussian matrices, arXiv:2403.09521 (to appear in Bernoulli ). 28

    arXiv

  22. [30]

    Lopatto and M

    P. Lopatto and M. Otto, Maximum gap in complex Ginibre matrices, arXiv:2501.04611

  23. [31]

    Marceca and J.L

    F. Marceca and J.L. Romero, Improved discrepancy for the planar Coulomb gas at low temperatures, arXiv:2212.14821. (to appear in J. Anal. Math. )

    arXiv

  24. [32]

    Marzo, L.D

    J. Marzo, L.D. Molag and J. Ortega-Cerd` a, Universality for fluctuations of counting statistics of random normal matrices, arXiv:2508.04386

    arXiv

  25. [33]

    Molag, Edge behavior of higher complex-dimensional determinantal point processes, Ann

    L.D. Molag, Edge behavior of higher complex-dimensional determinantal point processes, Ann. Henri Poincar´ e 24 (2023), no. 12, 4405–4437

    2023

  26. [34]

    Olver, A.B

    F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller and B.V. Saunders, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.13 of 2016-09-16

    2016

  27. [35]

    Sakai, Regularity of a boundary having a Schwarz function, Acta Math

    M. Sakai, Regularity of a boundary having a Schwarz function, Acta Math. 166 (1991), no. 3–4, 263–297

    1991

  28. [36]

    E. B. Saff and V. Totik, Logarithmic potentials with external fields , Grundlehren der Mathematischen Wis- senschaften, Springer-Verlag, Berlin, 1997

    1997

  29. [37]

    Soshnikov, Determinantal random point fields, Russian Math

    A. Soshnikov, Determinantal random point fields, Russian Math. Surveys 55 (2000), no. 5, 923–975

    2000

  30. [38]

    Soshnikov, Statistics of extreme spacing in determinantal random point processes, Mosc

    A. Soshnikov, Statistics of extreme spacing in determinantal random point processes, Mosc. Math. J. 5 (2005), no. 3, 705–719

    2005

  31. [39]

    Shi and Y

    D. Shi and Y. Jiang, Smallest Gaps Between Eigenvalues of Random Matrices With Complex Ginibre, Wishart and Universal Unitary Ensembles, arXiv:1207.4240

    arXiv

  32. [40]

    Thoma, Overcrowding and separation estimates for the Coulomb gas, Comm

    E. Thoma, Overcrowding and separation estimates for the Coulomb gas, Comm. Pure Appl. Math. 77 (2024), no. 7, 3227–3276

    2024

  33. [41]

    Vinson, Closest spacing of eigenvalues, Ph.D

    J. Vinson, Closest spacing of eigenvalues, Ph.D. thesis, Princeton University, 2001

    2001

  34. [42]

    Zhang, Quantitative gap universality for Wigner matrices, arXiv:2507.20442

    A. Zhang, Quantitative gap universality for Wigner matrices, arXiv:2507.20442

  35. [43]

    ˙Zyczkowski and H.-J

    K. ˙Zyczkowski and H.-J. Sommers, Truncations of random unitary matrices, J. Phys. A 33 (2000), no. 10, 2045–2057. 29

    2000

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.