Poisson Statistics for Coulomb Gases at Intermediate Temperature Regimes

David Padilla-Garza; Eric Thoma; Luke Peilen

REVIEW 2 major objections 5 minor 1 cited by

Poisson Statistics for Coulomb Gases at Intermediate Temperature Regimes

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read At intermediate temperatures, the local point process of a 2D Coulomb gas converges weakly to a Poisson point process with intensity equal to the equilibrium density at the zoom point.

desk verdict New intermediate-temperature Poisson result; the fixed-k correlation proof is sound but the uniform-in-k step in Prop 3.7 is not justified. read the letter →

arxiv 2507.08198 v2 pith:FZPP3QRT submitted 2025-07-10 math.PR math-phmath.MP

David Padilla-Garza , Luke Peilen , Eric Thoma This is my paper

classification math.PRmath-phmath.MP MSC 60F0560G5582B21

keywords CoulombgasintermediatetemperaturePoissonpointprocesscorrelationfunctionslocalequilibriummeasureconcentrationestimatelogarithmicpotential

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

The paper proves that in the intermediate temperature regime, the two-dimensional Coulomb gas loses all microscopic correlations: around a point $z$ inside the droplet, the rescaled particle configuration $Q_{z,N}$ converges weakly to a homogeneous Poisson point process with intensity equal to the equilibrium density $\mu_V(z)$. The temperature window is $\beta N \to \infty$ and $\beta \sqrt{N} \log N \to 0$, which places the gas hotter than any fixed-temperature regime but colder than the previously understood high-temperature regime. This fills the main gap in the microscopic picture for 2D Coulomb gases: at low temperatures particles are correlated and ordered, at very high temperatures they are independent, and the paper shows that independence already emerges throughout the intermediate window. The proof goes through a quantitative asymptotic description of all correlation functions, not just a soft compactness argument.

What carries the argument

The carrying mechanism is the splitting formula that rewrites the Gibbs density relative to the thermal equilibrium measure $\mu_\theta$: $P_{N,\beta}(dX_N) = K_{N,\beta}^{-1} e^{-\beta N^2 F_N(X_N,\mu_\theta)} \mu_\theta^{\otimes N}(dX_N)$, with $F_N$ the next-order jellium energy. On top of that sits a new concentration estimate for the logarithmic potential of the fluctuation field $\mathrm{fluct} = \mathrm{emp}_N - \mu_\theta$: with high probability, $\left|\sum_i h_{\mathrm{fluct}}(y_i)\right|$ is $O(N^{-1/2}\log N)$, with exponential tails. The proof of this estimate regularizes point charges by uniform measures on circles of radius $\eta$, uses the explicit Bessel Fourier transform of the circle measure to lower-bound the Coulomb energy when the smeared potential is large, and then uses isotropic-averaging overcrowding and conditional-density bounds to remove the smearing. Finally, a correlation-function computation expresses each $k$-point marginal in terms of exponential moments of $h_{\mathrm{fluct}}$ and shows these moments converge to one, yielding Poisson statistics via a standard Laplace-functional criterion.

What would settle it

Take the quadratic potential $V(x)=|x|^2$ and a sequence $\beta_N = N^{-1/2}/\log N$, well inside the window. Compute the two-point correlation function $R_2(y_1,y_2)$ of $Q_{0,N}$; the theorem predicts $R_2 \to \mu_V(0)^2 = 1/\pi^2$ uniformly in $y_1,y_2$. A persistent deviation as $N$ grows, such as a depression of $R_2$ when $|y_1-y_2|$ is of order one, would falsify the Poisson claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 1: assume the temperature and growth conditions (Temp) and (Grow), and let $z$ lie in the interior of $\operatorname{supp}(\mu_V)$. Then, under the Gibbs law $P_{N,\beta}$, the local point process $Q_{z,N} = \sum_i \delta_{N^{1/2}(x_i-z)}$ converges weakly to a Poisson point process whose intensity is $\mu_V(z) \mathrm{Leb}$. Equivalently, for every fixed $k$, the $k$-point correlation functions of $Q_{z,N}$ converge to $\mu_V(z)^k$, uniformly in the arguments, with the factorial-moment bounds needed for tightness. The quantitative form states that $R_k(Y_k) = e^{-\beta k^2 F_k(X_k,\mu_\theta)} \prod_i \mu_\theta(x_i)\left(1+O(\beta N^{1/2}\log N)\right)$, so the exponential Boltzmann factor from the next-order energy is the only finite-$N$ correction that matters. The identification of the deterministic intensity uses the thermal equilibrium measure $\mu_\theta$ approximating $\mu_V$ in the interior.

Load-bearing premise

The whole proof leans on two imported control estimates about how many particles can crowd into a tiny ball and how the conditional density of particles can vary; if either estimate fails, the concentration step that drives the argument collapses.

Editorial extensions

If this is right

In the intermediate window, the empirical statistics on any bounded microscopic window converge to $\mathrm{Poisson}(\mu_V(z)|A|)$, so particles become asymptotically uncorrelated at scale $N^{-1/2}$.
The quantitative correlation asymptotics give finite-$N$ error rates of order $\beta N^{1/2}\log N$ for factorial moments and hence for local statistics such as void probabilities at the microscopic scale.
The first marginal satisfies $\rho_1(x) = \mu_\theta(x)(1+O(\beta N^{(1+\gamma)/2}))$, implying exponential-in-$\beta N$ confinement of particles to the droplet.
Because the intensity is the equilibrium density $\mu_V(z)$, the result upgrades earlier mixed-Poisson limits to a genuine Poisson process with a deterministic, explicitly identified intensity.
The theorem opens a temperature window previously unexplored for 2D Coulomb gases, connecting very-high-temperature Poisson behavior with the energy-dominated low-temperature regime.

Reading between the lines

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

A natural stress test is to optimize the temperature exponent: the proof requires $\beta \sqrt{N} \log N \to 0$, and the bottleneck is the control of exponential moments of $h_{\mathrm{fluct}}$; improving that control could plausibly extend Poisson convergence to $\beta \sqrt{N} \to 0$.
The concentration estimate behind the theorem is likely to transfer to other functionals of the fluctuation field, giving a route to local laws or rigidity statements at intermediate temperatures that the paper does not pursue.
The restriction to $d=2$ is structural: the smeared-energy lower bound degenerates as $\eta \to 0$ in $d \geq 3$, so the same strategy would need a new idea for higher-dimensional Coulomb gases.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear [Colleague],

Here's my read on arXiv:2507.08198. The headline: the main theorem is new and probably correct, but the proof has a real gap in the final uniform-integrability step. The stress-test note is right; the reader's diagonal concern is a minor distraction.

What is new: for the 2D Coulomb gas, Theorem 1 claims Poisson statistics in the intermediate temperature window βN→∞ and β√N log N→0, with deterministic intensity μ_V(z). Lambert handled β≈1/N, and Thoma handled βN→0 but only got a mixed Poisson limit. This intermediate regime was genuinely open. The paper's main technical contribution, the concentration estimate for the logarithmic potential of fluctuations (Prop 1.3 / 2.6), is solid. The smearing argument, the energy lower bound of Lemma 2.2, and the correlation expansion all check out structurally. The fixed-k asymptotics are convincing.

The soft spot is Proposition 3.7. The proof establishes (3.15) for each fixed k, which gives pointwise convergence R_k → μ_V(z)^k. But then it asserts a uniform bound |R_k| ≤ (C μ_V(z))^k for all N and k, which is exactly what Proposition 3.2(2) needs for the summation over k to pass. That conclusion does not follow from the displayed estimates. The error term in Corollary 3.4 is 4kC0 βN^{1/2} log N times ‖F‖_{L^2}, and Proposition 3.6 only gives ‖F‖_{L^2} ≤ e^{Ck}. The relative error is therefore O(k e^{Ck} βN^{1/2} log N), which is o(1) only when k ≲ log N, not uniformly in k. Without a k-uniform bound, the supremum over N of the summed correlation integrals is uncontrolled. The diagonal behavior of e^{-βk^2F_k} is irrelevant; the k-dependence of the error is the real issue.

This looks fixable, but it needs a separate argument, for example a uniform bound on the correlation functions that does not grow exponentially in k, or a direct control of the exponential moments of the local counting function. As things stand, Theorem 1 is not fully proven.

One more thing: several load-bearing inputs come from very recent or unpublished work ([Tho24], [Tho25], and lecture notes [Ser24]). They seem reasonable, but a referee should check them carefully. The constants are not verified, which is minor.

Who is this for? Coulomb gas and random matrix theorists. The concentration estimate alone is worth keeping around even if the main theorem needs repair. I would send it to a serious referee, with instructions to focus on the final step.

Best, [Your name]

Referee Report

2 major / 5 minor

Summary. The paper studies the two-dimensional Coulomb gas with inverse temperature β_N in the regime β_N√N log N → 0, with the intended constraint Nβ_N → ∞. The main theorem claims that the microscopic point process Q_{z,N} converges weakly to a homogeneous Poisson point process with intensity μ_V(z) at bulk points z. The method is quantitative comparison of correlation functions: the authors prove a new concentration estimate for the logarithmic potential of fluctuations, use isotropic averaging and a regularization of the Coulomb kernel to control exponential moments, and derive an asymptotic formula for the k-point correlation functions. A corollary gives a first-marginal estimate and a confinement estimate.

Significance. If the proof is completed, this result fills a genuine gap in the literature: it extends Poisson statistics for the un-averaged local point process of the two-dimensional Coulomb gas from the high-temperature regime β_N ≍ 1/N to intermediate temperatures, with quantitative error rates. The concentration estimate for h_fluct (Propositions 1.3 and 2.6) is a useful technical contribution in its own right. The paper builds on genuine lemmas from prior work—overcrowding estimates and conditional density bounds—and does not appear to assume the conclusion. However, two load-bearing issues remain: the theorem statement omits the required Nβ_N → ∞ assumption, and the uniform-in-k bound needed for the point-process convergence criterion is not proven.

major comments (2)

[Section 1.2, Theorem 1] The statement of Theorem 1 assumes only (Temp) and (Grow), but the intended intermediate regime also requires Nβ_N → ∞. As written, (Temp) — β_N = o(1/(N^{1/2} log N)) — allows β_N ≍ 1/N; then θ = Nβ_N is bounded and μ_θ does not converge to μ_V. The proof of Proposition 3.7 invokes μ_θ(y_i) → μ_V(z) from [AS22, Theorem 1], which is a θ → ∞ limit. For bounded θ the correct limiting intensity is μ_θ(z), not μ_V(z), so the theorem as stated is false. The hypothesis Nβ_N → ∞ (equivalently β_N ≫ 1/N) must be included explicitly, as in the abstract.
[Proposition 3.7, final paragraph] The uniform-in-k bound |R_N^k(y_1,...,y_k)| ≤ (C μ_V(z))^k is asserted without proof. The preceding computation proves (3.15) only for each fixed k, with an O(β N^{1/2} log N) error whose implicit constant may depend on k. The displayed estimates (3.16)–(3.17) do not control the k-dependence of Corollary 3.4 (which requires k ≤ (N−k)^{1/2}/10 and β ≤ 1/(k C_0 (N−k)^{1/2} log(N−k))) nor the factor e^{−βk²F_k}, which can be as large as exp(C β k⁴ log N). Consequently condition (2) of Proposition 3.2, namely sup_N ∑_{k≥1} (1/k!)∫_Ω R_N^k < ∞, is not established. Pointwise convergence of each fixed-k correlation function is insufficient for weak convergence of point processes, so this gap is load-bearing. A separate k-uniform estimate, or an alternative tightness argument, is needed.

minor comments (5)

[Section 1.2] The notation βN is ambiguous: in the abstract and introduction it denotes the product β·N (e.g., 'βN → ∞'), while in (Temp) it appears to denote the sequence β_N. This clash makes the hypotheses of Theorem 1 unclear; please write β_N for the sequence and β N for the product.
[Corollary 1.2, Eq. (1.15)] The error term O(β N^{(1+γ)/2}) need not tend to 0 under (Temp) alone; for example β_N = (N^{1/2}(log N)^2)^{-1} satisfies (Temp) but β N^{(1+γ)/2} → ∞ for every γ > 0. The sharper error O(β N^{1/2} log N) from Proposition 3.7 would make the statement meaningful, or the corollary should specify the additional regime γ < 2α when combined with β ≤ N^{−1/2−α}.
[Proposition 3.6, proof] The lower-bound step is garbled: λ is defined twice, and the final displayed bound contains β log β, which is negative for β < 1. Please rewrite this step with β |log β| and clarify the Young/Jensen argument.
[Proposition 3.7, statement] The uniformity claim 'for any Y_k ∈ (R²)^k' is too strong as stated: for unbounded Y_k, the factor e^{−βk²F_k} is not uniformly 1 + O(β log N). State the uniformity over compact sets, or track the dependence on |Y_k|.
[Affiliations] The first affiliation contains a typo: 'Einsten Institute' should be 'Einstein Institute'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the Poisson limit is derived from new correlation-function and concentration estimates; prior self-citations are genuine lemmas that do not assume the conclusion.

full rationale

The central derivation is self-contained in the sense required for a circularity audit. Theorem 1 is proved by computing k-point correlation functions (Lemma 3.3 and Proposition 3.7) and invoking the standard convergence criterion in Proposition 3.2, which comes from [Lam21b, Lemma A.8]. The limiting intensity mu_V(z) is the equilibrium density determined by the potential V, not a parameter fitted to data. The self-citations to [Tho24] and [Tho25] supply overcrowding and conditional-density bounds (equations (2.14) and (2.15)), and [PG23] supplies the lower bound K_{N,beta} >= 1; these are stated lemmas with their own assumptions and do not contain the target Poisson-convergence result. The paper even notes that its proof is largely independent of [Tho25], using only a confinement bound that could be replaced. There is no fitted input renamed as a prediction, no uniqueness theorem imported from the authors to force the choice of process, and no known result merely renamed. The skeptical concern about the k-uniform bound in Proposition 3.7 is a potential correctness gap in the proof of uniform summability, not a circular reduction: the asserted bound is not derived from the target conclusion, and the issue is one of missing control rather than equivalence-by-definition. Accordingly, the paper shows no significant circularity; the score of 2 reflects the presence of multiple self-citations for technical inputs, but none is load-bearing in a circular way.

Assumptions & free parameters 6 free parameters · 10 assumptions · 0 invented entities

No numerical fits or physical entities are introduced. The proof depends on a chain of external theorems, several from the same research group, which are legitimate but load-bearing. The free parameters listed are proof devices, not part of the theorem statement.

free parameters (6)

eta (smearing radius) = N^{-100}
Chosen in Proposition 2.6 Step 1 to regularize the point charges. Must be much smaller than the microscopic scale; the power 100 is arbitrary and the theorem does not depend on its exact value.
delta_inv (first moment cutoff) = N^{100}
Used in Proposition 2.6 Step 3.1 so that the smeared fluctuation measure has controlled first moment and Lemma 2.2 applies. The exponent is arbitrary.
T (concentration threshold) = T >= C log N
Threshold appearing in Proposition 2.6; the tail bound requires T to grow logarithmically. C is a universal constant.
R (overcrowding scale) = N^{-1/2} beta^{-1/2}
Chosen at the minimal scale where the overcrowding estimate (2.14) holds; used throughout the proof of Proposition 2.5.
R_cut (cutoff radius in Proposition 3.6) = large enough so that (1-chi)|h_mu_theta| <= zeta_V
Cutoff in the exponential moment bound for h_mu_theta. Existence follows from the growth of V; the exact value is irrelevant.
lambda (interpolation parameter in Proposition 3.6) = 1/sqrt(N) for the lower bound
Balances Young and Jensen inequalities in the exponential moment estimates; any decaying choice works.

assumptions (10)

domain assumption Confinement bound (1.13) from Thoma 2025, Theorem 3
Used in Proposition 2.6 Step 3.1 to control the probability that particles escape to distance N^{99}. Load-bearing for the concentration estimate.
domain assumption Overcrowding estimate (2.14) from Thoma 2024, Theorem 1
Controls the probability of many particles in a small ball; used in Proposition 2.5 Steps 1 and 4 to bound exponential moments.
domain assumption Conditional density bound (2.15) from Thoma 2025, Proposition 3.1
Gives uniform bounds on conditional densities; used to transfer exponential moments from smeared to unsmeared potentials.
domain assumption Exponential moment bound for H^1 test functions (Serfaty 2024, Corollary 5.21)
Used in Proposition 3.6 to control fluctuations of regularized potentials. Requires the test function gradient to lie in L^2 cap L^infty.
domain assumption Average localization bound (Serfaty 2024, Corollary 5.26)
Used in Proposition 3.6 Step 2 to control the contribution of h_mu_theta outside a large ball.
domain assumption Partition function lower bound K_N,beta >= 1 (Padilla-Garza 2023, Proposition 5.10)
Used in Proposition 2.6 to convert energy upper bounds into probability bounds.
domain assumption Thermal equilibrium measure approximation mu_theta -> mu_V (Armstrong-Serfaty 2022, Theorem 1)
Used to identify the limiting Poisson intensity as mu_V(z) for z in the interior of the support.
standard math Equilibrium measure existence and Euler-Lagrange equation (Frostman 1935, Saff-Totik 1997)
Background for the definition of mu_V and the confinement function zeta_V.
standard math Coulomb kernel Fourier representation \hat{g}(xi) = 1/(2 pi |xi|^2)
Used in Lemma 2.2 to lower-bound the Coulomb energy in terms of the potential.
standard math Bessel function asymptotics (Abramowitz-Stegun)
Used in Lemma 2.2 to bound high-frequency contributions in the energy lower bound.

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Pith. "Pith review of Poisson Statistics for Coulomb Gases at Intermediate Temperature Regimes." pith.science (2026). https://pith.science/paper/FZPP3QRT

@misc{pith2026250708198,
  author       = {Pith},
  title        = {Pith review of: Poisson Statistics for Coulomb Gases at Intermediate Temperature Regimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FZPP3QRT}},
  note         = {Machine review of arXiv:2507.08198}
}

abstract

We consider the microscopic statistics of a Coulomb gas in $\mathbb{R}^2$ at intermediate temperatures. In particular, we show that the microscopic point process associated to the Coulomb gas converges to a homogeneous Poisson point process at intermediate temperature regimes $\beta N \rightarrow \infty$ and $\beta \sqrt{N} \log N \rightarrow 0$, extending previous results. Our approach relies on a novel quantitative asymptotic description of correlation functions, which is of its own interest.

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Works this paper leans on

12 extracted references · 7 canonical work pages · cited by 1 Pith paper

[1]

6, 1043–1065

[AB19] Gernot Akemann and Sung-Soo Byun, The high temperature crossover for general 2d coulomb gases, Journal of Statistical Physics 175 (2019), no. 6, 1043–1065. [ABG12] Romain Allez, Jean-Philippe Bouchaud, and Alice Guionnet, Invariant beta ensembles and the gauss-wigner crossover, Physical Review Letters 109 (2012), no. 9, 094102–. [AD14] Romain Allez...

work page 2019
[5]

[Caf98] L. A. Caffarelli, The obstacle problem revisited , Journal of Fourier Analysis and Applications 4 (1998), no. 4, 383–402. [CHM18] Djalil Chafa¨ ı, Adrien Hardy, and Myl` ene Ma¨ ıda,Concentration for coulomb gases and coulomb transport inequalities, Journal of Functional Analysis 275 (2018), no. 16, 1447–1483. [CS07] Luis Caffarelli and Luis Silve...

work page 1998
[10]

3, 635–743

[SS12] Etienne Sandier and Sylvia Serfaty, From the ginzburg-landau model to vortex lattice problems , Communications in Mathematical Physics 313 (2012), no. 3, 635–743. [SS15] Etienne Sandier and Sylvia Serfaty, 2d coulomb gases and the renormalized energy , The Annals of Probability 43 (2015), no. 4, 2026–2083. [ST97] Edward B. Saff and Vilmos Totik, Lo...

work page 2012
[399]

[Lam21b] , Poisson statistics for gibbs measures at high temperature , Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques57 (2021), no

[Lam21a] Gaultier Lambert, Mesoscopic central limit theorem for the circular β-ensembles and applications , Electronic Journal of Probability 26 (2021), 1–33. [Lam21b] , Poisson statistics for gibbs measures at high temperature , Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques57 (2021), no. 1, 326–350. [Leb16] Thomas Lebl´ e, Logari...

arXiv 2021
[1401]

[Joh98] Kurt Johansson, On fluctuations of eigenvalues of random hermitian matrices , Duke Mathematical Journal 91 (1998), no

[GZPG24] David Garc´ ıa-Zelada and David Padilla-Garza,Generalized transport inequalities and concentration bounds for riesz-type gases , Electronic Journal of Probability 29 (2024), 1–35. [Joh98] Kurt Johansson, On fluctuations of eigenvalues of random hermitian matrices , Duke Mathematical Journal 91 (1998), no. 1, 151–204. [KS09] Rowan Killip and Mihai...

work page 2024
[1935]

3, 1377 –

[GZ19] David Garc´ ıa-Zelada, A large deviation principle for empirical measures on Polish spaces: Appli- cation to singular Gibbs measures on manifolds , Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques 55 (2019), no. 3, 1377 –

work page 2019
[1971]

A maximum principle for the Coulomb gas: microscopic density bounds, confinement estimates, and high temperature limits

[Tho24] Eric Thoma, Overcrowding and separation estimates for the coulomb gas , Communications on Pure and Applied Mathematics 77 (2024), no. 7, 3227–3276. [Tho25] Eric Thoma, A maximum principle for the coulomb gas: microscopic density bounds, confinement estimates, and high temperature limits , arXiv:2501.02733, January

work page Pith review arXiv 2024
[1974]

1, 46–121

[AS21] Scott Armstrong and Sylvia Serfaty, Local laws and rigidity for coulomb gases at any temperature , Annals of Probability 49 (2021), no. 1, 46–121. [AS22] , Thermal approximation of the equilibrium measure and obstacle problem , Annales de la Facult´ e des sciences de Toulouse : Math´ ematiques, Serie 631 (2022), no. 4, 1085–1110. [BBNY19] Roland Ba...

arXiv 2021

Show all 12 references

[2022]

3, 633–656

[BGP15] Florent Benaych-Georges and Sandrine P´ ech´ e, Poisson statistics for matrix ensembles at large temperature, Journal of Statistical Physics 161 (2015), no. 3, 633–656. [BL18] Florent Bekerman and Asad Lodhia, Mesoscopic central limit theorem for general β-ensembles, A...

2015 arXiv
[2023]

[Bou23] Jeanne Boursier, Optimal local laws and clt for the circular riesz gas , arXiv:2112.05881v3, February

[BMP22] Paul Bourgade, Krishnan Mody, and Michel Pain, Optimal local law and central limit theorem for β-ensembles, Communications in Mathematical Physics 390 (2022), 1017–1079. [Bou23] Jeanne Boursier, Optimal local laws and clt for the circular riesz gas , arXiv:2112.05881v3...

2022 arXiv
[2024]

Ram´ ırez, Brian Rider, and B´ alint Vir´ ag,Beta ensembles, stochastic airy spectrum, and a diffusion, Journal of the American Mathematical Society 24 (2011), no

[RRV11] Jos´ e A. Ram´ ırez, Brian Rider, and B´ alint Vir´ ag,Beta ensembles, stochastic airy spectrum, and a diffusion, Journal of the American Mathematical Society 24 (2011), no. 4, 919–944. [RV07] Brian Rider and B´ alint Vir´ ag, The noise in the circular law and the gaus...

2011 arXiv
[2025]

[VV09] Benedek Valk´ o and B´ alint Vir´ ag,Continuum limits of random matrices and the brownian carousel , Inventiones mathematicae 177 (2009), 463–508. (D. Padilla-Garza) Einsten Institute of Mathematics, Hebrew University of Jerusalem Email address: David.Padilla-garza@mail...

2009

Pith tools

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

[1] [1]

6, 1043–1065

[AB19] Gernot Akemann and Sung-Soo Byun, The high temperature crossover for general 2d coulomb gases, Journal of Statistical Physics 175 (2019), no. 6, 1043–1065. [ABG12] Romain Allez, Jean-Philippe Bouchaud, and Alice Guionnet, Invariant beta ensembles and the gauss-wigner crossover, Physical Review Letters 109 (2012), no. 9, 094102–. [AD14] Romain Allez...

work page 2019

[2] [5]

[Caf98] L. A. Caffarelli, The obstacle problem revisited , Journal of Fourier Analysis and Applications 4 (1998), no. 4, 383–402. [CHM18] Djalil Chafa¨ ı, Adrien Hardy, and Myl` ene Ma¨ ıda,Concentration for coulomb gases and coulomb transport inequalities, Journal of Functional Analysis 275 (2018), no. 16, 1447–1483. [CS07] Luis Caffarelli and Luis Silve...

work page 1998

[3] [10]

3, 635–743

[SS12] Etienne Sandier and Sylvia Serfaty, From the ginzburg-landau model to vortex lattice problems , Communications in Mathematical Physics 313 (2012), no. 3, 635–743. [SS15] Etienne Sandier and Sylvia Serfaty, 2d coulomb gases and the renormalized energy , The Annals of Probability 43 (2015), no. 4, 2026–2083. [ST97] Edward B. Saff and Vilmos Totik, Lo...

work page 2012

[4] [399]

[Lam21b] , Poisson statistics for gibbs measures at high temperature , Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques57 (2021), no

[Lam21a] Gaultier Lambert, Mesoscopic central limit theorem for the circular β-ensembles and applications , Electronic Journal of Probability 26 (2021), 1–33. [Lam21b] , Poisson statistics for gibbs measures at high temperature , Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques57 (2021), no. 1, 326–350. [Leb16] Thomas Lebl´ e, Logari...

arXiv 2021

[5] [1401]

[Joh98] Kurt Johansson, On fluctuations of eigenvalues of random hermitian matrices , Duke Mathematical Journal 91 (1998), no

[GZPG24] David Garc´ ıa-Zelada and David Padilla-Garza,Generalized transport inequalities and concentration bounds for riesz-type gases , Electronic Journal of Probability 29 (2024), 1–35. [Joh98] Kurt Johansson, On fluctuations of eigenvalues of random hermitian matrices , Duke Mathematical Journal 91 (1998), no. 1, 151–204. [KS09] Rowan Killip and Mihai...

work page 2024

[6] [1935]

3, 1377 –

[GZ19] David Garc´ ıa-Zelada, A large deviation principle for empirical measures on Polish spaces: Appli- cation to singular Gibbs measures on manifolds , Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques 55 (2019), no. 3, 1377 –

work page 2019

[7] [1971]

A maximum principle for the Coulomb gas: microscopic density bounds, confinement estimates, and high temperature limits

[Tho24] Eric Thoma, Overcrowding and separation estimates for the coulomb gas , Communications on Pure and Applied Mathematics 77 (2024), no. 7, 3227–3276. [Tho25] Eric Thoma, A maximum principle for the coulomb gas: microscopic density bounds, confinement estimates, and high temperature limits , arXiv:2501.02733, January

work page Pith review arXiv 2024

[8] [1974]

1, 46–121

[AS21] Scott Armstrong and Sylvia Serfaty, Local laws and rigidity for coulomb gases at any temperature , Annals of Probability 49 (2021), no. 1, 46–121. [AS22] , Thermal approximation of the equilibrium measure and obstacle problem , Annales de la Facult´ e des sciences de Toulouse : Math´ ematiques, Serie 631 (2022), no. 4, 1085–1110. [BBNY19] Roland Ba...

arXiv 2021