{"id":"435b3ccc-dcbc-48ac-8e0a-1ae508912d7d","arxiv_id":"1909.00613","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A planar jellium with uniform disc background has a non-neutrality barrier between existence and determinantal structure, a disc or crossover equilibrium measure, and edge fluctuations that switch from heavy-tailed to Gumbel with the background charge.","lead":"This paper analyzes a planar Coulomb gas whose confining field comes from a smeared opposite-charge disc, with particles allowed to roam the whole plane. It derives the exact temperature and charge condition for the model to exist, identifies equilibrium densities, and shows the farthest particle has heavy-tailed or Gumbel fluctuations depending on charge.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's λ=1 proof is not established as written: the relative-entropy term claimed to vanish in Section 3.2 is positive for admissible sequences with γ_n↓2, and the LDP transfer to [32] is unverified.","rationale":"The reader's weakest assumption already identifies the reliance on [32] as the main risk, and I agree that this is the load-bearing point. My review sharpens the concern: at λ=1 the relative-entropy term does not actually vanish for admissible sequences with γ_n ↓ 2, so the proof is incomplete even before asking whether [32] applies. However, this is a gap in proof presentation rather than a demonstrated counterexample to the theorem: the divergent log Z_{σ_n} term is independent of µ and would cancel in probability ratios, so the claimed limit may be repairable by tracking additive constants. The internal proofs of Lemma 1.1 (up to a sign/parenthesis typo in the displayed condition), Theorem 1.4, and Theorem 1.5 have concrete supporting arguments, and the edge theorems do not depend on the problematic λ=1 entropy step. Therefore the appropriate verdict remains CONDITIONAL: the central claims are plausible and mostly supported, but the proof of Theorem 1.2 at λ=1 needs correction or a verified reference to [32] before the paper can be accepted as is. I do not see grounds to reject or to mark unverdictable, since the issues are local and fixable.","tokens_in":18785,"tokens_out":23941,"duration_ms":227830,"concrete_test":"Fix β_n = 1 and α_n = n + 1 + e^{−n}, so that γ_n = 2 + e^{−n} and all hypotheses of Theorem 1.2 hold at λ = 1. Compute Z_{σ_n} = ∫ e^{−γ_n Wρ} dℓ exactly in polar coordinates and evaluate (1/(nβ_n)) log Z_{σ_n}; it equals 1 + o(1). Then check whether the hypotheses of [32] are satisfied for this sequence, in particular exponential tightness of σ_n and uniform integrability of e^{−γ_n Wρ}. If a hypothesis fails, Theorem 1.2 as stated needs a repaired proof or a restriction on the admissible sequences; if all hypotheses hold, quote the verification explicitly so the black-box step is documented.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The low-temperature global asymptotics in Theorem 1.2 are the backbone of the paper, and at λ=1 the proof in Section 3.2 uses a sequence of reference measures σ_n with exponent γ_n = β_n(α_n − n + 1) > 2. The text asserts that (1/(nβ_n)) D(µ | σ_n^{⊗n}) → 0 for compactly supported smooth densities ρ. This is false as stated. The displayed formula contains the term (1/(nβ_n)) log Z_{σ_n}, where Z_{σ_n} = ∫ e^{−γ_n Wρ} dℓ. For the admissible sequence β_n = 1, α_n = n + 1 + e^{−n}, one has nβ_n → ∞, α_n/n → 1, and γ_n = 2 + e^{−n} > 2, so Lemma 1.1 applies. At infinity the density of σ_n behaves like |x|^{−γ_n}, hence Z_{σ_n} ~ C/(γ_n − 2) = C e^n, and therefore (1/(nβ_n)) log Z_{σ_n} = 1, not 0. Thus the claimed vanishing of the entropy term fails for an allowed sequence, and the proof does not justify the LDP transfer at λ=1. Separately, the proof of Theorem 1.3 contains the impossible literal bound 2 < δ < β(λ−1), where β = β_n ~ κ/n, so the right side tends to 0; the intended bound δ < κ(λ−1) is clear from the next line but not stated. The edge theorems are better supported, but the macroscopic theorems that anchor them rest on this unchecked step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a planar Coulomb gas (jellium) with external potential generated by a uniform disc background, allowing particles outside the disc and non-neutral total charge. The main results are: (i) an integrability condition (Lemma 1.1) determining when the Gibbs measure is well-defined; (ii) low-temperature macroscopic convergence of the empirical measure to a uniform disc (Theorem 1.2) for α_n/n → λ ≥ 1; (iii) high-temperature crossover convergence to a density solving a nonlinear equation (Theorem 1.3) for nβ_n → κ and κ(λ−1)>2; and (iv) edge fluctuations in the determinantal case β=2: a heavy-tailed product law when λ=1 (Theorem 1.4) and Gumbel fluctuations when λ>1 (Theorem 1.5). The proofs of Theorems 1.4 and 1.5 rely on determinantal structure and partition-function comparisons with Ginibre ensembles, while Theorems 1.2 and 1.3 are proved via a large-deviation transfer to a known result for fixed potentials.","tokens_in":19051,"tokens_out":13765,"duration_ms":120344,"significance":"If fully established, the results provide a unified picture of a planar jellium interpolating between Ginibre-type (uniform equilibrium, Gumbel edge) and spherical-ensemble-type (heavy-tailed edge) behavior, with a crossover regime and a phase transition at λ=1. The explicit integrability condition and the exact heavy-tailed edge distribution are concrete and falsifiable predictions. The paper is clearly written and situates the model well in the literature. The edge theorems (1.4 and 1.5) are supported by essentially complete arguments: Theorem 1.4 uses a careful dominated-convergence argument on the determinantal radial product, and Theorem 1.5 reduces to Ginibre via a partition-function ratio with a localization lemma. The macroscopic theorems (1.2 and 1.3) are the weak part: their proofs delegate the load-bearing large-deviation step to [32] without verifying the hypotheses for the n-dependent potentials, and one displayed claim in the λ=1 case is demonstrably false. The significance of the paper remains high, but the central asymptotic results require substantial proof repair.","major_comments":[{"comment":"The claim that (1/(nβ_n)) D(µ | σ_n^{⊗n}) → 0 for bounded compactly supported smooth densities ρ is false. For the admissible sequence β_n=1, α_n = n+1+e^{-n}, we have γ_n = β_n(α_n−n+1) = 2+e^{-n} > 2, so Lemma 1.1 applies and nβ_n → ∞, α_n/n → 1. The displayed decomposition contains the term (1/(nβ_n)) log Z_{σ_n}, where Z_{σ_n} = ∫ e^{-γ_n Wρ} dℓ. Since Wρ ~ log|z| at infinity, Z_{σ_n} ~ C/(γ_n−2) = C e^n, so (1/(nβ_n)) log Z_{σ_n} → 1, not 0. Thus the relative-entropy term is not negligible for allowed parameter sequences, and the subsequent transfer of the large-deviation principle from [32] is not justified. This leaves the λ=1 case of Theorem 1.2, which is a central part of the paper's low-temperature statement, without a valid proof.","section":"Section 3.2 (Proof of Theorem 1.2, λ=1 case)"},{"comment":"The parameter choice is impossible as written. The proof begins \"Choose any δ > 0 such that 2 < δ < β(λ−1)\", but β = β_n satisfies nβ_n → κ, so β_n(λ−1) → 0. No δ > 2 can satisfy the displayed inequality for large n. The intended bound appears to be δ < κ(λ−1), as used in the subsequent display, but the paper does not state this. More importantly, the proof delegates the key LDP step to [32] without verifying that the hypotheses of [32] hold for the sequence of n-dependent potentials V_n = −(α_n/n)Uρ. Since the crossover regime is one of the paper's main new results, this gap is load-bearing.","section":"Section 3.3 (Proof of Theorem 1.3)"},{"comment":"The paper states an LDP in (10) for a fixed potential V and then applies it to sequences V_n = −(α_n/n)Uρ. The hypotheses required to extend (10) to n-dependent potentials (e.g., uniformity in n of the lower-semicontinuity, growth, and relative-entropy bounds) are not checked in Sections 3.2 or 3.3; the text repeatedly says \"We refer to [32] for the details.\" Without an explicit verification that [32] or its method applies to the sequences considered here, the almost-sure convergence conclusions of Theorems 1.2 and 1.3 are not established. This is a structural gap in the proof of the macroscopic results, independent of the λ=1 computation issue.","section":"Section 2.1, Eq. (10) and Theorem 1.2/1.3 proofs"}],"minor_comments":[{"comment":"The sentence \"The proof of Theorem 1.4 is given in Section 1.4\" should refer to Section 3.4.","section":"Section 1.2, after Theorem 1.4"},{"comment":"The stated scale a_n = √(n c_n)/C_n appears to be missing a factor 2 relative to the standard Ginibre edge scaling in Section 2.2 (where a_n = 2√(n c_n)) after rescaling the potential V_Gin = λ_n/(2R^2)|z|^2. Please verify the constants; the Gumbel limit in Theorem 1.5 requires the correct normalization.","section":"Theorem 1.5"},{"comment":"The notation P(C_n) in the variational formula should be P(C^n) for clarity, though the meaning is inferable from context.","section":"Section 3.2"},{"comment":"The condition \"α − n > 2/β − 1\" is written awkwardly; rewriting as \"β(α − n + 1) > 2\" would match the proof and avoid confusion.","section":"Lemma 1.1"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the unverified transfer of the LDP in [32] to the n-dependent potentials, and the demonstrably false vanishing claim in the λ=1 case of Theorem 1.2. These are fixable in principle (by providing a full proof of the LDP for the sequence V_n or by an alternative argument), but as written the macroscopic theorems are not proven. The edge theorems appear sound. The possible factor-2 error in Theorem 1.5's a_n should also be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know about this paper if you care about planar Coulomb gases with n-dependent potentials. The model is a jellium on the full plane with a uniform disc background of opposite charge, not a priori confined. The main new content is a sharp integrability condition (Lemma 1.1), a macroscopic limit theorem that interpolates between the Ginibre disc and the spherical ensemble, a crossover PDE at higher temperature, and two edge fluctuation theorems: a heavy-tailed product law at λ=1, and Gumbel fluctuations for λ>1. The edge results are the strongest part—Theorem 1.4 is a clean, self-contained Kostlan computation, and Theorem 1.5 reduces to Ginibre via a partition-function ratio handled carefully, with the right use of Ameur's localization theorem for the λ>1 condition. The observation that determinantal structure forces non-neutrality (since β=2 needs α>n) is simple but worth stating.\n\nThe soft spot is the global asymptotic proofs. Theorems 1.2 and 1.3 are sketched via a transfer of the large deviation principle from [García-Zelada], but no hypothesis check is given for the n-dependent potentials V_n = -(α_n/n)Uρ. At λ=1 the proof claims the relative entropy term (1/(nβ_n))D(μ|σ_n^⊗n) vanishes, but that is not true for allowed sequences: take β_n=1, α_n=n+1+e^{-n}; then γ_n=2+e^{-n} and log Z_{σ_n} ~ n, so the term is 1, not 0. This doesn't necessarily break the theorem—the offending term is μ-independent, so it just shifts the rate function—but it does mean the proof as written does not establish the LDP transfer for the full stated range. The theorem may still be true, but it needs a proper argument or a mild additional hypothesis (e.g., γ_n-2 not too small). There is also a clear typo in the proof of Theorem 1.3: the bound 2<δ<β(λ-1) uses β_n~κ/n and is impossible; it should be δ<κ(λ-1).\n\nOverall, the paper is a solid, well-written contribution with concrete and plausible results. The edge theorems are rigorous, and the macro theorems are likely correct but currently under-proved. I'd send it to a serious referee, but the referee should be asked to check whether the LDP in [32] really applies, or whether the authors should restrict the theorem's hypotheses.","headline":"Solid, well-written paper with concrete edge results and plausible macroscopic theorems, but the global asymptotic proofs have a real gap at λ=1 and a typo; worth refereeing with requests for a proper LDP justification.","tokens_in":19649,"tokens_out":11618,"would_cite":true,"duration_ms":285239,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","60B20","82B05","82B26","31A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A planar jellium with a uniform opposite-sign disc has a uniform disc equilibrium, and its farthest particle switches from heavy-tailed to Gumbel as background charge crosses a threshold.","keywords":["planar Coulomb gas","Wigner jellium","logarithmic potential","equilibrium measure","large deviations","determinantal point process","edge fluctuations","Gumbel law"],"falsifier":"Run the determinantal simulation with β=2 and α_n=n+κ for a fixed positive κ and record the largest radius over many n: the claimed law L predicts that the survival function 1−L(t) behaves like (R/t)^{2κ} for large t, so a log-log plot of the empirical tail against t/R should become a straight line of slope −2κ. A slope different from −2κ, or a finite-n sequence that fails to approach the infinite product, would refute Theorem 1.4. Separately, checking the hypotheses of the cited large-deviation theorem for V_n at λ=1 would settle whether Theorem 1.2's proof transfers.","tokens_in":18497,"feed_emoji":"⚛️","tokens_out":10915,"duration_ms":97316,"temperature":0.7,"pith_summary":"The paper studies a planar Coulomb gas of n repelling unit charges held together by the field of a uniform disc of opposite charge, with particles allowed outside the disc. It proves that when the background charge ratio λ=lim α_n/n is at least 1, the empirical density converges almost surely to a uniform disc of radius R/√λ in the low-temperature regime, and to a crossover density solving Δlog φ=2πκ(φ−λ/(πR²)1_{|·|≤R}) in the high-temperature regime. In the determinantal case β=2, it proves a phase transition for the farthest particle: a heavy-tailed product law when α_n−n→κ>0, and a Gumbel law after rescaling when λ>1. The value is that this simple jellium connects two canonical models: Ginibre-type bulk behavior and spherical-ensemble-type heavy-tailed edge fluctuations.","feed_headline":"Farthest particle flips from heavy tail to Gumbel","feed_subtitle":"A planar jellium has Ginibre bulk; its edge law is a product law at λ=1 and Gumbel above.","key_machinery":"The central object is the logarithmic potential U_ρ of the uniform disc background, written explicitly as quadratic on the disc and logarithmic outside. The proofs rewrite the Gibbs weight using G(x,y)=g(x−y)+W_ρ(x)+W_ρ(y) with W_ρ=−U_ρ, which separates the n-dependent confining potential from the singular pair interaction and makes the large-deviation transfer possible. In the determinantal radially symmetric case, the key mechanism is the representation of the moduli of the particles as independent random variables with densities $t^{{2k−1}}$$e^{{-2nQ(t)}}$, which turns the farthest-particle distribution into a product of explicit one-dimensional integrals and leads directly to the infinite-product limit law and to the Ginibre comparison in the λ>1 regime.","core_discovery":"The central discovery is that this jellium, despite being non-neutral and unconfined, is exactly solvable enough to exhibit both macroscopic and edge phase transitions controlled by the charge ratio λ and the temperature parameter κ=lim nβ_n. Lemma 1.1 identifies the exact integrability boundary: the Gibbs measure is well defined if and only if β_n(α_n−n+1)>2, which in the determinantal case β=2 means α_n>n, so the system cannot be simultaneously charge-neutral and determinantal. Theorems 1.2 and 1.3 give the almost-sure limit of the empirical measure in the low-temperature and crossover regimes. Theorems 1.4 and 1.5 give the distribution of the modulus of the farthest particle in the determinantal case: when α_n=n+κ_n with κ_n→κ>0, the limit is the heavy-tailed law L(t)=∏_{k=0}^∞(1−(R/t)^{2(k+κ)}) on [R,∞), and when λ>1, an affine rescaling with explicit constants produces a Gumbel limit.","pith_inferences":["One could test Theorem 1.4 directly by simulating the β=2 jellium with α_n=n+κ for a fixed κ and checking that the empirical survival function of the largest radius follows a straight line of slope −2κ in a log-log plot against t/R.","The same G(x,y) rewriting should apply to other radially symmetric background charges whose potential is quadratic inside and logarithmic outside, suggesting analogues of Theorems 1.4 and 1.5 for a family of generalized jelliums.","The crossover equation Δlogφ=2πκ(φ−λ/(πR²)1_{|·|≤R}) may define a one-parameter family of random normal matrix models interpolating between the Ginibre disc and a free-field regime, which could be explored by studying the κ→∞ limit equation explicitly.","The heavy-tail-to-Gumbel transition at λ=1 might be a general mechanism for two-dimensional Coulomb gases with a background boundary: whenever the background potential outside the support is logarithmic, the farthest particle will have polynomial tail, and whenever the particles are strictly inside the smooth background, the tail becomes exponential. This is an inference beyond the paper's stateme"],"forward_implications":["For λ>1, the macroscopic support R/√λ lies strictly inside the background disc, so the limiting gas never occupies an outer ring of the disc: the outer edge of the background is a hard edge in the limit.","For β=2 and α_n−n→κ>0, the farthest-particle radius has a one-sided heavy-tailed limit on [R,∞), with survival probability decaying like (R/t)^{2κ} for large t.","For β=2 and λ>1, the rescaled farthest radius converges to the Gumbel law, matching the complex Ginibre ensemble edge, as expected because the particles only feel the quadratic part of the potential.","In the high-temperature crossover nβ_n→κ with κ(λ−1)>2, the limiting density is neither uniform nor the free Boltzmann density but solves a nonlinear elliptic equation with the background term λ/(πR²)1_{|·|≤R}.","The critical case κ(λ−1)=2 and the recovery of the uniform disc as κ→∞ are left open in the paper."],"supporting_citations":[{"why":"Supplies the large-deviation principle for singular Gibbs measures that the proofs of Theorems 1.2 and 1.3 transfer to the n-dependent potential V=-(α_n/n)U_ρ.","marker":"[32]"},{"why":"Provides the independent-radii representation for radially symmetric β=2 Coulomb gases and the Ginibre edge fluctuation asymptotics used in Theorems 1.4 and 1.5.","marker":"[21]"},{"why":"Gives the factorization of the determinantal point process into independent radial variables used to write the farthest-particle distribution as a product.","marker":"[45]"},{"why":"Gives the complex Ginibre edge limit theorem (Gumbel) that Theorem 1.5 is compared against after rescaling.","marker":"[62]"},{"why":"Used to interpolate the heavy-tailed product law to the Gumbel law as κ→∞ and to describe convergence of the edge point process.","marker":"[31]"},{"why":"Provides the quantitative exponential localization estimate used in Lemma 3.2 to show that the probability of the relevant scaled Ginibre event tends to one for λ>1.","marker":"[4]"}],"fun_headline_variants":["Jellium edge heavy tail flips to Gumbel at λ>1","Planar jellium: edge law goes from heavy tail to Gumbel","Non-neutral jellium edge: heavy-tail to Gumbel transition","Gumbel edge emerges in jellium beyond critical charge","Edge of planar jellium switches from heavy tail to Gumbel"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof leans on a borrowed large-deviations result being valid for these n-dependent potentials, but the paper never checks the result's hypotheses, and one displayed inequality in the proof of Theorem 1.3 asks for a δ with 2<δ<β_n(λ−1), which is impossible for large n since β_n→0, so the high-temperature theorem's derivation is not complete as written.","fun_headline_variants_meta":{"raw":{"variants":["Jellium edge heavy tail flips to Gumbel at λ>1","Planar jellium: edge law goes from heavy tail to Gumbel","Non-neutral jellium edge: heavy-tail to Gumbel transition","Gumbel edge emerges in jellium beyond critical charge","Edge of planar jellium switches from heavy tail to Gumbel"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000342,"raw_usage":{"total_tokens":1900,"prompt_tokens":981,"completion_tokens":919,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":821}},"tokens_in":597,"tokens_out":919,"duration_ms":8071,"temperature":1.0,"reasoning_tokens":821,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:43:04.158185+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the determinantal simulation with β=2 and α_n=n+κ for a fixed positive κ and record the largest radius over many n: the claimed law L predicts that the survival function 1−L(t) behaves like (R/t)^{2κ} for large t, so a log-log plot of the empirical tail against t/R should become a straight line of slope −2κ. A slope different from −2κ, or a finite-n sequence that fails to approach the infinite product, would refute Theorem 1.4. Separately, checking the hypotheses of the cited large-deviation theorem for V_n at λ=1 would settle whether Theorem 1.2's proof transfers.","supporting_citations":[{"cited_title":"A note on the second order universality at the edge of Coulo mb gases on the plane","cited_arxiv_id":null,"evidence_quote":"Provides the independent-radii representation for radially symmetric β=2 Coulomb gases and the Ginibre edge fluctuation asymptotics used in Theorems 1.4 and 1.5."},{"cited_title":"On the spectra of Gaussian matrices","cited_arxiv_id":null,"evidence_quote":"Gives the factorization of the determinantal point process into independent radial variables used to write the farthest-particle distribution as a product."},{"cited_title":"A limit theorem at the edge of a non-Hermitian random matri x ensemble","cited_arxiv_id":null,"evidence_quote":"Gives the complex Ginibre edge limit theorem (Gumbel) that Theorem 1.5 is compared against after rescaling."},{"cited_title":"Edge fluctuations for random normal matrix ensembles","cited_arxiv_id":"1812.11170","evidence_quote":"Used to interpolate the heavy-tailed product law to the Gumbel law as κ→∞ and to describe convergence of the edge point process."},{"cited_title":"A localization theorem for the planar Coulomb gas in an external field","cited_arxiv_id":"1907.00923","evidence_quote":"Provides the quantitative exponential localization estimate used in Lemma 3.2 to show that the probability of the relevant scaled Ginibre event tends to one for λ>1."}],"review_version":1}