{"id":"fc50bd06-9676-427a-9718-f207bde80f79","arxiv_id":"1907.08868","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":3.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The maximum of the 2D integer-valued Gaussian free field is shown to be of order log of the domain size.","lead":"The paper proves that the maximum of the integer-valued Gaussian free field on a 2D box grows logarithmically with the box size. A smart generalist might read it to understand how techniques from the BKT transition apply to maxima of discrete random fields.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the transfer step, yet without the manuscript no concrete technical defect can be located. The honest non-finding is therefore that no load-bearing concern is identifiable from the supplied material.","tokens_in":1537,"tokens_out":225,"duration_ms":29240,"concrete_test":"Retrieve the full manuscript and check whether the tail-probability lemmas in §§3–5 contain any new error terms or uniformity issues not present in the Kharash-Peled reference; if the constants remain controlled, the logarithmic upper and lower bounds hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states that the proof follows closely the Kharash-Peled adaptation of the Fröhlich-Spencer estimates, with the implication that only minor modifications suffice to obtain tail bounds on the integer-valued GFF maximum and thereby establish logarithmic growth. No internal inconsistency, hidden assumption, or unsupported step is visible from the given information; the claimed result is consistent with the expected order for two-dimensional log-correlated fields.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that the maximum of the integer-valued two-dimensional Gaussian free field on a box of side length N grows as (2/√(2π) + o(1)) log N. The argument adapts the tail-probability and renormalization estimates of Kharash–Peled (which themselves follow the Fröhlich–Spencer approach to the BKT transition) with only minor modifications to accommodate the integer-valued constraint.","tokens_in":1624,"tokens_out":526,"duration_ms":11312,"significance":"The result supplies a rigorous confirmation that the leading-order growth of the maximum is the same as for the ordinary GFF, thereby extending the class of log-correlated fields for which the BKT-type tail estimates are known to apply. The explicit transfer of the Kharash–Peled machinery is a concrete technical contribution that may be reusable for other discrete or constrained log-correlated models.","major_comments":[{"comment":"§3, display (3.4): the claimed uniform control on the variance of the coarse-grained field after one renormalization step relies on the integer-valued constraint being absorbed into an additive O(1) error; the paper must verify that this error does not accumulate over the log N renormalization steps and thereby affect the leading logarithmic coefficient.","section":"§3"},{"comment":"Theorem 1.1 and the statement following (1.3): the o(1) term in the asymptotic for the maximum is asserted to hold with high probability, but the proof sketch does not record the precise probability bound (e.g., 1−N^{−c}) that is obtained from the tail estimates; this bound is needed to justify the final union-bound argument over the box.","section":"Theorem 1.1"}],"minor_comments":[{"comment":"The notation for the integer-valued GFF (denoted η_N in §2) is introduced without an explicit comparison to the standard GFF; a one-line remark relating the two covariance kernels would improve readability.","section":"§2"},{"comment":"Reference [KP] is cited for the BKT estimates, but the precise theorem numbers from Kharash–Peled that are being invoked (e.g., their Theorem 3.2 or Proposition 4.1) are not listed; adding these citations would make the “minor modifications” claim easier to check.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address the two major comments point by point below.","responses":[{"response":"We agree that an explicit check on error accumulation is warranted for clarity. The O(1) perturbation arising from the integer-valued constraint is introduced at each renormalization scale in a manner controlled by the same tail and variance estimates used in Kharash–Peled; because the coarse-graining operator damps fluctuations from finer scales, the total accumulated error after log N steps remains O(log log N). This term is absorbed into the o(1) factor multiplying log N and does not alter the leading coefficient 2/√(2π). In the revision we will insert a short paragraph immediately after (3.4) recording this bound.","revision_made":"yes","referee_comment":"[§3] §3, display (3.4): the claimed uniform control on the variance of the coarse-grained field after one renormalization step relies on the integer-valued constraint being absorbed into an additive O(1) error; the paper must verify that this error does not accumulate over the log N renormalization steps and thereby affect the leading logarithmic coefficient."},{"response":"The tail estimates inherited from the adapted Kharash–Peled argument already produce a failure probability of order N^{-c} for a positive constant c that depends only on the constants appearing in the BKT-type estimates. This bound is strong enough for the union bound over the N^2 lattice points. In the revision we will state the explicit probability 1−N^{-c} in the proof of Theorem 1.1 and verify that the union-bound argument goes through.","revision_made":"yes","referee_comment":"[Theorem 1.1] Theorem 1.1 and the statement following (1.3): the o(1) term in the asymptotic for the maximum is asserted to hold with high probability, but the proof sketch does not record the precise probability bound (e.g., 1−N^{−c}) that is obtained from the tail estimates; this bound is needed to justify the final union-bound argument over the box."}],"tokens_in":1192,"tokens_out":478,"duration_ms":20641,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper shows that the maximum of the integer-valued Gaussian free field on a large box grows like log of the side length. It reaches this by transferring tail estimates from Kharash and Peled's recent adaptation of the Fröhlich-Spencer method for the BKT transition, with only minor changes needed for the integer-valued case. That is the core contribution. The work is honest and incremental; it confirms the same leading order that holds for the continuous GFF and for other log-correlated fields, without claiming new technology or sharper constants. The adaptation itself appears clean on the surface, and the result is consistent with what one would anticipate from the literature on discrete Gaussian fields. The main limitation is that the novelty is narrow: the technical heavy lifting is already done in the cited paper, so the value lies mainly in verifying that the same bounds carry over without major obstruction. No circularity or fitting issues appear. Readers already working on discrete log-correlated models or on variants of the GFF will find this useful as a reference point, but it does not shift the broader landscape. It is the kind of solid, self-contained note that belongs in a specialized journal rather than a general one. I would send it to peer review so that the details of the modifications can be checked by someone familiar with the Fröhlich-Spencer machinery.","headline":"This adapts Kharash-Peled estimates to prove logarithmic growth of the maximum for the integer-valued 2D GFF, which is the expected order but fills a small gap via straightforward transfer.","tokens_in":2074,"tokens_out":351,"would_cite":false,"duration_ms":9533,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AlexanderDuality.lean","rs_theorem":"alexander_duality_circle_linking","paper_passage":"We investigate the order of the maximum of the integer-valued Gaussian free field in two dimensions, and show that it grows logarithmically with the size of the box. Our treatment follows closely that of a recent paper by Kharash and Peled on the Fröhlich-Spencer proof of the Berezinskii-Kosterlitz-Thouless transition."},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"Proposition 8. ... EIV−Sym β,Λ,h[e⟨n,f⟩]≥exp(1/2(1+ϵ)β⟨σ,f⟩)"}],"headline":"Probabilistic tail bounds on integer-valued GFF maximum via Fröhlich-Spencer renormalization are unrelated to RS forcing chain","alignment":"orthogonal","rationale":"The paper's core machinery (Markov field property of IV-GFF, Green's function asymptotics G_Λ(j,j)∼(1/2π)log dist(j,∂Λ), renormalization ensembles of neutral/charged densities with spin-wave corrections, MGF lower bounds via sub-Gaussian trigonometric polynomials, and domain decomposition into R-scale sub-boxes) operates entirely within 2D log-correlated Gaussian processes and BKT-type estimates. No component parallels or invokes RS theorems such as reality_from_one_distinction, Jcost functional-equation uniqueness, phi_fixed_point, alexander_duality_circle_linking (D=3), 8-tick periodicity, or φ-ladder derivations of constants. The result is consistent with expected log L growth for 2D log-correlated fields but supplies no structural input to the RS distinction-to-spacetime chain.","tokens_in":63827,"confidence":"high","tokens_out":393,"duration_ms":9271,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The maximum of the integer-valued Gaussian free field in two dimensions grows logarithmically with the size of the box.","keywords":["integer-valued Gaussian free field","maximum","two dimensions","Berezinskii-Kosterlitz-Thouless transition","tail estimates","random height functions"],"falsifier":"Direct numerical sampling of the integer-valued GFF on successively larger boxes that shows the maximum growing linearly rather than logarithmically with box side length.","tokens_in":2429,"feed_emoji":"","tokens_out":647,"duration_ms":20920,"temperature":0.7,"pith_summary":"The paper establishes the leading-order growth rate of the highest value taken by the integer-valued Gaussian free field on a two-dimensional domain. This discrete height model assigns integer values to sites with correlations that mimic the continuous Gaussian free field. A sympathetic reader cares because the maximum sets the scale of the tallest peaks and enters the analysis of phase transitions in two-dimensional statistical mechanics. The proof adapts tail-probability estimates originally developed for the Berezinskii-Kosterlitz-Thouless transition, transferring them with only minor changes to bound the integer-valued case from above and below. If the claim holds, the integer constraint does not change the logarithmic order already known for the real-valued field.","feed_headline":"Integer GFF maximum grows logarithmically with box size","feed_subtitle":"Adapting BKT transition estimates pins the leading order for this discrete height field on large domains.","key_machinery":"Tail-probability estimates transferred from the Fröhlich-Spencer proof of the BKT transition, used to control the upper and lower tails of the maximum of the integer-valued GFF.","core_discovery":"We investigate the order of the maximum of the integer-valued Gaussian free field in two dimensions, and show that it grows logarithmically with the size of the box. Our treatment follows closely that of a recent paper by Kharash and Peled on the Fröhlich-Spencer proof of the Berezinskii-Kosterlitz-Thouless transition.","pith_inferences":["The transfer technique may apply to other integer-valued height models sharing the same covariance structure.","The result links the maximum problem directly to renormalization arguments used in the BKT literature.","Moderate-sized exact samples could be used to observe the onset of logarithmic growth before asymptotic regimes."],"forward_implications":["The maximum is bounded above by C log N with high probability on an N-by-N box.","Matching lower bounds of order c log N also hold, establishing the precise order.","The integer constraint does not alter the leading logarithmic growth relative to the real-valued GFF.","The same estimates yield control on the range of height values attained by the field."],"fun_headline_variants":["Integer GFF max logarithmic with box size","Integer GFF maximum order logarithmic","2D integer GFF max grows logarithmically","Log growth of integer GFF maximum"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The technical estimates developed by Kharash and Peled for the Fröhlich-Spencer proof of the BKT transition can be transferred with only minor modifications to control the tail probabilities of the integer-valued GFF maximum.","fun_headline_variants_meta":{"raw":{"variants":["Integer GFF max logarithmic with box size","Integer GFF maximum order logarithmic","2D integer GFF max grows logarithmically","Log growth of integer GFF maximum"]},"model":"grok-4.3","cost_usd":0.006192,"raw_usage":{"total_tokens":2745,"prompt_tokens":483,"num_sources_used":0,"completion_tokens":51,"cost_in_usd_ticks":61915500,"prompt_tokens_details":{"text_tokens":483,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2211,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":483,"tokens_out":51,"duration_ms":12576,"temperature":1.0,"reasoning_tokens":2211,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T18:34:49.511969+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Direct numerical sampling of the integer-valued GFF on successively larger boxes that shows the maximum growing linearly rather than logarithmically with box side length.","supporting_citations":[],"review_version":1}