Pith. sign in

REVIEW 2 major objections 2 minor 2 cited by

Maximum of the integer-valued Gaussian free field

T0 review · 2 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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

Pith's one-line read The maximum of the integer-valued Gaussian free field in two dimensions grows logarithmically with the size of the box.

desk verdict 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. read the letter →

arxiv 1907.08868 v1 pith:VBDSYNH4 submitted 2019-07-20 math.PR

Mateo Wirth This is my paper · ORCID
classification math.PR
keywords integer-valuedGaussianfreefieldmaximumtwodimensionsBerezinskii-Kosterlitz-Thoulesstransitiontailestimatesrandomheightfunctions
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

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.

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 (2)
  1. [§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.
  2. [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.
minor comments (2)
  1. [§2] 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.
  2. 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.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: [§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.

    Authors: 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: yes

  2. Referee: [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.

    Authors: 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: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation relies on external adaptation

full rationale

The paper presents a proof that the maximum of the integer-valued GFF grows logarithmically by adapting estimates from the independent Kharash-Peled work on the Fröhlich-Spencer BKT transition. No self-citations, fitted parameters renamed as predictions, self-definitional equations, or ansatzes smuggled via prior author work appear in the abstract or described structure. The central claim is established by transferring technical tail bounds with minor modifications, which constitutes an external methodological transfer rather than an internal reduction to the paper's own inputs. This is the standard case of a self-contained proof adapting prior independent results.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper adapts an existing proof technique; the ledger therefore contains only standard background facts from probability on graphs and no new free parameters or invented entities.

assumptions (2)
  • domain assumption The integer-valued GFF is a well-defined random field on the integer lattice whose covariance is given by the discrete Green's function.
    Invoked implicitly when the maximum is discussed; standard in the literature on discrete GFF.
  • ad hoc to paper The tail estimates and renormalization arguments of Kharash-Peled transfer to the integer-valued GFF with only minor changes.
    This is the load-bearing transfer assumption stated in the abstract.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Maximum of the integer-valued Gaussian free field." pith.science (2026). https://pith.science/paper/VBDSYNH4

@misc{pith2026190708868,
  author       = {Pith},
  title        = {Pith review of: Maximum of the integer-valued Gaussian free field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBDSYNH4}},
  note         = {Machine review of arXiv:1907.08868}
}
read the original abstract

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 Fr\"{o}hlich-Spencer proof of the Berezinskii-Kosterlitz-Thouless transition.

Discussion (0). Continue with ORCID to comment.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantitative delocalization for solid-on-solid models at high temperature and arbitrary tilt

    math.PR 2025-05 accept novelty 7.0 of 10

    For every p-SOS model with 0<p≤2 in two dimensions and small beta, the interface variance grows at least logarithmically with system size, for any boundary data.

  2. Quantitative delocalisation for the Gaussian and $q$-SOS long-range chains

    math.PR 2024-12 conditional novelty 7.0 of 10

    For the discrete Gaussian chain, the height variance grows like N^(α-2) for range exponent α in (2,3), like N/log N at α=3, and the same graph-surgery method gives new bounds for the q-SOS chain.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 2 Pith papers

  1. [1]

    Berezinskii

    V.L. Berezinskii. Destruction of long-range order in one-dimensio nal and two-dimensional systems having a continuous symmetry group I. classical systems . Sov. Phys. JETP , 32(3): 493-500, 1971

    work page 1971

  2. [2]

    Berezinskii

    V.L. Berezinskii. Destruction of long-range order in one-dimensio nal and two-dimensional systems possessing a continuous symmetry group. II. quantum s ystems. Sov. J. of Exp. Theoret. Phys. 34: 610, 1972

    work page 1972

  3. [3]

    Biskup and O

    M. Biskup and O. Louidor. Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian free field Adv. Math. 330: 598-687, 2016. 34

    work page 2016

  4. [4]

    Bolthausen, J.D

    E. Bolthausen, J.D. Deuschel, and G. Giacomin. Entropic repulsion and the maximum of the two-dimensional harmonic crystal Ann. Probab. 29(4): 1670-1692, 2001

    work page 2001

  5. [5]

    Bramson, J

    M. Bramson, J. Ding, and O. Zeitouni Convergence in law of the ma ximum of the two- dimensional discrete Gaussian free field. Comm. Pure Appl. Math. , 69(1):62-123, 2016

    work page 2016

  6. [6]

    Fr¨ ohlich, and T

    J. Fr¨ ohlich, and T. Spencer. The Kosterlitz-Thouless transitio n in two-dimensional abelian spin systems and the Coulomb gas. Comm. Math. Phys. , 81(4):527-602, 1981

    work page 1981

  7. [7]

    Georgii, O

    H-O. Georgii, O. H¨ aggstr¨ om, C. Maes. The random geometry o f equilibrium phases. Phase transitions and critical phenomena , 18:1-142. Academic Press, 2001

    work page 2001

  8. [8]

    Katznelson

    Y. Katznelson. An introduction to harmonic analysis . Cambridge University Press, 2004

    work page 2004

Show all 13 references
  1. [9]

    Kharash, R

    V. Kharash, R. Peled. The Frhlich-Spencer Proof of the Berezin skii-Kosterlitz-Thouless Tran- sition. Preprint, arXiv:1706.07737

  2. [10]

    Kosterlitz, and D.J

    J.M. Kosterlitz, and D.J. Thouless. Long range order and metas tability in two dimensional solids and superfluids. (Application of dislocation theory). Journal of Physics C: Solid State Physics, 5(11): L124, 1972

    1972

  3. [11]

    Kosterlitz, and D.J

    J.M. Kosterlitz, and D.J. Thouless. Ordering, metastability and p hase transitions in two- dimensional systems. Journal of Physics C: Solid State Physics , 6(7): 1181, 1973

    1973

  4. [12]

    G. F. Lawler and V. Limic. Random Walk: A Modern Introduction . Cambridge University Press, 2010

    2010

  5. [13]

    Milos, R

    P. Milos, R. Peled. Delocalization of two-dimensional random surf aces with hard-core con- straints. Commun. Math. Phys. 340: 146, 2015. 35

    2015

Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.