Maximum of the integer-valued Gaussian free field

Mateo Wirth

arxiv: 1907.08868 · v1 · pith:VBDSYNH4new · submitted 2019-07-20 · 🧮 math.PR

Maximum of the integer-valued Gaussian free field

Mateo Wirth This is my paper

Pith reviewed 2026-05-24 18:34 UTC · model grok-4.3

classification 🧮 math.PR

keywords integer-valued Gaussian free fieldmaximumtwo dimensionsBerezinskii-Kosterlitz-Thouless transitiontail estimatesrandom height functions

0 comments

The pith

The maximum of the integer-valued Gaussian free field in two dimensions grows logarithmically with the size of the box.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

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

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.

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)

[§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.
[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)

[§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.
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

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
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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 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.

axioms (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.

pith-pipeline@v0.9.0 · 5563 in / 1281 out tokens · 15578 ms · 2026-05-24T18:34:49.511969+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 8. ... EIV−Sym β,Λ,h[e⟨n,f⟩]≥exp(1/2(1+ϵ)β⟨σ,f⟩)

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 1 Pith paper

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

The impact of disorder and non-convex interactions on delocalisation of height functions
math.PR 2026-04 unverdicted novelty 7.0

Phase transitions in XY/Villain models and dual height functions persist under quenched disorder, and rough phases exist for annealed non-convex potentials like |∇h|^p with p≤2.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper

[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]

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]

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]

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]

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]

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]

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]

Katznelson

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

work page 2004
[9]

Kharash, R

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

work page arXiv
[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

work page 1972
[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

work page 1973
[12]

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

work page 2010
[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

work page 2015

[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

[9] [9]

Kharash, R

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

work page arXiv

[10] [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

work page 1972

[11] [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

work page 1973

[12] [12]

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

work page 2010

[13] [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

work page 2015