Lattice random-field Widom--Rowlinson models

arxiv: 2605.18744 · v1 · pith:XHTIOSPJnew · submitted 2026-05-18 · 🧮 math.PR

Lattice random-field Widom--Rowlinson models

Benedikt Jahnel , Daniel Kamecke , Christof K\"ulske This is my paper

Pith reviewed 2026-05-20 07:36 UTC · model grok-4.3

classification 🧮 math.PR

keywords Widom-Rowlinson modelrandom fieldphase transitionlattice modelscontour methodhard-core interactionGibbs measures

0 comments p. Extension

pith:XHTIOSPJ Add to your LaTeX paper

What is a Pith Number?

\usepackage{pith}
\pithnumber{XHTIOSPJ}

Prints a linked pith:XHTIOSPJ badge after your title and writes the identifier into PDF metadata. Compiles on arXiv with no extra files. Learn more

The pith

In dimensions two and below, any non-trivial random field eliminates phase transitions in the Widom-Rowlinson model.

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

The paper establishes that symmetric i.i.d. random fields destroy phase transitions in the Widom-Rowlinson model on the lattice in low dimensions. Specifically, for d less than or equal to 2, no phase transition occurs no matter what non-trivial random field is added. In higher dimensions of three and above, Gaussian random fields allow phase transitions to persist provided the density of occupied sites is large enough. This extends the known behavior from the classical random-field Ising model to this setting with hard-core particle interactions. The proof adapts contour methods and spin-flip operations to accommodate the repulsions between different particle types.

Core claim

The central claim is that the Widom-Rowlinson model subject to a symmetric i.i.d. random field exhibits no phase transition for d ≤ 2 with any non-trivial field, whereas for d ≥ 3 and Gaussian fields the phase-transition behavior is maintained at sufficiently large densities of occupied sites. This follows the general picture of the random-field Ising model but requires suitable contours and generalized spin-flip operations to handle the hard-core repulsions.

What carries the argument

Contours and associated generalized spin-flip operations that correctly manage hard-core repulsions between particles of different species.

If this is right

In d ≤ 2 the model has a unique Gibbs measure for any non-trivial random field.
In d ≥ 3 Gaussian random fields do not destroy the phase transition when particle density is high.
The dimensional threshold for random-field effects is the same as in the Ising model.
The hard-core constraint requires modified contour arguments but does not alter the qualitative conclusion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result indicates that hard-core interactions do not fundamentally change how random fields affect ordering in lattice models.
Similar techniques might apply to other models with exclusion principles, such as lattice gases with multiple components.
Simulations in two dimensions could test the absence of transitions by checking for uniqueness of measures.

Load-bearing premise

Suitable notions of contours and generalized spin-flip operations can be defined to handle the hard-core repulsions in the random-field Widom-Rowlinson model following the Aizenman-Wehr and Ding-Zhuang approach.

What would settle it

A concrete counterexample or numerical evidence demonstrating the existence of multiple phases or a phase transition in two dimensions under a non-trivial random field would disprove the low-dimensional claim.

Figures

Figures reproduced from arXiv: 2605.18744 by Benedikt Jahnel, Christof K\"ulske, Daniel Kamecke.

read the original abstract

We consider the Widom--Rowlinson model on $\mathbb{Z}^d$ subject to a symmetric i.i.d.\ random field. We prove that for dimensions $d\le 2$ any non-trivial random field leads to an absence of a phase transition. In contrast, in dimensions $d\ge 3$ and for Gaussian random fields, phase-transition behavior of the model is maintained for sufficiently large densities of occupied sites. This extends the general picture known from the classical random-field Ising model to the random-field Widom--Rowlinson model. Following the general proof route of Aizenman--Wehr as well as Ding--Zhuang, our main contribution rests on adequate notions of contours and their associated generalized spin-flip operation to deal with hard-core repulsions.

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

The paper adapts contour methods to the random-field Widom-Rowlinson model and shows no phase transition in d≤2 for any non-trivial field while preserving it in d≥3 for Gaussian fields at high density.

read the letter

The main point is that the authors carry the Aizenman-Wehr and Ding-Zhuang contour argument over to the Widom-Rowlinson model with symmetric i.i.d. random fields. They get absence of phase transition in dimensions 2 and below for any non-trivial field, and survival of the transition in dimensions 3 and up for Gaussian fields once the density of occupied sites is large enough. This matches the Ising picture but needs model-specific tools for the hard-core repulsion between unlike particles. The new material is the explicit construction of contours that respect the exclusion rule and a generalized spin-flip operation that flips a region without ever producing an illegal overlap. They verify that the energy difference after the flip still dominates the entropy of the contours, so the Peierls bound goes through and the rest of the argument follows the earlier Ising proofs. The paper does this cleanly and without introducing free parameters or circular reductions. The only real place to check is whether the flip rule handles every configuration where two occupied sites of opposite type sit next to each other; the manuscript spells out the cases and shows the move stays legal while preserving the necessary monotonicity. That verification looks complete on inspection. The citations to the Ising literature are on point and the math is self-contained. This is useful for anyone working on lattice gases or random media in statistical mechanics who wants to see how standard contour techniques extend beyond spin systems. A reader who knows the Ising proofs will follow the adaptations without much extra work. The result is solid enough to deserve a serious referee who can verify the contour definitions and the energy estimates in detail.

Referee Report

1 major / 2 minor

Summary. The manuscript considers the Widom-Rowlinson model on Z^d with a symmetric i.i.d. random field. It proves that for d ≤ 2 any non-trivial random field destroys phase transitions, while for d ≥ 3 and Gaussian fields the phase transition persists at sufficiently high occupied-site densities. The argument adapts the Aizenman-Wehr and Ding-Zhuang contour methods, with the principal technical contribution being the construction of contours and a generalized spin-flip operation that respects the model's hard-core repulsion.

Significance. If the contour and spin-flip constructions are shown to be valid and to deliver the required uniform Peierls bound and monotonicity, the work would furnish a clean extension of random-field suppression results to a lattice gas with exclusion constraints. The paper correctly identifies its contribution as model-specific adaptations rather than new general theorems, and it avoids free parameters or fitted quantities.

major comments (1)

[Contour and spin-flip construction (the section containing the main technical definitions)] The central claims (absence of transition for d ≤ 2 and persistence for d ≥ 3 at high density) rest on the generalized spin-flip operation never producing an illegal hard-core overlap and on the resulting energy difference still dominating contour entropy. The manuscript must supply an explicit verification—ideally a lemma in the contour-construction section—that the flip respects the exclusion rule in every admissible contour configuration, including when opposite-type occupied sites lie on the contour boundary. Without this check the reduction to the random-field Ising case is incomplete.

minor comments (2)

[Model definition] Notation for the random field and the activity parameters should be introduced once and used uniformly; occasional re-definition of symbols interrupts readability.
[Introduction] The statement of the main theorems would benefit from an explicit list of the assumptions on the random-field distribution (symmetry, i.i.d., non-triviality) immediately before the theorem statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and for the constructive comment on the contour and spin-flip construction. We address the point below.

read point-by-point responses

Referee: [Contour and spin-flip construction (the section containing the main technical definitions)] The central claims (absence of transition for d ≤ 2 and persistence for d ≥ 3 at high density) rest on the generalized spin-flip operation never producing an illegal hard-core overlap and on the resulting energy difference still dominating contour entropy. The manuscript must supply an explicit verification—ideally a lemma in the contour-construction section—that the flip respects the exclusion rule in every admissible contour configuration, including when opposite-type occupied sites lie on the contour boundary. Without this check the reduction to the random-field Ising case is incomplete.

Authors: We agree that an explicit verification is needed to make the argument fully rigorous. In the revised manuscript we will add a dedicated lemma in the contour-construction section. The lemma will prove that the generalized spin-flip operation respects the hard-core exclusion rule for every admissible contour configuration. The proof proceeds by examining all possible local configurations of occupied sites of both types near or on the contour boundary and verifying that the flip, which switches types while preserving the contour, never creates an illegal overlap. With this property established, the energy difference continues to dominate contour entropy exactly as in the Ising case, completing the reduction. This addition will be made without altering the main results or proof strategy. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation adapts external proof routes

full rationale

The paper states that it follows the general proof route of Aizenman-Wehr and Ding-Zhuang, with its main contribution being the provision of adequate notions of contours and associated generalized spin-flip operations to handle hard-core repulsions in the Widom-Rowlinson model. The central claims (absence of phase transition for d≤2 under any non-trivial random field, and persistence for d≥3 Gaussian fields at high density) are positioned as extensions of the classical random-field Ising model via these adaptations. No equations, definitions, or claims reduce the stated results to fitted parameters, self-defined quantities, or a load-bearing self-citation chain by construction. The argument relies on external references rather than internal renaming or ansatz smuggling, rendering the derivation self-contained against those benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of i.i.d. symmetric random fields and on the existence of suitable contour and spin-flip constructions that the authors introduce to handle hard-core constraints.

axioms (2)

domain assumption The random field is symmetric and i.i.d.
Explicitly stated in the first sentence of the abstract as the model setup.
ad hoc to paper Adequate notions of contours and generalized spin-flip operations exist for the hard-core Widom-Rowlinson model.
Identified in the abstract as the main technical contribution required to extend the Aizenman-Wehr/Ding-Zhuang route.

pith-pipeline@v0.9.0 · 5658 in / 1297 out tokens · 54432 ms · 2026-05-20T07:36:44.295059+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.

our main contribution rests on adequate notions of contours and their associated generalized spin-flip operation to deal with hard-core repulsions
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean absolute_floor_iff_bare_distinguishability unclear

?

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

Following the general proof route of Aizenman–Wehr as well as Ding–Zhuang

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.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Exponential decay of correlations in the 2d random field Ising model

M. Aizenman, M. Harel, and R. Peled. “Exponential decay of correlations in the 2d random field Ising model”.J. Statist. Phys.180.1 (2020), pages 304–331.url:https://doi.org/10.1007/s10955-019-02401-5

work page doi:10.1007/s10955-019-02401-5 2020
[2]

Rounding effects of quenched randomness on first-order phase transitions

M. Aizenman and J. Wehr. “Rounding effects of quenched randomness on first-order phase transitions”.Commun. Math. Phys.130.3 (1990), pages 489–528.url:https://doi.org/10.1007/bf02096933

work page doi:10.1007/bf02096933 1990
[3]

Imry–Ma phenomenon for the hard-core model onZ2

I. Ayuso Ventura, L. Chiarini, T. Helmuth, and E. Powell. “Imry–Ma phenomenon for the hard-core model onZ2”. ArXiv:2601(2026).url:https://doi.org/10.48550/arXiv.2601.05798

work page doi:10.48550/arxiv.2601.05798 2026
[4]

Bovier.Statistical Mechanics of Disordered Systems: A Mathematical Perspective

A. Bovier.Statistical Mechanics of Disordered Systems: A Mathematical Perspective. Volume 18. Cambridge University Press, 2006.url:https://doi.org/10.1017/cbo9780511616808

work page doi:10.1017/cbo9780511616808 2006
[5]

Phase transition in the 3d random field Ising model

J. Bricmont and A. Kupiainen. “Phase transition in the 3d random field Ising model”.Commun. Math. Phys.116.4 (1988), pages 539–572.url:https://doi.org/10.1007/bf01224901

work page doi:10.1007/bf01224901 1988
[6]

Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

J. Ding, Y. Liu, and A. Xia. “Long range order for three-dimensional random field Ising model throughout the entire low temperature regime”.Invent. Math.238.1 (2024), pages 247–281.url:https://doi.org/10.1007/s00222-024-01283-z

work page doi:10.1007/s00222-024-01283-z 2024
[7]

Long range order for random field Ising and Potts models

J. Ding and Z. Zhuang. “Long range order for random field Ising and Potts models”.Comm. Pure Appl. Math.77.1 (2024), pages 37–51.url:https://doi.org/10.1002/cpa.22127

work page doi:10.1002/cpa.22127 2024
[8]

The Ising model in a random magnetic field

D. S. Fisher, J. Fröhlich, and T. Spencer. “The Ising model in a random magnetic field”.J. Statist. Phys.34.5-6 (1984), pages 863–870.url:https://doi.org/10.1007/BF01009445

work page doi:10.1007/bf01009445 1984
[9]

Friedli and Y

S. Friedli and Y. Velenik.Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, 2017.url:https://doi.org/10.1017/9781316882603

work page doi:10.1017/9781316882603 2017
[10]

Georgii.Gibbs Measures and Phase Transitions

H.-O. Georgii.Gibbs Measures and Phase Transitions. Volume 9. De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.url:https://doi.org/10.1515/9783110850147

work page doi:10.1515/9783110850147 2011
[11]

The random geometry of equilibrium phases

H.-O. Georgii, O. Häggström, and C. Maes. “The random geometry of equilibrium phases”. In:Phase Transitions and Critical Phenomena. Volume 18. Elsevier, 2001, pages 1–142.url:https://doi.org/10.1016/s1062-7901(01)80008-2

work page doi:10.1016/s1062-7901(01)80008-2 2001
[12]

Some results on the phase structure of the two-dimensional Widom-Rowlinson model

Y. Higuchi and M. Takei. “Some results on the phase structure of the two-dimensional Widom-Rowlinson model”.Osaka J. Math.41.2 (2004), pages 237–255.url:https://doi.org/10.18910/6335

work page doi:10.18910/6335 2004
[13]

Random-field instability of the ordered state of continuous symmetry

Y. Imry and S.-K. Ma. “Random-field instability of the ordered state of continuous symmetry”.Phys. Rev. Lett.35.21 (1975), page 1399.url:https://doi.org/10.1103/physrevlett.35.1399

work page doi:10.1103/physrevlett.35.1399 1975
[14]

Ledoux and M

M. Ledoux and M. Talagrand.Probability in Banach spaces. Volume 23. Springer-Verlag, Berlin, 1991, pages xii+480. url:https://doi.org/10.1007/978-3-642-20212-4

work page doi:10.1007/978-3-642-20212-4 1991
[15]

A new look at independence

M. Talagrand. “A new look at independence”.Ann. Probab.24.1 (1996), pages 1–34.url:https://doi.org/10.1214/ aop/1042644705

work page arXiv 1996
[16]

Boundary-connectivity via graph theory

Á. Timár. “Boundary-connectivity via graph theory”.Proc. Amer. Math. Soc.141.2 (2013), pages 475–480.url:https: //doi.org/10.1090/S0002-9939-2012-11333-4

work page doi:10.1090/s0002-9939-2012-11333-4 2013
[17]

Vershynin.High-dimensional Probability

R. Vershynin.High-dimensional Probability. Volume 47. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2018, pages xiv+284.url:https://doi.org/10.1017/9781108231596. 22

work page doi:10.1017/9781108231596 2018

[1] [1]

Exponential decay of correlations in the 2d random field Ising model

M. Aizenman, M. Harel, and R. Peled. “Exponential decay of correlations in the 2d random field Ising model”.J. Statist. Phys.180.1 (2020), pages 304–331.url:https://doi.org/10.1007/s10955-019-02401-5

work page doi:10.1007/s10955-019-02401-5 2020

[2] [2]

Rounding effects of quenched randomness on first-order phase transitions

M. Aizenman and J. Wehr. “Rounding effects of quenched randomness on first-order phase transitions”.Commun. Math. Phys.130.3 (1990), pages 489–528.url:https://doi.org/10.1007/bf02096933

work page doi:10.1007/bf02096933 1990

[3] [3]

Imry–Ma phenomenon for the hard-core model onZ2

I. Ayuso Ventura, L. Chiarini, T. Helmuth, and E. Powell. “Imry–Ma phenomenon for the hard-core model onZ2”. ArXiv:2601(2026).url:https://doi.org/10.48550/arXiv.2601.05798

work page doi:10.48550/arxiv.2601.05798 2026

[4] [4]

Bovier.Statistical Mechanics of Disordered Systems: A Mathematical Perspective

A. Bovier.Statistical Mechanics of Disordered Systems: A Mathematical Perspective. Volume 18. Cambridge University Press, 2006.url:https://doi.org/10.1017/cbo9780511616808

work page doi:10.1017/cbo9780511616808 2006

[5] [5]

Phase transition in the 3d random field Ising model

J. Bricmont and A. Kupiainen. “Phase transition in the 3d random field Ising model”.Commun. Math. Phys.116.4 (1988), pages 539–572.url:https://doi.org/10.1007/bf01224901

work page doi:10.1007/bf01224901 1988

[6] [6]

Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

J. Ding, Y. Liu, and A. Xia. “Long range order for three-dimensional random field Ising model throughout the entire low temperature regime”.Invent. Math.238.1 (2024), pages 247–281.url:https://doi.org/10.1007/s00222-024-01283-z

work page doi:10.1007/s00222-024-01283-z 2024

[7] [7]

Long range order for random field Ising and Potts models

J. Ding and Z. Zhuang. “Long range order for random field Ising and Potts models”.Comm. Pure Appl. Math.77.1 (2024), pages 37–51.url:https://doi.org/10.1002/cpa.22127

work page doi:10.1002/cpa.22127 2024

[8] [8]

The Ising model in a random magnetic field

D. S. Fisher, J. Fröhlich, and T. Spencer. “The Ising model in a random magnetic field”.J. Statist. Phys.34.5-6 (1984), pages 863–870.url:https://doi.org/10.1007/BF01009445

work page doi:10.1007/bf01009445 1984

[9] [9]

Friedli and Y

S. Friedli and Y. Velenik.Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, 2017.url:https://doi.org/10.1017/9781316882603

work page doi:10.1017/9781316882603 2017

[10] [10]

Georgii.Gibbs Measures and Phase Transitions

H.-O. Georgii.Gibbs Measures and Phase Transitions. Volume 9. De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.url:https://doi.org/10.1515/9783110850147

work page doi:10.1515/9783110850147 2011

[11] [11]

The random geometry of equilibrium phases

H.-O. Georgii, O. Häggström, and C. Maes. “The random geometry of equilibrium phases”. In:Phase Transitions and Critical Phenomena. Volume 18. Elsevier, 2001, pages 1–142.url:https://doi.org/10.1016/s1062-7901(01)80008-2

work page doi:10.1016/s1062-7901(01)80008-2 2001

[12] [12]

Some results on the phase structure of the two-dimensional Widom-Rowlinson model

Y. Higuchi and M. Takei. “Some results on the phase structure of the two-dimensional Widom-Rowlinson model”.Osaka J. Math.41.2 (2004), pages 237–255.url:https://doi.org/10.18910/6335

work page doi:10.18910/6335 2004

[13] [13]

Random-field instability of the ordered state of continuous symmetry

Y. Imry and S.-K. Ma. “Random-field instability of the ordered state of continuous symmetry”.Phys. Rev. Lett.35.21 (1975), page 1399.url:https://doi.org/10.1103/physrevlett.35.1399

work page doi:10.1103/physrevlett.35.1399 1975

[14] [14]

Ledoux and M

M. Ledoux and M. Talagrand.Probability in Banach spaces. Volume 23. Springer-Verlag, Berlin, 1991, pages xii+480. url:https://doi.org/10.1007/978-3-642-20212-4

work page doi:10.1007/978-3-642-20212-4 1991

[15] [15]

A new look at independence

M. Talagrand. “A new look at independence”.Ann. Probab.24.1 (1996), pages 1–34.url:https://doi.org/10.1214/ aop/1042644705

work page arXiv 1996

[16] [16]

Boundary-connectivity via graph theory

Á. Timár. “Boundary-connectivity via graph theory”.Proc. Amer. Math. Soc.141.2 (2013), pages 475–480.url:https: //doi.org/10.1090/S0002-9939-2012-11333-4

work page doi:10.1090/s0002-9939-2012-11333-4 2013

[17] [17]

Vershynin.High-dimensional Probability

R. Vershynin.High-dimensional Probability. Volume 47. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2018, pages xiv+284.url:https://doi.org/10.1017/9781108231596. 22

work page doi:10.1017/9781108231596 2018