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The near-critical random bond FK-percolation model

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Pith's one-line read Random centred bond disorder up to the cube root of the deterministic critical window still leaves two-dimensional FK-percolation critical at every scale; Bernoulli percolation tolerates logarithmic disorder at every scale and macroscopic c

desk verdict Genuinely new results on quenched disorder in FK-percolation, with a clean deformation framework, but the flagship cube-root window theorem rests on an unproved generalization of the scaling-relation toolbox; the percolation results, including quenched Cardy, look right. read the letter →

arxiv 2509.08938 v1 pith:LMA6GXWP submitted 2025-09-10 math.PR math-phmath.MP

Emile Avérous , Rémy Mahfouf This is my paper
classification math.PRmath-phmath.MP MSC 60K3582B43
keywords FK-percolationrandombondenvironmentquenchedcriticalityRSWboxcrossingcriticalwindowscalingrelationsmixingratesnoisesensitivity
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 asks whether quenched disorder in the bond probabilities destroys criticality in two-dimensional FK-percolation, or averages out. It argues that if independent random edge parameters are centred (in a precise second-order sense) at the self-dual critical point, the quenched model typically keeps the strong Russo–Seymour–Welsh box-crossing property at every scale, even when the standard deviation of the disorder is as large as the cube root of the deterministic critical window, up to logarithmic corrections. That is a regime where a deterministic offset of the same size would have pushed the model far off criticality. The mechanism is a continuous deformation of the environment from the homogeneous critical point to the random one, controlled by Itô calculus and stability of mixing rates; the special case q=1 is pushed further, to logarithmic disorder at every scale and, via noise sensitivity, to macroscopic centered disorder for large-scale crossings and Cardy's formula.

What carries the argument

The central objects are the quenched FK measure on the torus with per-edge parameters p_e and its local mixing rates Δ_p^{(e)}(R), defined as the difference between wired and free connection probabilities for an edge inside a box of radius R. The argument is carried by a continuous deformation p(t) from the homogeneous critical environment to the random target, built from i.i.d. Brownian motions (or Skorokhod-stopped Brownian motions with drift), together with a breaking time T_b defined as the first time the deformed measure leaves RSW(δ,N), the stability classes for arm events, or the stability class for mixing rates. Itô's formula splits the evolution of crossing probabilities, arm probab

What would settle it

For q=2, compute the deterministic critical window W(N) from the critical mixing-rate relation (or from known FK-Ising exponents), then simulate quenched FK-Ising on the torus with i.i.d. Gaussian edge parameters centred at p_c with standard deviation σ_N = c W(N)^{1/3} log(N)^{-2}. Estimate the probability over the environment that some 2-by-1 rectangle has primal or dual crossing probability outside a fixed window [δ,1−δ]. Theorem 1.6 predicts this probability decays like exp(−c (W(N)^{1/3}/σ_N)^{1/2}); if it does not decay to zero, or decays at a qualitatively slower rate, the central quenc

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Extended reading notes

Core claim

For 1<q≤4, Theorem 1.6 states that if independent random edge parameters satisfy the natural centering condition (⋆)_N^B — E[p_e]=p_c − μ(q) σ_e^2 + O(σ_e^3) with exponential tails — and Σ_p ≤ c fW(N)^{1/3} log(N)^{-2}, then with probability at least 1−exp(−c(fW(N)^{1/3}/Σ_p)^{1/2}) the quenched FK measure on the torus T_N sits in the strong RSW class RSW(δ,N), meaning every 2-by-1 rectangle has primal and dual crossing probabilities bounded uniformly away from 0 and 1. Interpreting the conjectured comparison fW(N) ≍ W(N), this permits random standard deviations of order W(N)^{1/3}, whereas a deterministic offset of that size would be thoroughly off-critical. With only first-moment centering

Load-bearing premise

The argument hinges on Proposition 2.4 — that a stronger RSW box-crossing bound alone forces covariances of edges to factor as products of local mixing rates, forces quasimultiplicativity of those rates, and forces comparability of nearby mixing rates even without translation invariance — which the paper states with a proof sketch rather than a complete proof; the advertised W^{1/3} scaling additionally assumes the conjectured mixing-rate exponent bound ι(q) ≥ 1/2 so that fW(

Editorial extensions

If this is right

  • If Theorem 1.6 is right, random bond disorder does not round off the critical phase: the quenched model inherits critical box-crossing behaviour on scales up to N even when individual bonds deviate from p_c by far more than the deterministic critical window.
  • For Bernoulli percolation, centered independent disorder of size log(N)^{-2} preserves criticality at every scale, and even macroscopic centered disorder preserves large-scale crossing probabilities and the Cardy formula.
  • The W^{1/3} window is presented as essentially optimal for independent bonds: any generic self-duality-compatible centering function cannot push the window beyond W^{1/3} (Section 4.3).
  • Allowing a small dependent correction to the environment, of order σ^3 Ξ(N)^{-1/2}, enlarges the window to Ξ(N)^{1/2}; for 1<q≤2 this is expected to be constant or logarithmic, giving a random-environment critical window of logarithmic order.
  • The natural centering condition (⋆)_N^B is the second-order self-duality condition, matching primal and dual first and second moments of the annealed environment; it is the correct centering to use in any future test or extension of these results.

Reading between the lines

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

  • A consequence the authors leave implicit: if the conjectured mixing-rate bound ι(q) ≥ 1/2 fails, the quantitative gain from W^{1/2} to W^{1/3} shrinks; the qualitative conclusion of quenched criticality would survive, but with a smaller disorder window.
  • The deformation-plus-breaking-time scheme is portable: the same Itô-calculus stability proof should apply to other scale-invariant lattice models whenever one has an analogue of Proposition 2.4, making the tool a general template for 'quenched disorder is irrelevant' statements.
  • For Bernoulli percolation, the noise-sensitivity argument suggests that the large-scale Cardy formula should hold for much wider classes of dependent centered environments; testing whether bounded dependent disorder also preserves the formula is a natural next step.
  • The W^{1/3} barrier is specific to independent bonds; the dependent-correction construction suggests that allowing weak long-range correlations among bond parameters could systematically enlarge the quenched critical window, pointing to a possible phase diagram in the strength-versus-correlation plane.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies quenched FK-percolation on the torus T_N, with independent random edge weights centered at or near the self-dual point p_c(q). Under a naive centering condition (⋆)_N^A it claims that if max_e σ_e ≤ c W(N)^{1/2}, then with high P_N-probability the quenched measure satisfies strong RSW box crossing on T_N (Theorem 1.5). Under a natural centering condition (⋆)_N^B it claims the same for max_e σ_e ≤ c fW(N)^{1/3} log^{-2} N, where fW is a modified critical window (Theorem 1.6). For q=1 it proves a logarithmic window and, via noise sensitivity, a Cardy-formula statement for environments with macroscopic fluctuations (Theorems 1.8 and 1.10). The method is a continuous deformation from the homogeneous critical environment to the random environment, implemented with stopped Brownian motions, Itô semimartingale decompositions, a stopping time T_b that controls RSW, arm-event stability and mixing-rate stability, and large-deviation estimates. The main quantitative bridge is Proposition 2.4, which is a non-translation-invariant version of the scaling-relations technology of [25]; the paper states it but gives only a sketch.

Significance. If the technical gaps are filled, this is a substantial contribution. It identifies a regime in which quenched disorder does not destroy criticality and in fact permits random fluctuations much larger than the deterministic near-critical window; the q=1 results via noise sensitivity are elegant and likely correct. The paper is transparent about its two main caveats: the unproved Proposition 2.4 and the conjectural identification fW ≍ W. The proof architecture is coherent, with careful Itô computations, detailed tail estimates, and a legitimate bootstrap stopping-time argument. The main obstruction to acceptance is the status of Proposition 2.4, on which all later stability lemmas depend.

major comments (2)
  1. [Section 2.2, Proposition 2.4] Proposition 2.4 is the only bridge from the homogeneous scaling-relations technology of [25] to non-translation-invariant random environments, and it is not proved. The text explicitly says a complete proof would require reproducing [25] almost verbatim; item 1 is skipped and items 2-3 are only sketched. Every later stability result (Lemmas 2.14, 3.3, 3.4, 4.3, 4.4) uses this proposition as a black box. In particular, Lemma 3.4 Step 4 reconstructs the local mixing rate from a ratio of three edge covariances; this requires item 1 with constants depending only on δ, not on e,f or on the environment. The sketch of item 3 contains an unjustified inference (from equal boundary conditions at one annulus to equality of configurations inside), and RSW(δ,N) alone does not obviously give the asserted uniform comparability of mixing rates at different centers. If item 1 or 3 fails, the propagation
  2. [Section 1.3, Theorem 1.6] The headline 'W^{1/3}' statement is conditional in a way that is easy to miss. Theorem 1.6 is stated for fW(N)^{1/3}, not W(N)^{1/3}; the identification fW ≍ W requires the conjectural bound Δ_{pc}(r) ≍ r^{-ι(q)} with ι(q) ≥ 1/2, while the only rigorous input cited is ι > 1/4 from [5]. Thus, as a theorem, the paper establishes RSW for fluctuations of order fW(N)^{1/3}, and the stronger claim 'standard deviations of order W(N)^{1/3}' is conditional on a conjecture. The abstract and introduction should be reworded so that the theorem, the conjecture, and the conditional corollary are clearly separated.
minor comments (5)
  1. [Definition 1.1 and 1.2] In Definition 1.1, '|ω| denotes the number of vertices in the configuration ω' should be 'number of open edges'; otherwise the weight p^{|ω|}(1-p)^{|E|-|ω|} does not make sense. Definition 1.2 inherits the notation.
  2. [Theorem 1.9 and Theorem 1.10] There is an x vs 1-x inconsistency. With x_N = ((1/2-x)N,0), the event [C_N A_N] connected to [x_N B_N] has limiting probability 1-x, not x; the proof of Theorem 1.10 uses 1-x while the statement and Theorem 1.9 use x. One of these must be changed.
  3. [Section 4.3] The optimality discussion is not a theorem. The construction of a 'generic' f_q and an environment-dependent tilde f_q is heuristic; in particular the bound on α is asserted without proof. I suggest labeling Section 4.3 explicitly as heuristic, or making the assumptions formal, since as written it sits between a proof and a remark.
  4. [Section 4.4 / Theorem 1.7] Proposition 4.6 and Theorem 1.7 are only sketched; for instance, existence and uniqueness of the SDE (4.5) before T_b is not discussed, and Remark 4.7 explicitly disclaims rigor. If these results are kept as theorems, the proof should be completed; otherwise they should be presented as conjectures or heuristics.
  5. [Appendix B.1 and assorted typos] The definition of θ(s) as argmin is later referred to as an argmax; the sign convention should be checked. There are also small typos: 'percolation percolation' near the end of Section 1.1, 'to of the four corners' in Lemma 2.7, 'sort for each vertex' in the Introduction.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the deformation/bootstrap proof is a legitimate stability argument; the main theorems are conditional on an external sketched generalization (Prop. 2.4), not reduced to their inputs.

full rationale

The paper's central theorems (1.5, 1.6, 1.8, 1.10) are derived by continuously deforming the homogeneous critical environment into the random one and controlling, via Itô's formula, the probabilities that the deformed measure leaves the RSW/stability classes. This is a bootstrap: the stopping time T_b is defined as the first exit from RSW(δ,N) ∩ Stab_#(δ,N) ∩ Stab_Δ(ρ,N), and Lemmas 3.3/3.4/4.3/4.4 show that before T_b the relevant quantities change by at most a controlled amount. The conclusion (T_b large) is not an input of the proof; it is obtained from estimates whose constants depend only on δ, ρ and on external critical-mixing-rate facts. No equation in the paper identifies a 'predicted' quantity with a fitted or assumed quantity by construction. The only genuinely load-bearing step that is not fully proved in this paper is Proposition 2.4, which generalizes [25]'s homogeneous mixing-rate/covariance estimates to arbitrary RSW environments. The paper states: 'Providing a complete proof of this proposition would require introducing a lot of additional side tools involved in [25], but ending up doing the same proof, almost verbatim.' This is an acknowledged sketch: item 1 is skipped and items 2–3 are only sketched, and all later stability lemmas use the proposition as a black box. This makes the main theorems conditional, but it is not circularity: [25] is an external result for the homogeneous model, and Proposition 2.4 is stated for every p satisfying RSW(δ,N), not for measures assumed to satisfy the conclusion of Theorem 1.5/1.6. If Proposition 2.4 were false, the proof would fail, but failure of a lemma is not reduction of the conclusion to the hypothesis. Self-citations are present but not load-bearing. The deformation strategy is attributed to [42]: 'We apply an approach similar to the one in [42], moving continuously the random environment from the deterministic critical point to the desired random one.' The approach is then implemented in full in the present paper, so the citation is methodological. Reference [5] is cited only for the interpretive bound ι(q)>1/4 ('Let us mention an ongoing work [5]...'); Theorem 1.6 itself is formulated in terms of fW(N) and does not need the bound. The Bernoulli-percolation results rest on external noise-sensitivity theorems [10,35] and an explicit coupling, not on a self-citation chain. Thus the paper's central claims have independent content; the main risk is the missing complete proof of Prop. 2.4, which i

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central theorems do not fit any free numerical parameters: all constants depend only on q and δ, and the environmental standard deviations σ_e are inputs, not fitted values. The load-bearing imported content is the [25] scaling relations and the new non-translation-invariant extension asserted in Proposition 2.4. The W^{1/3} headline additionally uses a conjectural exponent bound. No new physical entities are introduced.

assumptions (8)
  • domain assumption Critical FK measures with q in [1,4] satisfy a strong RSW box-crossing property with a uniform constant δ(q).
    Invoked throughout (Definition 1.4, Eq. (1.1)) from Beffara-Duminil-Copin [7] and Duminil-Copin et al. [28].
  • domain assumption Scaling relations and mixing-rate estimates of Duminil-Copin-Manolescu [25] hold for homogeneous FK-percolation in the full plane and on the torus.
    Theorems 2.2 and 2.3 are imported from [25]; they give derivative bounds for crossing probabilities in terms of Δ_pc and W(N).
  • ad hoc to paper Proposition 2.4: the [25] mixing-rate/covariance estimates extend to non-translation-invariant environments satisfying RSW(δ,N), with constants depending on δ.
    Stated in Section 2.2 with only a sketch; the text says a complete proof would reproduce [25] almost verbatim. This proposition is used in every stability lemma.
  • domain assumption The conjectured mixing-rate exponent bound ι(q) ≥ 1/2 (equality iff q=4), implying fW(N) ≍ W(N) up to a subpolynomial factor.
    Used in Section 1.3 and Section 4 to interpret Theorem 1.6 as a W(N)^{1/3} random critical window. The theorem statement itself is conditional on fW.
  • standard math Noise sensitivity of critical Bernoulli percolation crossing events (Benjamini-Kalai-Schramm [10], Garban-Steif [35]).
    Used in Lemma 5.4 and Theorem 1.10 to reduce a quenched crossing probability to its annealed critical value.
  • standard math Cardy-Smirnov formula for critical site percolation on the triangular lattice (Smirnov [52]).
    Used as the annealed limit in Theorem 1.10.
  • standard math Skorokhod embedding with tail estimates for variables with stretched exponential tails (Lemmas 3.6 and 4.5).
    Used to pass from Gaussian/Brownian environments to general independent environments satisfying (⋆).
  • domain assumption The edge environments satisfy the independence, centering, and exponential tail hypotheses (⋆)_N^A, (⋆)_N^B, or (⋆)_N^{q=1}.
    These are the theorem hypotheses, not derived; all results are conditional on them.

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Pith. "Pith review of The near-critical random bond FK-percolation model." pith.science (2026). https://pith.science/paper/LMA6GXWP

@misc{pith2026250908938,
  author       = {Pith},
  title        = {Pith review of: The near-critical random bond FK-percolation model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMA6GXWP}},
  note         = {Machine review of arXiv:2509.08938}
}
abstract

We study FK-percolation where the edge parameters are chosen as independent random variables in the near-critical regime. We show that if these parameters satisfy a natural centering condition around the critical point, then the quenched model typically exhibits critical behaviour at scales much larger than the deterministic characteristic length. More precisely, in a box of size $N$, if the homogeneous model with deterministic edge parameter $p$ looks critical in the regime $|p-p_c|\le \textrm W$, then the quenched model with random edge parameters $\mathbf p$ that typically satisfy $|\mathbf p-p_c|\le \textrm W^{1/3}$ looks critical, assuming some conjectured inequality on critical exponents, and up to logarithmic corrections. We also treat the special case of Bernoulli percolation, where we show that if one first samples non-degenerate independent random edge parameters centered around $\frac12$, and then a percolation configuration on these edges, the quenched model almost surely looks critical at large scales.

Figures

Figures reproduced from arXiv: 2509.08938 by the authors.

Figure 1
Figure 1. An illustration of the event C ([CN ; AN ]; [xN ; BN ]), the blue curve depicts a path of blue hexagons. site percolation on the triangular lattice is a random colouring of the vertices of TN , where each site is coloured in blue or yellow with probability 1 2 and indepen￾dently from the others. Fix some positive number x ∈ [0; 1] and consider the point xN = (⌊( 1 2 −x)N⌋, 0). In this context, denote ϕT , 1 2 ,1 the… view at source ↗
Figure 2
Figure 2. An illustration of the events A (e) 4 (r, R) (left) and A (e) 3+ (r, R) (right). Primal paths are represented by solid red curves and dual paths are represented by dashed red curves. upper half-plane in TN centred at e as H (e) N := (Z/NZ) × [0,(N − 1)/2]. One can define similarly the lower, left and right half-planes. This allows us to define (see also [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Illustration of the event C 0 h (R). Primal paths are depicted in full red lines and dual paths are in dashed red lines. The greyed out region is the domain D introduced after Definition 3.2. see Proposition 3.1 as a consequence of the fact that, with high PN probability, the breaking time Tb(δ, ρ) is not too small. In fact, it will suffice to show that there exist constants δ, ρ, c, ≲ such that for all σN ≤ c · W(N… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: An illustration of the pivotality event Piv [PITH_FULL_IMAGE:figures/full_fig_p058_4.png]
Figure 5
Figure 5. Figure 5: An illustration of the four cases in the proof. Once again, any length denoted by [PITH_FULL_IMAGE:figures/full_fig_p059_5.png]

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Pith tools

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