Mixing rate exponent of planar Fortuin-Kasteleyn percolation

Baojun Wu; Haoyu Liu; Zijie Zhuang

REVIEW 3 major objections 3 minor 1 cited by

Mixing rate exponent of planar Fortuin-Kasteleyn percolation

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The mixing rate exponent of FK percolation is 3κ/8 − 1, matching CLE.

desk verdict Genuinely new CLE computation that answers Duminil-Copin and Manolescu's Question 3; the unconditional κ∈(8/3,4] claim runs ahead of the proof, which is sketched there. read the letter →

arxiv 2502.09950 v2 pith:XAB7KHB4 submitted 2025-02-14 math.PR math-phmath.MP

Haoyu Liu , Baojun Wu , Zijie Zhuang This is my paper

classification math.PRmath-phmath.MP MSC 60K3560J6782B20

keywords FKpercolationrandom-clustermodelmixingrateexponentconformalloopensembleLiouvillequantumgravityweldingnear-criticalexponentsscalingrelations

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

This paper answers Question 3 of the scaling-relations program for planar FK percolation by computing the mixing rate exponent—the power-law rate at which a boundary condition's influence on a far-away edge decays—and proving it equals 3κ/8 − 1, where κ is the CLE parameter attached to the cluster weight q. The lattice computation is conditional on the standard but unproved conjecture that critical FK interfaces converge to nested CLE_κ; for the FK-Ising case q = 2, where convergence is proved, the result is unconditional and gives exponent 1. The paper's genuinely unconditional result lives in the continuum: for CLE_κ with κ ∈ (8/3, 8) \ {6}, the CLE mixing rate exponent exists and equals 3κ/8 − 1, with the odd-even discrepancy positive below κ = 6, zero at κ = 6, and negative above. If the conjectural convergence holds, plugging this exponent into the earlier scaling relations yields the near-critical exponents of FK percolation.

What carries the argument

The central object is the conformal loop ensemble (CLE_κ), a random collection of noncrossing loops parameterized by κ ∈ (8/3, 8); the mixing rate is the renormalized difference in probability that a thin annulus contains an odd-level versus even-level surrounding loop. The key identity is Theorem 1.5: the Radon-Nikodym derivative of the odd-level loop law against the even-level law equals a ratio of alternating sine-weighted partition functions in the annulus modulus r, with leading term 4 cos((κ − 4)π/4) $r^{{3κ/8 − 1}}$, which determines the exponent. This formula is derived by welding CLE loops to Liouville quantum gravity surfaces: each loop is realized as the conformal welding interface between a generalized quantum disk and a generalized quantum annulus, and the boundary-length laws of these annuli are computed using exact solvability of Liouville field theory. The FK-to-CLE comparison is carried by an annulus event of a primal circuit surrounding a dual circuit, which converges to the CLE odd/even loop events under the conjectured convergence.

What would settle it

Simulate nested CLE_κ for κ = 4.5 on a fine grid and estimate, for a thin annulus [ε, (1 + δ)ε], the probability difference P[Aodd] − P[Aeven] renormalized by P[Aeven] across ε → 0; if the log-log slope fails to approach 3(4.5)/8 − 1 = 0.6875, Theorem 1.3 is false. For the lattice statement, repeat the analogous estimate on critical FK at q = 3 (κ = 4.8) in Λ_R and check whether log ∆(R) / log R tends to 0.8; a different limit would contradict Theorem 1.1 if the convergence conjecture holds.

Watch

Extended reading notes

Core claim

Theorem 1.3 is the unconditional anchor: for each κ ∈ (8/3, 8) \ {6}, the nested CLE_κ mixing rate has the power-law behavior Δ_κ(r; δ) ≍ $r^{{3κ/8 − 1}}$, with the sign of Δ_κ matching the sign of 6 − κ, and Δ_κ = 0 at κ = 6. Theorem 1.1 then transfers this to FK percolation: for q ∈ (1, 4] with κ = 4π / arccos(−√q / 2) ∈ [4, 6), assuming the convergence conjecture, the lattice mixing rate exponent exists and equals 3κ/8 − 1. The bridge is Proposition 1.7, which shows that a CLE-friendly annulus event—a primal circuit enclosing a dual circuit—captures the same mixing rate as the original edge and crossing events, and Theorem 1.8, which uses the conjectured convergence to compare the two rates. The CLE result comes from an exact formula for the Radon-Nikodym derivative between the laws of odd- and even-nesting CLE loops, expressed as a ratio of $\theta$-like partition functions.

Load-bearing premise

The transfer from the unconditional CLE result to FK percolation rests on the conjecture that critical FK_q interfaces converge to nested CLE_κ, which has been proved only in the FK-Ising case q = 2.

Editorial extensions

If this is right

For q ∈ (1, 4], assuming the convergence conjecture, the FK mixing rate exponent is ι(q) = 3κ/8 − 1, settling Question 3 of [DCM22].
Combined with the scaling relations of [DCM22], the near-critical exponents β, γ, and δ are determined for q ∈ (1, 4]; α is also determined for q ∈ [2, 4], and for q ∈ (1, 2) if a further scaling relation holds.
For the FK-Ising model (q = 2), the mixing rate exponent is unconditionally 1.
For CLE_κ with κ ∈ (8/3, 8) \ {6}, the mixing rate exponent exists and equals 3κ/8 − 1, and the odd-even probability difference changes sign at κ = 6.
The exact comparison of odd/even loop laws reproduces the Coulomb-gas annulus partition functions of the O(n) loop model.

Reading between the lines

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

If the same LQG welding machinery were pushed to boundary-touching loops and to loops at arbitrary nesting levels, the exact Radon-Nikodym formula would likely yield explicit annulus crossing probabilities and multi-loop correlation functions; the paper only sketches that the approach extends.
A numerical extraction of the log-log slope of the lattice mixing rate for a value of q without a convergence proof, e.g., q = 3, compared with 3κ/8 − 1 would give an indirect test of the convergence conjecture.
The sign flip at κ = 6 means that odd nesting levels dominate in thin annuli when κ < 6 and even levels dominate when κ > 6; this parity asymmetry vanishes exactly at Bernoulli percolation, matching the known zero mixing rate there.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, the CLE mixing rate exponent 3κ/8−1 and the exact Radon-Nikodym formula (1.5) are real, new results, and the LQG welding derivation is the first rigorous confirmation of the Coulomb gas/CFT exponent. Second, the unconditional Theorem 1.3 is not yet fully proved on the range κ∈(8/3,4]: Section 5 is an admitted sketch with omitted calculations, and κ=4 is deferred to a cited continuity result. The headline claim runs ahead of the proof on that range.

What is actually new: the exact formula comparing odd- and even-level CLE loop laws, the mixing rate exponent, and the transfer theorem (Proposition 1.7 plus Theorem 1.8) linking the FK mixing rate to the CLE event A(r;δ). The FK-side reduction is solid—the modified coupling in Proposition 2.7 that handles the non-increasing event A(r;δ) is a genuinely clever piece of work. For FK-Ising, this yields an unconditional new proof of ι=1. The exponent is not fitted: it comes out of the r→0 expansion of an exact formula, so the circularity burden is low. The paper is also honest about its dependence on Conjecture 3.1.

Soft spots, in order of seriousness. The stress-test note lands: Section 5 explicitly says the detailed steps are omitted, and the κ=4 case is dispatched by "sending κ↑4" with continuity cited from [ACSW24, Appendix A]. Since Theorem 1.3 is stated unconditionally for this whole range, the paper needs that calculation written out, or the theorem restated with Section 5 flagged as a proof outline. This is a proof-completeness gap, not an error I can see, and it is closable. Second, the use of [SXZ24, Lemma 5.3] beyond its stated γ range is acknowledged in a footnote but not proved; that dependency should be either proved or explicitly conjectured. Third, the paper leans on several very recent preprints by overlapping authors; the cited results look like prior theorems rather than circular inputs, so I would not call this a circularity problem, but the referee should verify those dependencies carefully.

Who this is for: people working on CLE, LQG, and two-dimensional critical exponents. The FK-Ising corollary and the conditional FK result matter for the Duminil-Copin–Manolescu program. It deserves a serious referee: the core computation is credible and important, and the gaps are concrete and fixable. A referee should insist on a complete Section 5 and a proof of the extended [SXZ24] lemma before acceptance.

Referee Report

3 major / 3 minor

Summary. The paper defines a conformal-loop-ensemble (CLE) analog of the Fortuin-Kasteleyn mixing rate and proves that, for κ ∈ (8/3,8) \ {6}, the CLE mixing rate exponent exists and equals 3κ/8−1, with zero mixing rate at κ=6. Assuming convergence of critical FK_q percolation loop configurations to CLE_κ (Conjecture 3.1), the authors derive the FK mixing rate exponent for q ∈ (1,4] and thereby answer Question 3 of Duminil-Copin and Manolescu; for q=2 the result is unconditional. The CLE computation is based on an exact Radon-Nikodym derivative between the laws of odd-level and even-level CLE loops obtained via Liouville quantum gravity conformal welding and Liouville CFT partition functions, together with a lattice argument (Proposition 1.7) relating the FK mixing rate to a primal/dual circuit event.

Significance. If fully substantiated, the main result is a significant and parameter-free derivation: the exponent 3κ/8−1 is extracted from the exact local expansion of the Radon-Nikodym derivative (1.5), not fitted to a target value. The paper also gives a complete-looking lattice proof of Proposition 1.7, and the q=2 corollary is genuinely unconditional. The connection to Cardy's Coulomb-gas annulus partition functions is appealing and likely to be influential. However, as written the unconditional continuum theorem is not proved on the whole claimed range: the simple-loop case κ ∈ (8/3,4] is only sketched, and the κ=4 case is passed over by a cited continuity argument. These gaps affect the headline claims, so the paper currently needs substantial revision before it can be accepted.

major comments (3)

[Section 5 and Theorem 1.3] Theorem 1.3 is stated as an unconditional result for all κ ∈ (8/3,8) \ {6}, and Theorem 1.1 uses κ=4 (q=4). The proof for κ ∈ (8/3,4] is not complete in the manuscript: Proposition 5.2 imports the key boundary-length identity (5.6) from [ARS22, Lemmas 4.8 and 5.5], Theorem 5.4 states that 'the detailed steps of calculation are omitted', and the κ=4 case is disposed of by 'sending κ↑4' with continuity cited from [ACSW24, Appendix A]. The omitted inverse-Laplace calculation that converts (5.4)–(5.5) into the modulus densities em1 and em2 is load-bearing: the exponent 3κ/8−1 and its sign come from the exact Zodd/Zeven ratio, and the κ=4 limit is needed for q=4. Please supply the missing calculation and a proof of the κ=4 limiting statement, or explicitly weaken the theorem to the range for which a complete proof is given.
[Footnote to Proposition 4.14] The proof of the non-simple case κ ∈ (4,8) rests on identity (4.15), quoted from [SXZ24, Lemma 5.3]. The footnote asserts that the lemma, originally stated for γ=√(8/3), is valid for all γ ∈ (√2,2), but no proof or reference for this extension is provided. This extension is used in Proposition 4.14 and then in Propositions 4.15–4.16 and Theorem 4.20 to obtain the exact ratio (4.32)/(1.5). Since no alternative derivation is supplied, this is a load-bearing unproved input; it should be proved in this paper or replaced by a precise citation covering the full range γ ∈ (√2,2).
[Proof of Theorem 1.8, Section 1.3] The passage from Conjecture 3.1 to the inclusions Aeven_{ε^{1−θ},(1+δ)ε^{1+θ}} ⊆ A(εR;δ) ⊆ Aeven_{ε^{1+θ},(1+δ)ε^{1−θ}} is asserted without proof. The loop metric d defined in Section 3.1 gives proximity of individual loops, but it does not by itself control the correspondence of nesting parity for all loops surrounding the origin, nor does the statement of Conjecture 3.1 provide the simultaneous 'all loops ... corresponding CLE loop' coupling used in the argument. Since Theorem 1.8 is the bridge between the CLE computation and the FK mixing exponent, the proof needs either a stronger convergence statement or a separate lemma establishing convergence of the events Aodd and Aeven under the assumed topology; otherwise the comparison is not checkable.

minor comments (3)

[Definition 4.2] The sentence 'Let h2 be a random generalized function with the same law as h2 defined above' is self-referential as written; the notation for the mean-zero component should be clarified.
[Section 1.3] After (1.6), the notation Δp,q(r,R) ≍ Δp,q(R)/Δp,q(r) uses Δp,q(R) without explicitly recalling that it is the same as Δp,q(R) in (1.1); please make the notation consistent.
[Equation (4.30)] The displayed identity after the Poisson summation contains expressions such as 'gπ/τ' and 'τ/gπ' that are easy to misread; inserting parentheses, for example exp(−(πg/τ)p^2 + ((1−g)π/τ + i(2χ+2πm))p), would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found: the CLE mixing-rate exponent emerges from the r→0 expansion of the exact Radon-Nikodym derivative (1.5), and the FK comparison is conditional on the explicitly stated Conjecture 3.1.

full rationale

The central CLE derivation is not circular. The target exponent 3κ/8−1 is never used as an input; it emerges from the r→0 expansion of the exact Radon-Nikodym derivative (1.5), which itself is obtained in Section 4 from LQG welding and modulus computations, not from the claimed conclusion. The proof of Theorem 1.3 then compares odd/even CLE loop events by inclusion-exclusion and Lemma 3.6, using only this stronger formula. The FK part is explicitly conditional: Theorem 1.1 assumes Conjecture 3.1 (FK→CLE convergence), and Proposition 1.7 provides a genuine equivalence proof between the FK mixing rate and the event A(r;δ), with upper and lower bounds proved via [DCM22] couplings and RSW estimates. No observable is defined in terms of the exponent it is said to predict, and no fitted parameter is renamed as a prediction. The remaining concerns are rigor/completeness issues, not circularity: Section 5 is an explicit sketch for κ∈(8/3,4], with 'the detailed steps of calculation are omitted' and κ=4 dispatched by 'sending κ↑4' with continuity cited from [ACSW24, Appendix A]; and Proposition 4.14 extends [SXZ24, Lemma 5.3] beyond its stated γ=√(8/3) by a footnote assertion. These make the unconditional Theorem 1.3 not fully established as written, but they do not make the derivation equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central unconditional CLE claim is a parameter-free derivation, but it borrows a substantial LQG and welding toolkit from recent preprints, mostly by overlapping authors. The FK claim additionally assumes the unproved CLE convergence conjecture. No numerical fitting or ad hoc fitted constants appear.

assumptions (6)

domain assumption Convergence of critical FK_q loop configuration to nested CLE_κ (Conjecture 3.1)
The transfer Theorem 1.8 from FK to CLE is conditional on this unproved conjecture; without it, ι(q) = ι(κ) is not established for q ≠ 2.
standard math Increasing-coupling and quasi-multiplicativity results of Duminil-Copin and Manolescu [DCM22, Theorems 1.6 and 2.5]
Proposition 1.7 uses the coupled exploration and the bound P[τ < ∞] ≍ Δ(r,R), and Theorem 1.8 uses quasi-multiplicativity; these are imported from the cited paper.
standard math Conformal-welding and quantum-surface toolkit: [SXZ24, Lemma 5.3 and Proposition 4.21], [ARS22, Theorem 1.6 and Proposition 4.4], [ACSW24, Proposition 7.6], [MS17, Proposition 4.9]
The proof of Theorem 1.5 for κ ∈ (4,8) is built on these formulas for annulus partition functions, moduli of quantum annuli, and non-touching of second-level CLE loops.
ad hoc to paper Extension of [SXZ24, Lemma 5.3] from γ = √(8/3) to all γ ∈ (√2,2)
The footnote to Proposition 4.14 asserts the formulas remain valid for all γ in the non-simple range, but no proof is given in this paper.
domain assumption Continuity in κ used to reach κ = 4 via [ACSW24, Appendix A]
Theorem 1.3 includes κ = 4; the paper sends κ ↑ 4 citing an appendix of another preprint rather than deriving the limit here.
standard math Conformal radius increments of nested CLE loops are i.i.d. [SSW09, Theorem 1]
Lemma 3.5 uses this to bound the number of even loops hitting a small domain and to choose δ0.

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Cite this review

Pith. "Pith review of Mixing rate exponent of planar Fortuin-Kasteleyn percolation." pith.science (2026). https://pith.science/paper/XAB7KHB4

@misc{pith2026250209950,
  author       = {Pith},
  title        = {Pith review of: Mixing rate exponent of planar Fortuin-Kasteleyn percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XAB7KHB4}},
  note         = {Machine review of arXiv:2502.09950}
}

abstract

Duminil-Copin and Manolescu (2022) recently proved the scaling relations for planar Fortuin-Kasteleyn (FK) percolation. In particular, they showed that the one-arm exponent and the mixing rate exponent are sufficient to derive the other near-critical exponents. The scaling limit of critical FK percolation is conjectured to be a conformally invariant random collection of loops called the conformal loop ensemble (CLE). In this paper, we define the CLE analog of the mixing rate exponent. Assuming the convergence of FK percolation to CLE, we show that the mixing rate exponent for FK percolation agrees with that of CLE. We prove that the CLE$_\kappa$ mixing rate exponent equals $\frac{3 \kappa}{8}-1$, thereby answering Question 3 of Duminil-Copin and Manolescu (2022). The derivation of the CLE exponent is based on an exact formula for the Radon-Nikodym derivative between the marginal laws of the odd-level and even-level CLE loops, which is obtained from the coupling between Liouville quantum gravity and CLE.

Figures

Figures reproduced from arXiv: 2502.09950 by the authors.

**Figure 1.** Left: Primal and dual clusters are shown in gray and white, respectively. The event A(r; δ) requires a non-contractible dual circuit ηe (in dashed blue) surrounded by a non-contractible primal circuit η (in red) in Λr,(1+δ)r. Equivalently, the outer boundary L of a dual cluster lies in Λr,(1+δ)r and surrounds the origin, which should converge to a CLE loop of odd or even nesting level depending on the boundary condi… view at source ↗

**Figure 2.** Construction of CLEκ via a branching SLEκ(κ − 6) process. In this case m = 2, the outermost loop L 0 1 is the concatenation of the bold green and orange curves. By continuing η beyond σm towards the origin, we can construct a series of nested loops in the green region. where the infimum is taken over all homeomorphisms ψ from ∂D to itself. For two countable collections Γ1, Γ2 of loops, we say d(Γ1, Γ2) ≤ ε if and on… view at source ↗

**Figure 3.** Left: The graph of the 4 γ2 -stable L´evy process (Xt)t≥0 with only upward jumps. We draw a blue curve on the right of a red vertical line which corresponds to a jump, and identify the points that are on the same green horizontal line below the graph. Right: The 4 γ2 -stable looptree of disks constructed from (Xt)t≥0. The points on the red vertical line are collapsed to a single point. By replacing each topological … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: An illustration of Proposition 4.13. Top: A forested pinched thin quantum annulus from QAf T . Its conformal welding with the yellow forested disk from QDf 1,0 gives another sample from QDf 1,0 decorated with a boundary-touching CLE loop. The quantum surface QAT consid…

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