Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions

Jes\'us Salas; Tao Chen; Yihao Xu; Youjin Deng; Zongzheng Zhou

REVIEW 2 major objections 5 minor 60 references

Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper derives an exact formula for the correction-to-scaling exponent of the cluster-size distribution in the two-dimensional Fortuin–Kasteleyn Potts model, equal to $\Omega=8/[(2g+1)(2g+3)]$ in the Coulomb-gas coupling $g$…

desk verdict A clean, plausible exact formula for the correction-to-scaling exponent in 2D Potts clusters, with honest numerics and one load-bearing universality assumption that deserves referee scrutiny. read the letter →

arxiv 2411.12646 v2 pith:7NXSXPQH submitted 2024-11-19 cond-mat.stat-mech

Yihao Xu , Tao Chen , Zongzheng Zhou , Jesús Salas , Youjin Deng This is my paper · ORCID

classification cond-mat.stat-mech

keywords correction-to-scalingexponentpercolationFortuin-KasteleynPottsmodelcluster-sizedistributionCoulombgasO(n)loopMonteCarloclusteralgorithmtwodimensions

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 derives a closed-form expression for the subleading correction in the critical cluster-size distribution of the two-dimensional Fortuin–Kasteleyn Potts model: $n_s \simeq s^{-\tau} A (1+B s^{-\Omega})$ with $\Omega = 8/[(2g+1)(2g+3)]$, where $g$ is the Coulomb-gas coupling and $\sqrt{Q} = -2\cos(\pi g)$. The formula covers both the critical branch ($g\in(0,1]$) and the tricritical branch ($g\in[1,2]$) and contains the known exact percolation value $\Omega=72/91$ as the special case $g=2/3$. It is derived from the annulus partition function of the $O(n)$ loop model and tested with high-precision cluster Monte Carlo simulations of the loop model on the hexagonal and triangular lattices, with agreement within two standard deviations for $Q=1,2,3,4$ on the critical branch and $Q=1,2$ on the tricritical branch. If correct, it converts a nuisance parameter in numerical scaling fits into a known rational function of $g$, and it fixes the subleading magnetic exponent through $y_{h1}-y_{h2} = \Omega d_f$.

What carries the argument

The load-bearing object is the partition function of the $O(n)$ loop model on an annulus, expressed as a Coulomb-gas series in the modulus $\tilde{q} = e^{-2\pi L/\ell}$. The derivation isolates the crossing probability that a cluster connects the two boundaries, whose leading scaling is $R^{d_f-2}$ and whose first correction is $\tilde{q}^{1/g}$; equating $R\sim s^{1/d_f}$ turns this into the cluster-size correction $s^{-\Omega}$ with $\Omega d_f = 1/g$. The numerical check uses the induced-subgraph cluster algorithm on the dual spin representation of the $O(n)$ loops, chosen because the subleading thermal exponent is irrelevant or marginal there, so finite-size and correction effects are much smaller than in the Potts spin representation.

What would settle it

Measure $s^{\tau}n_s$ directly for FK clusters of the critical $Q=3$ Potts model on the square lattice over a wide range of $s$; if the subleading exponent is not compatible with $\Omega=9/14$ after finite-size corrections are removed, the formula or the loop-to-FK universality transfer fails.

Watch

Extended reading notes

Core claim

The central claim is that the correction-to-scaling exponent for the FK cluster-size distribution is exactly $\Omega = 1/(g d_f) = 8/[(2g+1)(2g+3)]$. Starting from the Coulomb-gas form of the $O(n)$ annulus partition function, the authors expand the crossing probability and isolate the leading correction $\tilde{q}^{1/g}$ in the modular parameter; using the cluster size–radius relation $s \sim R^{d_f}$ and the hyperscaling relation for the Fisher exponent, they identify this correction with $s^{-\Omega}$ and obtain $\Omega d_f = 1/g$. The same argument shows that the difference of the two leading magnetic eigenvalues is $y_{h1} - y_{h2} = 1/g$, so the subleading magnetic exponent is not independent of the cluster-size correction. The formula is claimed for both the critical Potts line ($0<Q\le 4$, $g\in(0,1]$) and the tricritical line ($1\le Q\le 4$, $g\in[1,2]$), with $Q=4$ at $g=1$ where the two lines meet. Numerical measurements of $s^{\tau}n_s$ in the $O(n)$ loop model for $n=1,\sqrt2,\sqrt3$ on the critical branch and $n=1,\sqrt2$ on the tricritical branch, plus $n=2$, are stated to be consistent with the formula.

Load-bearing premise

The load-bearing premise, stated in Section I, is that Fortuin–Kasteleyn clusters of the Potts model and the domains enclosed by $O(n)$ loops have the same critical behavior down to the subleading $s^{-\Omega}$ correction; if that identification fails, the formula would not transfer to Potts clusters.

Editorial extensions

If this is right

For $Q=1$ the formula reduces to the known exact percolation exponent $\Omega=72/91$, embedding the percolation result in a one-parameter family.
For the critical Ising, 3-state Potts, and 4-state Potts models it predicts $\Omega=32/45$, $9/14$, and $8/15$, respectively.
On the tricritical branch it predicts $\Omega=72/187$ for $Q=1$ and $\Omega=32/77$ for $Q=2$, with the $Q=4$ value $8/15$ common to both branches.
Because $\Omega d_f = 1/g = y_{h1}-y_{h2}$, a measurement of $\Omega$ determines the subleading magnetic eigenvalue $y_{h2}$ without a separate magnetization-correlation measurement.
The authors conjecture that the formula holds on the full branches, i.e., for $Q\in[0,4]$ in both the critical and tricritical Potts models and $n\in[-2,2]$ in the $O(n)$ loop model.

Reading between the lines

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

Testable extension: simulate FK clusters directly in the $Q=3$ Potts model on a square lattice and extract $\Omega$; agreement with $9/14$ would close the loop-to-FK assumption, while disagreement would localize the failure.
Analytic continuation: because the formula is rational in $g$, it gives concrete predictions for $Q<1$ on the critical branch, where the $O(n)$ loop weight is negative; exact-enumeration or conformal-field-theory checks could probe that regime.
General principle: the identity $y_{h1}-y_{h2}=\Omega d_f$ may be a general consequence of hyperscaling rather than a special property of the annulus computation, in which case subleading magnetic exponents in other two-dimensional models could be inferred from cluster-size corrections alone.

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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

I'll be brief. The paper gives a closed form for the correction-to-scaling exponent of the cluster-size distribution in 2D Potts: Omega(g) = 8/[(2g+1)(2g+3)], which reproduces Ziff's percolation value 72/91 and gives new predictions on both the critical and tricritical branches. The derivation follows Ziff's strategy using the Saleur-Bauer/Cardy annulus partition function, and the Monte Carlo tests are done on an O(n) loop model with almost no critical slowing down. The numerical data support the formula within 1--2 standard deviations essentially everywhere, and Q=4 is essentially exact.

What is genuinely new is the explicit formula as a function of the Coulomb-gas coupling g. The ingredients are in the literature—the theta' = 1/g correction for cluster mass versus radius appears in Refs. [6,52], and df is known—but nobody had written down the resulting Omega for n_s. The paper is honest about this, which I appreciate.

Two soft spots. First, the derivation's "most relevant correction" step is only valid for g in (1/2, 3/2). For g < 1/2 the e^2 term from Eq. (34) dominates; for g > 3/2 the e^{4/g−2} term dominates. The abstract and Eq. (8) claim g in (0, 2]. The tested points all lie in the safe interval, so the numerics are fine, but the full-range claim is not actually derived. That is an easy fix: restrict the claim or analyze the crossover.

Second, the conceptually bigger issue is the transfer from O(n) loop domains to FK Potts clusters. The paper states this as its main assumption, and prior evidence covers df, backbone, and shortest-path exponents, but not the subleading magnetic operator's amplitude in n_s. The simulations test the loop domains, not the FK clusters directly. I see no obvious reason the amplitude would vanish, and the percolation case—where the loop model maps to site percolation on the dual lattice—comes out right, so I lean toward the formula being correct in the tested range. But a referee should push for a direct FK-model measurement at Q=2 or Q=3, or at least a sharper universality argument for the subleading magnetic operator.

A minor issue: the tricritical fits are delicate, and the quoted error bars are probably optimistic given the systematic spread with cutoffs. That is typical for this kind of analysis and not disqualifying.

Bottom line: the paper is careful, the derivation is plausible, and the numerics are good. The soft spots are real but addressable. My recommendation: send it to a serious referee, with specific attention to the O(n)-to-FK transfer and the g-range restriction.

Referee Report

2 major / 5 minor

Summary. The paper claims an exact formula for the correction-to-scaling exponent Ω of the cluster-size distribution n_s ~ s^{-τ}(1+B s^{-Ω}) for the Fortuin-Kasteleyn representation of the two-dimensional Q-state Potts model, Ω = 8/[(2g+1)(2g+3)], where the Coulomb-gas coupling g is related to Q by Q = 2 + 2 cos(2πg). The derivation extends Ziff's annulus argument using the Saleur-Bauer/Cardy partition function for the O(n) loop model, and the claim is tested by Monte Carlo simulations of a generalized Ising representation of the O(n) loop model on the triangular lattice for Q = 1,2,3 on the critical branch, Q = 1,2 on the tricritical branch, and Q = 4, where both branches meet. The numerical estimates are reported to be consistent with the conjectured values within roughly one to two standard deviations after accounting for systematic corrections.

Significance. If the formula is correct, it is a substantial exact result: it generalizes Ziff's exact percolation exponent Ω = 72/91 to the full Potts family, it predicts the combination y_{h1} - y_{h2} = Ω d_f of magnetic exponents, and it gives concrete falsifiable predictions for both critical and tricritical branches. The paper also has clear methodological strengths: the derivation uses established exact Coulomb-gas/annulus results rather than fitting parameters, the Monte Carlo algorithm is designed to suppress critical slowing-down, and the numerical analysis is transparent about cutoff choices and error bars. The main caveat is that the exact derivation and the simulations both concern O(n) loop domains, while the headline claim concerns FK clusters of the Potts model; the transfer between the two is explicitly an assumption, and the evidence adduced for it covers leading geometric exponents but not the subleading magnetic operator that controls Ω.

major comments (2)

[Sec. III, Eqs. (33)-(35)] The sentence 'As g ∈ (0,2], both series generically have a nonzero leading term, and the most relevant correction term is q^{1/g} from (33)' is not correct for the full stated range. Comparing the exponents appearing in (33)-(34), the term q^{1/g} is the most relevant (smallest exponent) only for 1/2 < g < 3/2; for g < 1/2 the q^2 term from (34) dominates, and for g > 3/2 the q^{4/g-2} term dominates. Since conjecture (14) is displayed for g ∈ (0,2], the derivation as written does not establish the formula outside (1/2,3/2). The physical positive-loop-weight cases n = √Q ∈ [0,2] do lie in g ∈ [1/2,3/2], so the tested data are unaffected, but the claimed full-range result needs either a corrected range or an additional argument explaining why the competing annulus corrections do not contribute to the cluster-size distribution.
[Sec. I and Sec. V] The central transfer from O(n) loop domains to FK clusters of the Q = n^2 Potts model is load-bearing and is not established for the correction-to-scaling exponent. The paper states this explicitly as 'the main assumption in this work' in Sec. I, and previous support (fractal dimension, backbone, shortest-path exponents) concerns leading geometric exponents. The Monte Carlo measurements in Sec. V are measurements of spin domains in the generalized Ising model (45), i.e., of the domains enclosed by O(n) loops, not of FK clusters of the Potts model. Equation (41) shows that Ω d_f equals y_{h1} - y_{h2}, so the claimed value for FK clusters requires not only that the two representations share the subleading magnetic exponent y_{h2}, but also that the corresponding operator has a nonzero amplitude in the FK cluster-size distribution. A direct FK-cluster simulation for at least one nontrivial Q, or an analytic argument pinning down the operator content of n_s in the FK representation, is needed to make the Potts-model claim fully supported.

minor comments (5)

[Sec. V B, tricritical Q = 2] The text states 'we conjecture Ω = 33/77 ≈ 0.428571', but the formula (14) with g = 5/4 gives Ω = 8 / [(7/2)(11/2)] = 32/77 ≈ 0.415584, and Table II uses 32/77. The number 33/77 is a typo and should be corrected.
[Introduction, last paragraph] The organizational sentence says 'Monte Carlo simulations and measurements are discussed in Sec. IV, and in Sec. IV, we present the numerical results'; the numerical results are presented in Sec. V, not Sec. IV. The cross-reference should be fixed.
[Abstract and Sec. I] The statement that the O(n) loop model is 'in the same universality class as the Q = n^2 Potts model' is stated as a fact in the abstract, while the body of the paper correctly labels the loop-domain/FK-cluster identification as the main assumption. The abstract should reflect this qualification, especially because for g outside (1/2,3/2) the loop weight n is negative and the loop model is not the physical O(n) model.
[Fig. 3 caption] The caption writes 'shows that n(s) on the x^- branch has no noticeable finite-lattice-size corrections'; the symbol n(s) is undefined there and should read n_s or s^τ n_s, consistent with the axis label.
[Sec. V B, first paragraph] The phrase 'we have assumes that the size distribution n_s ... exhibits' contains a grammatical error ('assumes' should be 'assumed'), and the sentence would benefit from a reference to the ansatz (1) rather than a repetition of the formula.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Ω formula is derived from the external Saleur–Bauer/Cardy annulus partition function plus standard Coulomb-gas exponents, and the Monte Carlo fits estimate the exponent freely before comparison.

full rationale

The derivation chain in Sec. III is not circular. The correction-to-scaling formula Ω = 8/[(2g+1)(2g+3)] follows from the annulus partition function (28)–(29) of Saleur–Bauer/Cardy: extracting the leading correction q^{1/g} in the crossing probability, using the fractal dimension df = yh1 = (2g+1)(2g+3)/(8g), and applying Ziff's scaling argument for P≥s yields Ω = 1/(g df). All inputs are external exact results or standard Coulomb-gas formulas; no parameter is fitted to produce the predicted value. The Monte Carlo verification in Sec. V is independent: it measures the cluster-size distribution of the generalized Ising representation of the O(n) loop model and fits the exponent y1 freely (Tables I–II), obtaining estimates consistent with the conjecture; the fixed-exponent fits are explicitly consistency checks rather than predictions. The paper's central caveat—'The main assumption in this work is that the FK clusters of the critical and tricritical Potts models have the same critical behavior as the domains enclosed by the O(n) loops'—is a genuine physical hypothesis rather than a definitional equivalence; it is supported by published results for df and backbone exponents ([27], [51], including a same-group citation) and is explicitly labeled as an assumption. That transfer is a correctness risk deserving a direct FK-model test, not a circular step. A minor qualification is that the annulus expansion's statement that q^{1/g} is the most relevant correction is only clearly valid on a subinterval of g∈(0,2] (the simulated points lie in that subinterval), but that is a derivation-completeness issue, not circularity. Accordingly, no step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The theoretical formula is parameter-free; it uses established Coulomb-gas exponents and the exact annulus partition function as external inputs. The key domain assumption is the O(n)-to-FK universality. No new entities are postulated.

assumptions (4)

domain assumption Coulomb-gas parametrization and exact exponents yt1, yt2, yh1, yh2, df as functions of g (Eqs. 9-11)
Taken from Nienhuis' Coulomb-gas solution of the O(n) model; standard in the field, not derived in this paper.
standard math Saleur-Bauer/Cardy annulus partition function for the O(n) model (Eqs. 28-29)
Exact external result used to extract the leading correction e^{1/g}; the paper relies on its small-e expansion.
domain assumption Universality of O(n) loop domains with FK clusters of the Q=n^2 Potts model
Stated as the main assumption in Sec. I; supported by prior results for df and backbone exponents but assumed here for the cluster-size distribution and its corrections.
domain assumption Scaling form ns ~ s^{-τ}(1 + B s^{-Ω}) and hyperscaling relation τ=1+d/df
Standard scaling hypothesis for the cluster-size distribution, used throughout the derivation and fits.

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Cite this review

Pith. "Pith review of Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions." pith.science (2026). https://pith.science/paper/7NXSXPQH

@misc{pith2026241112646,
  author       = {Pith},
  title        = {Pith review of: Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7NXSXPQH}},
  note         = {Machine review of arXiv:2411.12646}
}

abstract

The number $n_s$ of clusters (per site) of size $s$, a central quantity in percolation theory, displays at criticality an algebraic scaling behavior of the form $n_s\simeq s^{-\tau}\, A\, (1+B s^{-\Omega})$. For the Fortuin--Kasteleyn representation of the $Q$-state Potts model in two dimensions, the Fisher exponent $\tau$ is known as a function of the real parameter $0\le Q\le4$, and, for bond percolation (the $Q\rightarrow 1$ limit), the correction-to-scaling exponent is derived as $\Omega=72/91$. We theoretically derive the exact formula for the correction-to-scaling exponent $\Omega=8/[(2g+1)(2g+3)]$ as a function of the Coulomb-gas coupling strength $g$, which is related to $Q$ by $Q=2+2\cos(2 \pi g)$. Using an efficient Monte Carlo cluster algorithm, we study the O($n$) loop model on the hexagonal lattice, which is in the same universality class as the $Q=n^2$ Potts model, and has significantly suppressed finite-size corrections and critical slowing-down. The predictions of the above formula include the exact value for percolation as a special case and agree well with the numerical estimates of $\Omega$ for both the critical and tricritical branches of the Potts model.

Figures

Figures reproduced from arXiv: 2411.12646 by the authors.

**Figure 2.** FIG. 2. Illustration of the correspondence between the O( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Finite-cluster-size effects on [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. Quantity [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Estimates of the correction-to-scaling exponent Ω for [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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Works this paper leans on

60 extracted references · 55 canonical work pages

[1]

Stauffer and A

D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor and Francis, London, 1994)

work page 1994
[2]

Bollob´ as and O

B. Bollob´ as and O. Riordan, Percolation (Cambridge University Press, Cambridge, 2006)

work page 2006
[3]

Adler, M

J. Adler, M. Moshe, and V. Privman, New method for an- alyzing confluent singularities and its application to two- dimensional percolation, Phys. Rev. B 26, 1411 (1982)

work page 1982
[5]

R. M. Ziff and F. Babalievski, Site percolation on the Penrose rhomb lattice, Physica A 269, 201 (1999), arXiv:1101.0807

work page Pith review arXiv 1999
[6]

Fractal dimensions and corrections to scaling for critical Potts clusters

A. Aharony and J. Asikainen, Fractal dimensions and corrections to scaling for critical Potts clusters, Fractals 11, 3 (2003), arXiv:cond-mat/0206367

work page Pith review arXiv 2003
[7]

Di Francesco, P

P. Di Francesco, P. Mathieu, and M. S´ en´ echal,Conformal Field Theory (Springer–Verlag, New York, 1997)

work page 1997
[8]

Grimmett, The Random-Cluster Model (Springer, Heidelberg, 2006)

G. Grimmett, The Random-Cluster Model (Springer, Heidelberg, 2006)

work page 2006
[9]

R. B. Potts, Some generalized order-disorder transforma- tions, Proc. Cambridge Philos. Soc. 48, 106 (1952)

work page 1952

Show all 60 references

[10]

F. Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982)

1982
[11]

F. Y. Wu, Erratum: The Potts model, Rev. Mod. Phys. 55, 315 (1983)

1983
[12]

F. Y. Wu, Potts model of magnetism (invited), J. Appl. Phys. 55, 2421 (1984)

1984
[13]

R. J. Baxter, Exactly Solved Models in Statistical Me- chanics (World Scientific, Singapore, 1985)

1985
[14]

P. W. Kasteleyn and C. M. Fortuin, Phase transitions in lattice systems with random local properties, J. Phys. Soc. Jpn. 26, 11 (1969), Supplement

1969
[15]

C. M. Fortuin and P. W. Kasteleyn, On the random- cluster model: I. Introduction and relation to other mod- els, Physica 57, 536 (1972)

1972
[16]

Nienhuis, A

B. Nienhuis, A. N. Berker, E. K. Riedel, and M. Schick, First- and second-order phase transitions in Potts mod- els: Renormalization-group solution, Phys. Rev. Lett. 43, 737 (1979)

1979
[17]

Nienhuis, E

B. Nienhuis, E. K. Riedel, and M. Schick, Variational renormalisation-group approach to the q-state Potts model in two dimensions, J. Phys. A: Math. Gen. 13, L31 (1980)

1980
[18]

K. K. Murata, Hamiltonian formulation of site perco- lation in a lattice gas, J. Phys. A: Math. Gen. 12, 81 (1979)

1979
[19]

Nienhuis, Analytical calculation of two leading expo- nents of the dilute Potts model, J

B. Nienhuis, Analytical calculation of two leading expo- nents of the dilute Potts model, J. Phys. A: Math. Gen. 15, 199 (1982)

1982
[20]

Janke and M

W. Janke and M. J. Schakel, Geometrical vs. Fortuin– Kasteleyn clusters in the two-dimensional q-state Potts model, Nucl. Phys. B 700, 385 (2004), arXiv:cond- mat/0311624

2004
[21]

Y. Deng, J. R. Heringa, and H. W. J. Bl¨ ote, Constrained tricritical phenomena in two dimensions, Phys. Rev. E 71, 036115 (2005)

2005
[22]

X. Qian, Y. Deng, and H. W. J. Bl¨ ote, Dilute Potts model 14 in two dimensions, Phys. Rev. E 72, 056132 (2005)

2005
[23]

Nienhuis and M

B. Nienhuis and M. Nauenberg, First-order phase tran- sitions in renormalization-group theory, Phys. Rev. Lett. 45, 777 (1975)

1975
[24]

Klein, D

W. Klein, D. J. Wallace, and R. K. P. Zia, Essential singularities at first-order phase transitions, Phys. Rev. Lett. 37, 639 (1976)

1976
[25]

M. E. Fisher and N. H. Berker, Scaling for first-order transituons in thermodynamic and finite systems, Phys. Rev. B 26, 2507 (1978)

1978
[26]

Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J

B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Stat. Phys. 34, 731 (1984)

1984
[27]

Nienhuis, Coulomb gas formulations of two- dimensional phase transitions, in Phase Transitions and Critical Phenomena, Vol

B. Nienhuis, Coulomb gas formulations of two- dimensional phase transitions, in Phase Transitions and Critical Phenomena, Vol. 11, edited by C. Domb and J. L. Lebowitz (Academic Press, London, 1987) pp. 1–53

1987
[28]

Nauenberg and D

M. Nauenberg and D. J. Scalapino, Singularities and scaling functions at the Potts-model multicritical point, Phys. Rev. Lett. 44, 837 (1980)

1980
[29]

J. L. Cardy, M. Nauenberg, and D. J. Scalapino, Macro- scopic loops in the loop O( n) model at Nienhuis’ critical point, Phys. Rev. B 22, 2560 (1980)

1980
[30]

Salas and A

J. Salas and A. D. Sokal, Logarithmic corrections and finite-size scaling in the two-dimensional 4-state Potts model, J. Stat. Phys. 88, 567 (1997), arXiv:hep- lat/9607030

1997
[31]

Saleur and B

H. Saleur and B. Duplantier, Exact determination of the percolation hull exponent in two dimension, Phys. Rev. Lett. 58, 2325 (1987)

1987
[32]

Nolin, W

P. Nolin, W. Qian, X. Sun, and Z. Zhuang, Backbone exponent for two-dimensional percolation, arXiv:2309.05050 (2023), preprint

2023 arXiv
[33]

D. P. Landau and K. Binder, A Guide to Monte-Carlo Simulations in Statistical Physics , 4th ed. (Cambridge University Press, Cambridge, 2014)

2014
[34]

A. D. Sokal, Monte Carlo methods in statistical mechan- ics: foundations and new algoriths, in Functional Inte- gration: Basics and Applications , edited by C. DeWitt- Morette, P. Cartier, and A. Folacci (Plenum, New York,
[35]

R. H. Swendsen and J.-S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58, 86 (1987)

1987
[36]

Li and A

X.-J. Li and A. D. Sokal, Rigorous lower bound on the dynamic critical exponent of the Swendsen–Wang algo- rithm, Phys. Rev. Lett. 63, 827 (1989)

1989
[37]

Chayes and J

L. Chayes and J. Machta, Graphical representations and cluster algorithms II, Physica A 254, 477 (1998)

1998
[38]

Y. Deng, T. M. Garoni, J. Machta, G. Ossola, G. Polin, and A. D. Sokal, Critical behavior of the Chayes– Machta–Swendsen–Wang dynamics, Phys. Rev. Lett.99, 055701 (2007), arXiv:0705.2751

2007 arXiv
[39]

Prokof’ev and B

N. Prokof’ev and B. Svistunov, Worm algorithms for classical statistical models, Phys. Rev. Lett. 87, 160601 (2001), arXiv:cond-mat/0103146

2001 arXiv
[40]

Y. Deng, T. Garoni, and A. D. Sokal, Dynamic critical behavior of the worm algorithm for the Ising model, Phys. Rev. Lett. 99, 110601 (2007), arXiv:cond-mat/0703787

2007 arXiv
[41]

Salas and A

J. Salas and A. D. Sokal, Universal amplitude ratios in the critical two-dimensional Ising model on a torus, J. Stat. Phys. 98, 551 (2000), The relevant part is contained in the first preprint version of this paper, arXiv:cond- mat/9904038v1

2000
[42]

Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys

B. Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys. Rev. Lett. 49, 1062 (1982)

1982
[43]

Batchelor and H

M. Batchelor and H. W. J. Bl¨ ote, Conformal invariance and critical behavior of the O( n) model on the honey- comb lattice, Phys. Rev. B 39, 2391 (1989)

1989
[44]

Peled and Y

R. Peled and Y. Spinka, Lectures on the spin and loop O(n) models, in Sojourns in Probability Theory and Sta- tistical Physics - I. Spin Glasses and Statistical Mechan- ics, A Festschrift for Charles M. Newman , Springer Pro- ceedings in Mathematics & Statistics, Vol. 298, edi...

2019 arXiv
[45]

Duminil-Copin, A

H. Duminil-Copin, A. Glazman, R. Peled, and Y. Spinka, Macroscopic loops in the loop O( n) model at Nien- huis’ critical point, J. Eur. Math. Soc. 23, 315 (2021), arXiv:1707.09335

2021 arXiv
[46]

R. J. Baxter, q colourings of the triangular lattice, J. Phys. A: Math. Gen. 19, 2821 (1986)

1986
[47]

R. J. Baxter, Chromatic polynomials of large triangular lattices, J. Phys. A: Math. Gen. 20, 5241 (1987)

1987
[48]

Y. Deng, T. M. Garoni, W. Guo, H. W. J. Bl¨ ote, and A. D. Sokal, Cluster simulations of loop models on two- dimensional lattices, Phys. Rev. Lett. 98, 120601 (2007)

2007
[49]

C. Ding, X. Qian, Y. Deng, W. Guo, and H. W. K. Bl¨ ote, Geometric properties of two-dimensional O( n) loop con- figurations, J. Phys. A: Math. Theor. 40, 3305 (2007), arXiv:cond-mat/0608547

2007 arXiv
[50]

Q. Liu, Y. Deng, and T. Garoni, Worm Monte Carlo study of the honeycomb-lattice loop model, Nucl. Phys. B 846, 283 (2011), arXiv:1011.1980

2011 arXiv
[51]

S. Fang, D. Ke, W. Zhong, and Y. Deng, Backbone and shortest-path exponents of the two-dimensional Q- state Potts model, Phys. Rev. E 105, 044122 (2022), arXiv:2112.10162

2022 arXiv
[52]

Asikainen, A

J. Asikainen, A. Aharony, B. B. Mandelbrot, E. M. Rauch, and J. P. Hovi, Fractal geometry of critical Potts clusters, Eur. Phys. J. B 34, 479 (2003), arXiv:cond- mat/0212216

2003
[53]

J. L. Cardy, Conformal invariance, in Phase Transitions and Critical Phenomena, Vol. 11, edited by C. Domb and J. L. Lebowitz (Academic Press, London, 1987) pp. 55– 126

1987
[54]

Duminil-Copin and S

H. Duminil-Copin and S. Smirnov, The connective con- stant of the honeycomb lattice equals p 2 + √ 2, Ann. Math. 175, 1653 (2012), arXiv:1007.0575

2012 arXiv
[55]

H. W. J. Bl¨ ote and B. Nienhuis, Fully packed loop model on the honeycomb lattice, Phys. Rev. Lett. 72, 1372 (1994)

1994
[56]

Batchelor, J

M. Batchelor, J. Suzuki, and C. M. Yung, Exact re- sults for Hamiltonian walks from the solution of the fully packed loop model on the honeycomb lattice, Phys. Rev. Lett. 73, 2646 (1994)

1994
[57]

Kondev, J

J. Kondev, J. de Gier, and B. Nienhuis, Operator spec- trum and exact exponents of the fully packed loop model, J. Phys. A: Math. Gen. 29, 6489 (1996), arXiv:cond- mat/9603170

1996
[58]

Saleur and M

H. Saleur and M. Bauer, On some relations between lo- cal height probabilities and conformal invariance, Nucl. Phys. B 320, 591 (1989)

1989
[59]

J. L. Cardy, The O( n) model on the annulus, J. Stat. Phys. 125, 1 (2006), arXiv:math-ph/0604043

2006 arXiv
[60]

Y. Deng, W. Guo, and H. W. J. Bl¨ ote, Cluster simula- 15 tion of the O( N ) loop model on the honeycomb lattice, arXiv:cond-mat/0605165 (2006), preprint

2006 arXiv
[61]

We use instead the standard Ising model notation given by Eq

Please note that the Ising model is not written as the 2-state Potts model (3)/(4). We use instead the standard Ising model notation given by Eq. (45)

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Stauffer and A

D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor and Francis, London, 1994)

work page 1994

[2] [2]

Bollob´ as and O

B. Bollob´ as and O. Riordan, Percolation (Cambridge University Press, Cambridge, 2006)

work page 2006

[3] [3]

Adler, M

J. Adler, M. Moshe, and V. Privman, New method for an- alyzing confluent singularities and its application to two- dimensional percolation, Phys. Rev. B 26, 1411 (1982)

work page 1982

[4] [5]

R. M. Ziff and F. Babalievski, Site percolation on the Penrose rhomb lattice, Physica A 269, 201 (1999), arXiv:1101.0807

work page Pith review arXiv 1999

[5] [6]

Fractal dimensions and corrections to scaling for critical Potts clusters

A. Aharony and J. Asikainen, Fractal dimensions and corrections to scaling for critical Potts clusters, Fractals 11, 3 (2003), arXiv:cond-mat/0206367

work page Pith review arXiv 2003

[6] [7]

Di Francesco, P

P. Di Francesco, P. Mathieu, and M. S´ en´ echal,Conformal Field Theory (Springer–Verlag, New York, 1997)

work page 1997

[7] [8]

Grimmett, The Random-Cluster Model (Springer, Heidelberg, 2006)

G. Grimmett, The Random-Cluster Model (Springer, Heidelberg, 2006)

work page 2006

[8] [9]

R. B. Potts, Some generalized order-disorder transforma- tions, Proc. Cambridge Philos. Soc. 48, 106 (1952)

work page 1952