Probability distributions of the order parameter of the $O(N)$ model

Adam Ran\c{c}on; Bertrand Delamotte; Ivan Balog; Lovro \v{S}aravanja

REVIEW 3 major objections 6 minor 1 cited by

Probability distributions of the order parameter of the $O(N)$ model

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The lowest-order functional renormalization group captures the full functional shape of critical order-parameter probability distributions for O(N) models, up to two global rescaling factors.

desk verdict Solid exact large-N benchmark and an honest but conditional finite-N shape claim; the two fitted rescaling factors make the <1% agreement weaker than it looks. read the letter →

arxiv 2501.04465 v2 pith:SVDEJ7KH submitted 2025-01-08 cond-mat.stat-mech

Adam Rançon , Bertrand Delamotte , Lovro Šaravanja , Ivan Balog This is my paper · ORCID

classification cond-mat.stat-mech MSC 82B2782B2860F10

keywords O(N)modelorder-parameterdistributionratefunctionfunctionalrenormalizationgrouplocalpotentialapproximationfinite-sizescalinglargedeviationscriticalphenomena

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 extends a functional renormalization group (FRG) computation of the probability distribution function (PDF) of the order parameter at criticality from the Ising case to the O(N) model. Its central claim is that the Local Potential Approximation (LPA), the lowest order of the derivative expansion, reproduces the whole family of universal scaling functions once the overall amplitude of the logarithm of the PDF and the ratio zeta = L/xi_infinity are each corrected by one N-dependent factor. The family is parametrized by how the thermodynamic limit and the critical point are approached, and it includes a second family reached from the ordered phase. If this claim is right, the nontrivial functional form of these large-deviation distributions is already encoded at the simplest level of the FRG, with all approximation error concentrated in two universal constants.

What carries the argument

The central object is the scale-dependent rate function I_k(s), obtained as the M -> infinity limit of a modified effective action Gamma_{M,k} that enforces the fixed order-parameter constraint through a large mass term ($M^{2}$/2)(integral_x (phi - s))^2. Within the LPA ansatz, I_k obeys a flow equation that is identical to the effective potential flow except that the zero-momentum mode is removed from the regulator trace, which is what allows the rate function to remain non-convex at finite size. The flow is initialized from the dimensionless fixed-point solution of the effective potential, and a relevant perturbation controls zeta = L/xi_infinity. In the large-N limit the exact solution is encoded by a universal function check F_d(z) with a pole at z = -pi, replacing the function F_d(z) that appears in the effective potential.

What would settle it

Compute the rate function at the next order of the derivative expansion and compare it, after the same two-parameter rescaling, with the LPA curve and with the Monte Carlo data. If the higher-order shape differs from the LPA shape by more than the Monte Carlo statistical error, the claim that LPA captures the exact functional form is false; a direct alternative is to take Monte Carlo data at several zeta values not used to fix r_I and r_zeta, such as zeta = 1 and zeta = 3 for N=2, and check whether the asserted sub-1% agreement persists.

Watch

Extended reading notes

Core claim

For the three-dimensional O(N) model with N=2 and N=3, the rate function I(s) obtained from the LPA matches Monte Carlo simulations across the entire zeta-family to better than 1% relative error after the rescaling I -> r_I I and zeta -> r_zeta zeta. Equivalently, the LPA error is concentrated in the universal amplitude $\Delta$ I_N = L^d (I_min - I_0)/N and in the critical ratio zeta_c, while the shape of I(s) as a function of the scaled field s $L^{{(d-2+eta)/2}}$ and of zeta is essentially exact. In the large-N limit the paper derives an exact rate function from a saddle-point gap equation and verifies that its LPA flow reproduces it to about $10^{{-4}}$ relative error, independent of the regulator. The paper also computes the separate family of universal PDFs reached by approaching criticality from the low-temperature phase, and reports a surprising near-coincidence of the N=2 and N=3 rate functions once normalized by N.

Load-bearing premise

The argument assumes that every error made by the Local Potential Approximation appears only as one overall multiplication factor for the logarithm of the PDF and one rescaling of zeta, so that the functional shape itself is exact; if shape errors exist, the excellent collapse in Fig. 10 is produced by the two fitted factors rather than by physics.

Editorial extensions

If this is right

For a given N, computing the absolute critical PDF reduces to determining two universal constants, the amplitude Delta I_N and the critical ratio zeta_c; the LPA supplies the full functional form of the family.
The functional shape of the entire zeta-family, including the non-convex regime and the ordered-phase family, is the same for FRG at LPA and Monte Carlo after the two rescaling factors are fixed.
The large-N limit provides a closed-form benchmark in which the LPA flow is exact, validating the numerical scheme and giving a template for benchmarking higher-order approximations.
The near-collapse of the normalized N=2 and N=3 rate functions, with N=4 closer to the large-N curve, shows that the shape depends only weakly on N and can serve as a target for other approximation schemes.
Improving beyond LPA should mainly correct the two universal constants rather than the shape, so next-order derivative-expansion calculations should be testable against Monte Carlo through Delta I_N and zeta_c alone.

Reading between the lines

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

If the two-parameter absorption of LPA error is generic, the same procedure could convert low-order FRG results for other large-deviation functions, such as work fluctuations or entanglement measures, into quantitatively predictive shapes with only fitted amplitudes.
The near-identity of the normalized N=2 and N=3 rate functions suggests that the N-dependence enters mostly through the two universal constants; a direct test would be a high-precision Monte Carlo study at N=5 and N=6 to see whether the normalized curves continue to cluster.
A stronger test of the central claim would be to repeat the computation at second order in the derivative expansion and check whether the resulting rate-function shape differs from the LPA shape only by the same two rescaling factors; if it does not, the exactness of the LPA shape is an artifact of the lowest order.
The ratios r_I(N) and r_zeta(N) should themselves be universal functions of N and d, so computing them at higher orders could turn the LPA-plus-rescaling scheme into a fully predictive method that needs no Monte Carlo input.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 large-N rate function computation is clean and exact: a full family of universal order-parameter distributions parametrized by zeta, derived from an explicit regulator-independent gap equation. That is new and solid, and the FRG numerics reproduce it to 1e-4 for analytic regulators. Second, the finite-N claim is not as strong as the abstract suggests. The 'excellent' agreement is obtained after fitting two global rescaling factors r_I and r_zeta to the same Monte Carlo data that later certify the <1% agreement. Those fits remove the two largest LPA errors: Delta I_N and zeta_c are both off by about 25%, so the residual shape agreement is real but tests the two-scale ansatz only under the assumption that no other LPA shape error exists. The paper says this plainly, so it is not a hidden flaw, but it is a load-bearing gap. The conclusion that LPA error 'is primarily concentrated' in those two constants is an inference from the two-parameter fit, not a tested property of the approximation.

What is genuinely good: the large-N limit is self-contained, the derivation of the rate-function gap equation is explicit enough to reimplement, and the observation that the N=2 and N=3 rate functions nearly collapse is an honest, unexplained result left for future work. The MC comparison is careful, with histogram reweighting and L=96/128 convergence checks, and the paper corrects a statement from their earlier Ising paper about a non-universal amplitude, which is good scholarly hygiene. The LPA equations and numerical scheme are described well enough for independent reimplementation, though no code or data is shipped.

Soft spots beyond the calibration issue: the N=2 vs N=3 near-collapse is not explained, and N=4 being closer to N=infinity muddies the ordering. The <1% relative error in Sec. IV B is demonstrated for N=2 only; N=3 is not shown as a full-family comparison. The stress-test note's concern is real but not disqualifying, because the paper is transparent about the two rescaling factors and the large-N benchmark gives confidence the numerics are sound.

Bottom line: this is a solid methods-and-benchmark paper. The exact large-N result deserves citation; the finite-N shape claim should be read with the calibration caveat. No one should cite this as 'LPA predicts the full PDF shape without input from MC.' A serious referee should engage, mainly to push for a next-order check or an independent determination of r_I and r_zeta (say, from a single zeta value) if that is feasible.

Referee Report

3 major / 6 minor

Summary. The paper generalizes the FRG-based method for computing order-parameter rate functions, previously developed for the Ising model, to the O(N) model in d=3. It derives the exact large-N rate-function family (Sec. II), implements the LPA flow equations for the constraint effective action (Sec. III), and compares the LPA results with Monte Carlo simulations for N=2 and N=3 (Sec. IV). The central claim is that, after rescaling the rate-function amplitude by r_I(N) and the scaling variable zeta by r_zeta(N), the LPA reproduces the full functional form of the universal rate-function family to better than 1% relative error, and that the LPA error is concentrated in the two universal constants Delta I_N and zeta_c.

Significance. The exact large-N solution and its FRG derivation are clean, self-contained, and carefully benchmarked; the numerical integration of the LPA flow is validated against this exact result to high precision, and the regulator dependence is small. If the two-scale hypothesis for finite N were independently established, the paper would provide a strong nonperturbative tool: the entire universal family of critical rate functions would be obtained from LPA, leaving only two global constants to be determined by other means. However, the finite-N evidence for this hypothesis is currently weakened by the fact that the two rescaling factors are fitted to the very Monte Carlo data that later certify the shape agreement, and the full rate-function family comparison is shown for N=2 only. The paper is honest about the 20-25% discrepancies in Delta I_N and zeta_c, and the residual shape agreement is genuine evidence, but the stronger conclusion about where the LPA error lives is an inference from a two-parameter fit rather than a tested property of the approximation.

major comments (3)

[Sec. IV B, Fig. 10] The central claim that 'the error induced by the LPA is primarily concentrated in the calculation of the two universal constants Delta I_N and zeta_c' is not supported by an independent test. The factors r_I(N) and r_zeta(N) are obtained by best-collapsing the FRG and MC curves of Delta I_N(zeta) in Fig. 9, i.e. they are fitted to the same Monte Carlo data that later certify the <1% agreement in Fig. 10. Table III shows that the two largest LPA errors are precisely the overall amplitude Delta I_N (about 20-25%) and the critical ratio zeta_c (about 25%), so the fit removes the two dominant discrepancies by construction. The residual 1% agreement in the rate-function shape is genuine evidence for the two-scale ansatz, but it tests that ansatz only under the assumption that no other LPA truncation error exists. The paper currently provides no out-of-sample check of this assumption: a next-order derivative-expansion result for Delta I_N or zeta_c is not given, the full zeta-family comparison in Fig. 10 is for N=2 only, and no train/test split of the MC data is made. I recommend either providing such an independent check (e.g., fixing r_I and r_zeta on a subset of zeta values or on N=3 and predicting the rest, or computing a next-order derivative-expansion estimate) or tempering the conclusion to state that the two-scale ansatz is consistent with, rather than established by, the present data.
[Sec. IV B, Fig. 10] The paper claims to compute 'the entire family of universal scaling functions' for finite N, but the full rate-function family comparison between FRG and MC is shown only for N=2 (Fig. 10). For N=3, only the zeta=0 rate function (Fig. 7) and the integrated quantity Delta I_N(zeta) (Fig. 9) are compared. Since the central claim is about the functional form of the whole family in zeta and rho, the N=3 case should either be shown with the same family plot or the claim should be restricted to N=2. This is particularly important because the fitted rescalings differ between N=2 and N=3, so the universality of the two-scale hypothesis across N is not yet demonstrated.
[Sec. IV B, Eq. (49)] The assessment of the 1% agreement in Sec. IV B is made after also normalizing the field variable by rho_0, the position of the minimum of the rate function at zeta=0. This normalization removes a third non-universal scale in addition to the two fitted factors r_I and r_zeta. The paper acknowledges that the field scale is non-universal, but the role of the rho_0 normalization should be made explicit in the error budget: the <1% relative error in Fig. 10 is a statement about the shape in the normalized variable rho/rho_0, not about the absolute field scale. I ask the authors to state clearly that three rescalings (r_I, r_zeta, rho_0) are used in the comparison, and to discuss how the conclusion would be affected if rho_0 were treated as an additional fitted parameter.

minor comments (6)

[Eq. (35)] Equation (58) appears to contain a typesetting error: the term should read (d-2) * check_rho * tilde_I'_k, but the tilde on I is missing on the right-hand side, making the equation dimensionally inconsistent as typeset.
[Fig. 5] In Eq. (35), the coefficient '2d' is ambiguous; it should be typeset as 2^d (or, for d=3, 8) divided by L^d (rho_0 - rho). As printed, '2d Ld(rho0 - rho)' could be misread as 2d times a logarithm or as a product, and the LaTeX is missing a fraction or parentheses.
[Sec. III C] The caption of Fig. 5 refers to 'Probability distributions P(check_rho)' but the text immediately below explains that the plotted quantity is actually <delta(s^2/2 - check_rho)>, which differs from P(s) by a Jacobian factor rho^{(N-2)/2}. The caption should be corrected to avoid confusion between the PDF and the histogram of the squared magnetization.
[Sec. IV, Table III] The statement that using the fixed-point potential as the initial condition 'ensures there will not be corrections to scaling' is only strictly true if the fixed point is exact. At LPA for finite N, U* is an approximate fixed point, and setting all irrelevant perturbations to zero at k_* does not by itself eliminate corrections to scaling from the truncation. The paper's regulator-dependence tests mitigate this concern, but the phrase is too strong as written.
[Sec. IV B] The MC values of zeta_c are quoted as 3.2 +/- 0.2 (N=2) and 2.7 +/- 0.2 (N=3). Because the fitted rescalings r_zeta(2) = 1.33 and r_zeta(3) = 1.12 are derived from the same data, the reader should be told explicitly in the table caption that the MC zeta_c values are not independent of the rescaling factors used in the shape comparison; otherwise the 25% discrepancy and the near-perfect collapse in Fig. 10 may appear contradictory.
[Sec. IV B] The sentence 'This has also been observed in the calculation of the rate function of the 3D Ising model using FRG [49] and perturbative RG [45]' is an important supporting reference, but since the present N=2 result is the main finite-N evidence in this paper, the phrasing could be sharpened to distinguish the N=1 and N=2 cases and to emphasize that the two-scale ansatz is an empirical observation at LPA for finite N, not a proven property of the exact theory.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the two fitted scales are transparent nuisance parameters; the <1% shape agreement is a genuine holdout comparison.

full rationale

The paper's central finite-N claim is that the LPA reproduces the functional form of the rate-function family once two non-universal scales are adjusted: the overall amplitude of the logarithm of the PDF (r_I) and the scale of zeta (r_zeta). These factors are explicitly fitted to the MC data of Delta I_N versus zeta, so the corrected values of Delta I_N and zeta_c are not predictions; however, the paper never presents them as predictions. The actual test is the full rate-function shape I(rho) at fixed zeta and its zeta-dependence after the global rescaling, shown in Fig. 10 for N=2. A global vertical rescaling and a global zeta-rescaling cannot manufacture agreement in the rho-dependence of I(rho) or in the way the whole family changes with zeta, so the reported <1% residual error is a genuine, non-tautological test of the shape ansatz. The large-N calculation is regulator-independent and exact, providing an independent internal benchmark for the numerical method. The self-citation to the Ising predecessor [49] is used only as background and as a supporting observation; the present paper's central comparison is carried out against external Monte Carlo data and the exact large-N limit. The only caveats are that the conclusion that LPA error is 'primarily concentrated' in the two constants is an inference from the success of the two-parameter fit, and the full-shape comparison is shown only for N=2; these are correctness/completeness concerns, not circularity.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central derivation rests on standard FRG and large-N techniques plus several methodological choices. The only fitted quantities are the two rescaling factors r_I and r_zeta, the correlation length amplitude xi_+,LPA, the regulator prefactor alpha, and the non-universal field scale rho_0. No new entities are introduced.

free parameters (5)

r_I(N), rate function amplitude rescaling = r_I(2) approx 0.79, r_I(3) approx 0.81
Fitted to best collapse between FRG-LPA and Monte Carlo curves of DeltaI_N versus zeta (Sec. IV B, Fig. 9); without this factor the LPA amplitude is off by about 20-25%.
r_zeta(N), correlation-length rescaling = r_zeta(2) approx 1.33, r_zeta(3) approx 1.12
Fitted to best collapse between FRG-LPA and Monte Carlo data; rescales zeta because the LPA correlation length amplitude differs from Monte Carlo.
xi_+,LPA, non-universal amplitude of correlation length = Table I: for N=2 theta 1.0444, exp 1.3029, Wett 1.4148; for N=3 theta 1.0458, exp 1.3794, Wett 1.5297
Fitted amplitude in Eq. (65) used to convert the perturbation delta_r to zeta via Eq. (67).
rho_0, non-universal scale of the field = Position of the minimum of the rate function at zeta=0, from Monte Carlo or FRG separately
Non-universal amplitude of rho_hat fixed by normalizing all curves to rho_0 (Sec. IV); this absorbs one unknown scale in the comparison.
alpha_opt, regulator prefactor = 1.0 (theta), 4.65 (exponential), 6.05 (Wetterich)
Regulator prefactor chosen by optimizing the LPA critical exponent nu (Sec. III C); results depend mildly on this choice.

assumptions (7)

domain assumption The O(N) model in 2<d<4 has a second-order transition and universal critical behavior described by the Wilson-Fisher fixed point.
Invoked in Sec. I and used to define the scaling form Eq. (3) and to justify flowing to a fixed point.
domain assumption The exact FRG flow equation Eq. (44) with regulator R_k is correct, and the LPA ansatz Eq. (49) is a valid truncation.
Standard FRG framework; no formal error bound at LPA is given.
standard math In the large N limit, the functional integral is dominated by a uniform saddle point for the auxiliary fields lambda and rho.
Used in Sec. II A and II B; standard large N argument, appropriate for N tending to infinity.
ad hoc to paper The fixed point potential U_star can be used as the initial condition at scale k_star, with all irrelevant perturbations set to zero.
Sec. III C; this assumes no corrections to scaling, which is checked only indirectly through the large N benchmark and regulator variation.
domain assumption Monte Carlo data at L=96 and L=128, with histogram reweighting and literature values of Tc and xi_+, are converged proxies for the exact universal rate function.
Sec. IV; L-convergence is shown for N=2 and N=3, but corrections to scaling are not quantified.
ad hoc to paper The LPA error is concentrated in the two universal constants DeltaI_N and zeta_c, leaving the functional form untouched.
Sec. IV B; this is the post hoc conclusion from the collapse after fitting r_I and r_zeta.
ad hoc to paper Choosing the regulator prefactor alpha that extremizes nu_LPA is a valid prescription for the LPA truncation.
Sec. III C; this is a heuristic based on convergence of the derivative expansion, with no proof at LPA.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Probability distributions of the order parameter of the $O(N)$ model." pith.science (2026). https://pith.science/paper/SVDEJ7KH

@misc{pith2026250104465,
  author       = {Pith},
  title        = {Pith review of: Probability distributions of the order parameter of the $O(N)$ model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SVDEJ7KH}},
  note         = {Machine review of arXiv:2501.04465}
}

abstract

We study the probability distribution function (PDF) of the order parameter of the three-dimensional $O(N)$ model at criticality using the functional renormalisation group. For this purpose, we generalize the method introduced in [Balog et al., Phys. Rev. Lett. {\bf 129}, 210602 (2022)] to the $O(N)$ model. We study the large $N$ limit, as well as the cases $N=2$ and $N=3$ at the level of the Local Potential Approximation (LPA), and compare our results to Monte Carlo simulations. We compute the entire family of universal scaling functions, obtained in the limit where the system size $L$ and the correlation length of the infinite system $\xi_\infty$ diverge, with the ratio $\zeta=L/\xi_\infty$ constant. We also generalize our results to the approach of criticality from the low-temperature phase where another infinite family of universal PDF exists. We find that the LPA describes very well the functional form of the family of PDFs, once we correct for a global amplitude of the (logarithm of the) PDF and of $\zeta$.

Figures

Figures reproduced from arXiv: 2501.04465 by the authors.

**Figure 2.** FIG. 2. Rate function in the large [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Rate functions at [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4. Relative error of the rate function obtained from numerically solving LPA flow equation in the large [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: FIG. 5. Probability distributions [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: FIG. 6. Rate functions of the [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: FIG. 7. Rate functions of the [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8. Universal amplitude ∆ [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: FIG. 9. Universal amplitude ∆ [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: FIG. 10. Rate functions as functions of ˇρ [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: FIG. 11. Rate functions for different [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Journey from the Wilson exact RG towards the Wegner-Morris Fokker-Planck RG and the Carosso field-coarsening via Langevin stochastic processes
cond-mat.stat-mech 2025-02 conditional novelty 5.0 of 10

Stochastic RG flows on a finite volume with frozen empirical magnetization yield formal Fokker-Planck equations for the magnetization's large-deviation rate function, but no new rate function is computed.

Reference graph

Works this paper leans on

67 extracted references · 62 canonical work pages · cited by 1 Pith paper

[1]

Fukuda and E

R. Fukuda and E. Kyriakopoulos, Nuclear Physics B 85, 354 (1975)

work page 1975
[2]

O’Raifeartaigh, A

L. O’Raifeartaigh, A. Wipf, and H. Yoneyama, Nuclear Physics B 271, 653 (1986)

work page 1986
[3]

Mukaida and Y

H. Mukaida and Y. Shimada, Nuclear Physics B 479, 663 (1996)

work page 1996
[4]

Bouchaud and A

J.-P. Bouchaud and A. Georges, Physics Reports 195, 127 (1990)

work page 1990
[5]

Privman and M

V. Privman and M. E. Fisher, Phys. Rev. B 30, 322 (1984)

work page 1984
[6]

Binder, Phys

K. Binder, Phys. Rev. Lett. 47, 693 (1981)

1981
[7]

A. D. Bruce and N. B. Wilding, Phys. Rev. Lett. 68, 193 (1992)

work page 1992
[8]

M. M. Tsypin, Phys. Rev. Lett. 73, 2015 (1994)

work page 1994

Show all 67 references

[9]

M. M. Tsypin and H. W. J. Bl¨ ote, Phys. Rev. E 62, 73 (2000)

2000
[10]

Hilfer, B

R. Hilfer, B. Biswal, H. G. Mattutis, and W. Janke, Phys. Rev. E 68, 046123 (2003)

2003
[11]

Hilfer, B

R. Hilfer, B. Biswal, H.-G. Mattutis, and W. Janke, Computer Physics Communications 169, 230 (2005)

2005
[12]

Malakis and N

A. Malakis and N. G. Fytas, Physica A 365, 197 (2006)

2006
[13]

J. Xu, A. M. Ferrenberg, and D. P. Landau, Phys. Rev. E 101, 023315 (2020)

2020
[14]

Hasenfratz, T

A. Hasenfratz, T. Neuhaus, K. Jansen, H. Yoneyama, and C. B. Lang, Physics Letters B 199, 531 (1987)

1987
[15]

Kuti and Y

J. Kuti and Y. Shen, Physical Review Letters 60, 85 (1988)

1988
[16]

Hasenfratz, K

A. Hasenfratz, K. Jansen, J. Jersa ´k, C. Lang, T. Neuhaus, and H. Yoneyama, Nuclear Physics B 317, 81 (1989)

1989
[17]

G¨ ockeler and H

M. G¨ ockeler and H. Leutwyler, Nuclear Physics B361, 392 (1991)

1991
[18]

G¨ ockeler and H

M. G¨ ockeler and H. Leutwyler, Physics Letters B253, 193 (1991)

1991
[19]

G¨ ockeler and H

M. G¨ ockeler and H. Leutwyler, Nuclear Physics B350, 228 (1991)

1991
[20]

Dimitrovi´ c, J

I. Dimitrovi´ c, J. Nager, K. Jansen, and T. Neuhaus, Physics Letters B 268, 408 (1991)

1991
[21]

G¨ ockeler, H

M. G¨ ockeler, H. A. Kastrup, T. Neuhaus, and F. Zimmermann, Nuclear Physics B 404, 517 (1993). 20

1993
[22]

S. T. Bramwell, P. C. W. Holdsworth, and J. F. Pinton, Nature (London) 396, 552 (1998)

1998
[23]

S. T. Bramwell, K. Christensen, J. Y. Fortin, P. C. W. Holdsworth, H. J. Jensen, S. Lise, J. M. L´ opez, M. Nicodemi, J. F. Pinton, and M. Sellitto, Phys. Rev. Lett. 84, 3744 (2000)

2000
[24]

Portelli and P

B. Portelli and P. C. W. Holdsworth, Journal of Physics A Mathematical General 35, 1231 (2002)

2002
[25]

Archambault, S

P. Archambault, S. T. Bramwell, and P. C. W. Holdsworth, J. Phys. A 30, 8363 (1997)

1997
[26]

Joubaud, A

S. Joubaud, A. Petrosyan, S. Ciliberto, and N. B. Garnier, Phys. Rev. Lett. 100, 180601 (2008)

2008
[27]

K. G. Wilson and J. B. Kogut, Phys. Rept. 12, 75 (1974)

1974
[28]

P. M. Bleher and J. G. Sinai, Communications in Mathematical Physics 33, 23 (1973)

1973
[29]

P. M. Bleher and Y. G. Sinai, Communications in Mathematical Physics 45, 247 (1975)

1975
[30]

P. M. Bleher and P. Major, The Annals of Probability 15, 431 (1987)

1987
[31]

P. M. Bleher and P. Major, Annales de l’I.H.P. Physique th´ eorique49, 7 (1988)

1988
[32]

P. M. Bleher and P. Major, Communications in Mathematical Physics 125, 43 (1989)

1989
[33]

Jona-Lasinio, Il Nuovo Cimento B (1971-1996) 26, 99 (1975)

G. Jona-Lasinio, Il Nuovo Cimento B (1971-1996) 26, 99 (1975)

1975
[34]

Cassandro and G

M. Cassandro and G. Jona-Lasinio, Advances in Physics 27, 913 (1978)

1978
[35]

Jona-Lasinio, Physics Reports 352, 439 (2001)

G. Jona-Lasinio, Physics Reports 352, 439 (2001)

2001
[36]

A. D. Bruce, T. Schneider, and E. Stoll, Phys. Rev. Lett. 43, 1284 (1979)

1979
[37]

Rudnick, H

J. Rudnick, H. Guo, and D. Jasnow, J. Stat. Phys. 41, 353 (1985)

1985
[38]

Eisenriegler and R

E. Eisenriegler and R. Tomaschitz, Phys. Rev. B 35, 4876 (1987)

1987
[39]

Hilfer, International Journal of Modern Physics B 07, 4371 (1993)

R. Hilfer, International Journal of Modern Physics B 07, 4371 (1993)

1993
[40]

Hilfer and N

R. Hilfer and N. B. Wilding, Journal of Physics A: Mathematical and General 28, L281 (1995)

1995
[41]

Esser, V

A. Esser, V. Dohm, and X. Chen, Physica A: Statistical Mechanics and its Applications 222, 355 (1995)

1995
[42]

X. S. Chen, V. Dohm, and N. Schultka, Phys. Rev. Lett. 77, 3641 (1996)

1996
[43]

X. S. Chen and V. Dohm, International Journal of Modern Physics B 12, 1277 (1998)

1998
[44]

Rudnick, W

J. Rudnick, W. Lay, and D. Jasnow, Phys. Rev. E 58, 2902 (1998)

1998
[45]

S. Sahu, B. Delamotte, and A. Ran¸ con, (2024), arXiv:2407.12603

2024 arXiv
[46]

Berges, N

J. Berges, N. Tetradis, and C. Wetterich, Physics Reports 363, 223 (2002)

2002
[47]

Dupuis, L

N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. Pawlowski, M. Tissier, and N. Wschebor, Physics Reports 910, 1 (2021)

2021
[48]

Moshe and J

M. Moshe and J. Zinn-Justin, Physics Reports 385, 69 (2003)

2003
[49]

Balog, A

I. Balog, A. Ran¸ con, and B. Delamotte, Phys. Rev. Lett. 129, 210602 (2022)

2022
[50]

Universal and non-universal large deviations in critical systems,

I. Balog, B. Delamotte, and A. Ran¸ con, “Universal and non-universal large deviations in critical systems,” (2024), arXiv:2409.01250

2024 arXiv
[51]

Pel´ aez and N

M. Pel´ aez and N. Wschebor, Phys. Rev. E94, 042136 (2016)

2016
[52]

Wetterich, Physics Letters B 301, 90 (1993)

C. Wetterich, Physics Letters B 301, 90 (1993)

1993
[53]

Wetterich, Zeitschrift f¨ ur Physik C60, 461 (1993)

C. Wetterich, Zeitschrift f¨ ur Physik C60, 461 (1993)

1993
[54]

Wetterich, Zeitschrift f¨ ur Physik C Particles and Fields57, 451 (1993)

C. Wetterich, Zeitschrift f¨ ur Physik C Particles and Fields57, 451 (1993)

1993
[55]

Fister and J

L. Fister and J. M. Pawlowski, Phys. Rev. D 92, 076009 (2015)

2015
[56]

Balog, H

I. Balog, H. Chat´ e, B. Delamotte, M. Marohni´ c, and N. Wschebor, Physical Review Letters123, 240604 (2019)

2019
[57]

De Polsi and N

G. De Polsi and N. Wschebor, Phys. Rev. E 106, 024111 (2022)

2022
[58]

De Polsi, I

G. De Polsi, I. Balog, M. Tissier, and N. Wschebor, Phys. Rev. E 101, 042113 (2020)

2020
[59]

F. Rose, I. Balog, A. Ran¸ con, to be published
[60]

Tetradis and D

N. Tetradis and D. Litim, Nuclear Physics B 464, 492 (1996)

1996
[61]

Knorr, Journal of Physics A: Mathematical and Theoretical 54, 275401 (2021)

B. Knorr, Journal of Physics A: Mathematical and Theoretical 54, 275401 (2021)

2021
[62]

Wolff, Phys

U. Wolff, Phys. Rev. Lett. 62, 361 (1989)

1989
[63]

Newman and G

M. Newman and G. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon Press, 1999)

1999
[64]

Butera and M

P. Butera and M. Comi, Phys. Rev. B 58, 11552 (1998)

1998
[65]

Hasenbusch, Journal of Statistical Mechanics: Theory and Experiment 2008, P12006 (2008)

M. Hasenbusch, Journal of Statistical Mechanics: Theory and Experiment 2008, P12006 (2008)

2008
[66]

Y. Deng, H. W. J. Bl¨ ote, and M. P. Nightingale, Phys. Rev. E 72, 016128 (2005)

2005
[67]

Kanaya and S

K. Kanaya and S. Kaya, Phys. Rev. D 51, 2404 (1995)

1995

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Fukuda and E

R. Fukuda and E. Kyriakopoulos, Nuclear Physics B 85, 354 (1975)

work page 1975

[2] [2]

O’Raifeartaigh, A

L. O’Raifeartaigh, A. Wipf, and H. Yoneyama, Nuclear Physics B 271, 653 (1986)

work page 1986

[3] [3]

Mukaida and Y

H. Mukaida and Y. Shimada, Nuclear Physics B 479, 663 (1996)

work page 1996

[4] [4]

Bouchaud and A

J.-P. Bouchaud and A. Georges, Physics Reports 195, 127 (1990)

work page 1990

[5] [5]

Privman and M

V. Privman and M. E. Fisher, Phys. Rev. B 30, 322 (1984)

work page 1984

[6] [6]

Binder, Phys

K. Binder, Phys. Rev. Lett. 47, 693 (1981)

1981

[7] [7]

A. D. Bruce and N. B. Wilding, Phys. Rev. Lett. 68, 193 (1992)

work page 1992

[8] [8]

M. M. Tsypin, Phys. Rev. Lett. 73, 2015 (1994)

work page 1994