Pith. sign in

REVIEW 3 major objections 4 minor 55 references

Long range to short range crossover in one dimension

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Numerical evidence places the 1D long-range to short-range crossover of Lévy self-avoiding walks at σ*=1, in support of Sak's scenario.

desk verdict New branching algorithm gives real data on 1D Lévy SAW, but the finite-size scaling evidence for Sak is overclaimed and internally inconsistent. read the letter →

arxiv 2507.08092 v1 pith:T6DIV4IR submitted 2025-07-10 cond-mat.stat-mech

Mrinal Sarkar , Nicolò Defenu , Tilman Enss This is my paper
classification cond-mat.stat-mech MSC 82B2782B4182B80 PACS 05.40.Fb64.60.Fr
keywords long-rangeinteractionsself-avoidingLévyflightsO(n)modeln→0SakscenarioLR–SRcrossovercriticalexponentsMonteCarlosimulationsfinite-sizescaling
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 tries to settle where long-range interactions stop controlling critical behavior in one-dimensional self-avoiding Lévy flights, which realize the $n \to 0$ limit of the $O(n)$ model. It claims the crossover happens at decay exponent $\sigma^* = 1$, following Sak: for $\sigma < 1$ the anomalous dimension is $\eta = 2 - \sigma$, and for $\sigma > 1$ it crosses continuously to the short-range value $\eta = 1$. This matters because earlier simulations reported a smooth crossover near $\sigma = 2$ and Flory-type scaling predicted different exponents. The paper's branching Monte Carlo algorithm reaches much longer walks and, after correcting finite-size effects, yields exponents that favor Sak over Fisher and Flory. If correct, the result fixes the LR–SR boundary in a model one-dimensional system and explains why effective-dimensionality arguments fail there.

What carries the argument

The central object is the self-avoiding Lévy flight on an infinite one-dimensional lattice, with step-length distribution $P(r) \sim r^{-(1+\sigma)}$, which maps to the $O(n \to 0)$ model through de Gennes' high-temperature expansion. To reach the asymptotic scaling regime, the authors use a branching SAW algorithm in which each node attempts additional steps with probability $z-1$, improving statistics for long walks. Critical exponents are extracted by extrapolating data to the critical fugacity $z_c$, applying a tail correction to $G(R)$, and then fitting $\gamma(N)$ and $\nu(N)$ as functions of inverse walk length with the correction form $\nu(N) = \nu(\infty) + bN^{-(\sigma-\sigma^*)}$. This correction exponent comes from a functional-RG flow equation for the ratio $J_\sigma$ of long-range to short-range couplings, whose scaling dimension is $\Delta = \sigma - \sigma^*$.

What would settle it

Simulate Lévy-SAW in d=1 with the same branching algorithm but push the maximum walk length to $10^{6}$ and study σ=1.5: the Sak claim requires γ(N) to extrapolate to 1 with a correction exponent near 0.5, whereas a crossover at σ=2 would keep γ visibly above 1 at the largest accessible N.

Watch

Extended reading notes

Core claim

The central claim is that the long-range to short-range crossover in the one-dimensional $O(n \to 0)$ model occurs at $\sigma^* = 1$, with $\eta = 2 - \sigma$ in the long-range regime $\sigma < 1$ and $\eta = 1$ in the short-range regime $\sigma > 1$, plus a weak logarithmic correction exactly at $\sigma = 1$. The susceptibility exponent $\gamma$ follows the hyperscaling relation $\gamma = (2 - \eta)\nu$ in the long-range regime, so $\gamma = \sigma\nu_{\mathrm{LR}}$ below $\sigma^*=1$ and $\gamma = 1$ above it. The authors report that $\gamma$ at $\sigma=1.5$ extrapolates to 1 with a correction exponent $\omega \approx \sigma - \sigma^*$, which they take as direct evidence against Fisher's crossover at $\sigma=2$ and against Flory-type predictions. In the long-range regime, measured exponents lie slightly below Flory's values, and the effective dimension approach holds only approximately because the short-range anomalous dimension $\eta_{\mathrm{SR}}=1$ is large in $d=1$.

Load-bearing premise

The extrapolation to asymptotic exponents assumes the functional-RG flow equations for O(n) models, derived for d≥2, still hold at n=0 and d=1, and that every observable follows the single correction form ν(N)=ν∞+$bN^{{-(σ−σ*)}}$ over the simulated range of $10^{3}$–$10^{4}$ steps.

Editorial extensions

If this is right

  • The LR–SR boundary in 1D $O(n\to0)$ is fixed at $\sigma^*=1$, so for $\sigma<1$ the LR universality class extends down to the mean-field boundary $\sigma=d/2=1/2$.
  • Hyperscaling $\gamma=(2-\eta)\nu$ holds across the LR regime, allowing $\gamma$ to be predicted from $\nu$ measurements.
  • Effective-dimension reasoning should not be used quantitatively in $d=1$ LR systems; deviations grow with $\eta_{\mathrm{SR}}$.
  • At $\sigma=1$, logarithmic corrections make apparent exponents converge slowly, so future studies should expect strong finite-size effects near the crossover.

Reading between the lines

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

  • The same branching algorithm could be adapted to measure the correction exponent $\omega = \sigma - \sigma^*$ in the 2D LR Ising model, where Sak's scenario has independent support, providing a cross-check on universality.
  • A re-analysis of Grassberger's published finite-walk data with the correction $N^{-(\sigma-1)}$ could reconcile the two numerical studies without new simulations.
  • If the correction exponent is universal, it should also appear in dynamical quantities such as relaxation times of LR spin systems near the crossover.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper uses Monte Carlo simulations of self-avoiding Lévy flights on a one-dimensional lattice to study the long-range to short-range crossover in the O(n→0) model, measuring the anomalous dimension η, the correlation length exponent ν, and the susceptibility exponent γ as functions of the long-range decay parameter σ. The central claim is that the data support Sak's scenario with a crossover at σ* = 1, meaning η = 2 − σ for σ < 1 and η = 1 for σ > 1, with γ approaching 1 as σ crosses into the short-range regime. The paper further claims that finite-size corrections to γ and ν behave as N^(−(σ−σ*)) and that this 'convincingly excludes' a crossover at σ = 2. The evidence is obtained from branching SAW simulations extending to walk lengths of 10^3–10^4 steps, with extrapolation to the asymptotic limit using correction-to-scaling ansätze.

Significance. If the central claim is correct, the paper resolves a long-standing controversy about the location and nature of the LR–SR crossover in the one-dimensional O(n→0) model, favoring Sak's scenario over Fisher's and Flory's alternatives. The branching SAW algorithm is a genuine methodological contribution that improves statistics for long walks, and the paper makes explicit falsifiable predictions for η, γ, and ν. However, the quantitative support for the crossover location currently rests on a finite-size correction exponent that is internally inconsistent across observables and is extracted under assumptions that effectively presuppose the conclusion. The qualitative trend of η(σ) is suggestive, but the 'strong evidence' claim is not yet established.

major comments (3)
  1. [Fig. 3(a) and SM Fig. S2(a)] At σ = 1.5, the manuscript quotes the finite-size correction exponent as ω = 0.533(1) from γ (main text) and ω = 0.417(1) from ν (Supplemental Material). These two values are mutually inconsistent by more than one hundred times the quoted statistical error, and they deviate from the predicted σ − σ* = 0.5 by 0.033 and 0.083, respectively. Since the same universal correction exponent is presented as the basis for the claim that the data 'convincingly exclude' a crossover at σ = 2, this internal inconsistency removes the quantitative support for Sak's scenario; the authors should either refit both observables with a common ω or explain why different correction exponents should appear for different observables.
  2. [Fig. 3(a), SM Fig. S2(a), and SM Eq. (S16)] The power-law fits in Fig. 3(a) and SM Fig. S2(a) fix the asymptotic exponents to Sak's values, γ(∞) = 1 and ν(∞) = 1, before extracting the correction exponent ω. The question under investigation is precisely whether the true asymptotes are Sak's values or, for example, Flory's values, so fixing the asymptote removes the discriminator between scenarios. The fitted quantity γ(N) − γ(∞) is the deviation from the very value whose validity is at issue, and therefore the extracted ω cannot by itself rule out a crossover at σ = 2. The authors should perform fits with free asymptotic exponents and compare the goodness of fit against the Sak and Flory predictions.
  3. [SM Eqs. (S10)–(S16)] The 'field-theoretic prediction' of the correction exponent Ω = σ − σ* rests on an explicit assumption stated in the SM: 'Assuming that Eqs. (S10), (S11), (S12) hold for n = 0 and d = 1.' These fRG flow equations were derived for O(n) models in d ≥ 2 (with n = 1 in the cited reference), so their validity at n = 0 and d = 1 is a nontrivial extrapolation, not a derivation. Moreover, the extension of the correction form to γ is justified only by the statement 'We expect similar finite-size corrections to apply to all thermodynamic observables.' Therefore the main-text assertion that 'Field theory predicts a finite-size correction to the scaling ∼ N^(−(σ−σ*))' overstates the support, and the numerical evidence for ω is less independent of the hypothesis being tested than the presentation suggests.
minor comments (4)
  1. [Main text, Results (σ = 1) vs. SM Fig. S1] The main text reports η = 1.011^{+0.042}_{-0.062} at σ = 1, while the SM log-correction fit reports η = 1.048(1); the two values should be reconciled or the difference explicitly explained.
  2. [SM Eq. (S15)–(S16)] The step from Eq. (S15) to Eq. (S16) sets b = a/log N_∞, but log N is not constant over the fitted N range; this approximation should be stated and its effect on the quoted ω values assessed.
  3. [Fig. 2] The hyperscaling check uses the same Monte Carlo data for γ and ν, so it is a consistency check rather than an independent validation; the text should state this explicitly and avoid presenting it as a separate confirmation.
  4. [Introduction and concluding paragraph, Ref. [38]] The description of Grassberger's result as not in agreement with Sak's prediction should be checked for precision: the concluding sentence refers to 'a smooth dependence of γ on σ up to σ = 2', but Ref. [38] is primarily about ν, so the distinction between γ and ν claims should be clarified.

Circularity Check

2 steps flagged · score 6.0 of 10

Partial circularity: the central correction-exponent evidence for Sak is imported from the authors' own fRG paper and from fits that fix the Sak asymptote, while the direct η measurement remains independent.

  1. self citation load bearing [Supplemental Material, 'Finite size correction to SAW critical exponents', Eqs. (S10)–(S14); used in main text Fig. 3 and SM Fig. S2.]
    "Defenu et. al.[S24] studied the O(n) model with LR interaction... Assuming that Eqs. (S10), (S11), (S12) hold for n=0 and d=1, and identifying σ∗ = 2−η2, one obtains from the evolution equation for Jσ in the regime σ>σ∗, ∂t ¯Jσ = (σ−σ∗) ¯Jσ... This shows that ∆ = σ−σ∗ is the scaling dimension associated with Jσ."

    The quantitative prediction used as 'strong evidence' for Sak—that finite-size corrections scale with exponent σ−σ*—is derived by citing [S24], whose author overlap includes present author Defenu. The cited derivation itself identifies σ* as 2−η_SR, which is precisely Sak's criterion. The SM then applies this self-cited result at n=0 and d=1 by assumption, and the main text converts it into an apparent confirmation that the crossover is at σ*=1. Thus the correction-exponent evidence reduces to the prior self-citation that already contains the target conclusion, rather than being an independent test.

  2. fitted input called prediction [Main text, 'Results', Fig. 3(a) caption and surrounding text; SM 'Behavior of the SAW critical exponent ν', Fig. S2(a).]
    "With γ(∞)=1, a fitting of the data using γ(N)−γ(∞)=bN^{−ω}, yields ω=0.533(1)≈σ−σ∗. This provides strong evidence in favor of the Sak scenario and convincingly excludes the crossover at σ=2. ... With νLR(∞)=1, a fitting of the data using a power-law νLR(N)−νLR(∞)=bN^{−ω} to incorporate finite-size correction to the scaling, yields ω=0.417(1)..."

    The fits fix the asymptotic exponents to their Sak values, γ(∞)=1 and ν_LR(∞)=1, before extracting the correction exponent ω. Whether the asymptotic value is 1 (Sak) or the Flory value is exactly the point at issue in the Fisher/Flory versus Sak debate. Fixing the Sak asymptote removes the discriminator, so the resulting ω is a fitted quantity rather than an independent prediction. The subsequent agreement of ω with σ−σ*, where σ* itself comes from the same Sak picture, is therefore a consistency check on an assumed input, not evidence that convincingly excludes a crossover at σ=2.

full rationale

The paper contains two genuinely independent, non-circular components. First, the direct measurement of η from the two-point correlation function G(R) at criticality is compared with the Sak curve η=2−σ for σ<1 and η=1 for σ>1 without fitting those asymptotes; this is an external benchmark and supports Sak over Fisher qualitatively. Second, the hyperscaling check in Fig. 2 uses γ and ν measured from the same simulations but from different observables (c_N and <log R_N>), so it is a consistency test rather than a circular prediction. However, the quantitative claim advertised as 'strong evidence' and 'convincingly excludes the crossover at σ=2' is not independent. The finite-size correction exponent ω≈σ−σ* is imported from the authors' own fRG work [S24] (self-citation) after assuming it holds at n=0 and d=1, and the fits in Fig. 3(a) and SM Fig. S2(a) fix the asymptotic exponents to the Sak values before extracting ω. The correction-exponent argument thus reduces by construction to the Sak input, although the η data alone would still give partial, weaker support. An additional correctness concern, noted by the skeptic, is that ω_γ=0.533(1) and ω_ν=0.417(1) at the same σ=1.5 differ from each other and from σ−σ*=0.5 by much more than the quoted statistical errors; this internal inconsistency is not itself circularity, but it further undermines the weight placed on the correction-exponent evidence. Overall, because the principal independent measurement (η) is externally benchmarked and not circular, the paper is not fully circular; but because the central 'strong evidence' for Sak reduces to a self-cited assumption and to fits that fix the target asymptote, a score of 6 is appropriate.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim relies on standard SAW-to-O(n) mappings, the assumed validity of hyperscaling, and an extrapolation of the authors' earlier fRG results to n=0, d=1. The fitted correction exponents and log-correction parameters are used to support the crossover location but are not independently constrained by external data.

free parameters (4)
  • correction exponent omega at sigma=1.5 (gamma fit) = 0.533(1)
    Fitted from gamma(N)-gamma(infinity)=b N^{-omega} with gamma(infinity)=1; compared to predicted sigma-sigma*=0.5.
  • correction exponent omega at sigma=1.5 (nu fit) = 0.417(1)
    Fitted from nu(N)-nu(infinity)=b N^{-omega} with nu(infinity)=1; deviates from predicted 0.5 by about 17 percent.
  • log-correction amplitude a at sigma=1 = 11.1(2)
    Fit parameter in G(R)=R^{1-eta} a/(b+log R); non-universal amplitude.
  • log-correction scale b at sigma=1 = 11.4(2)
    Fit parameter in G(R)=R^{1-eta} a/(b+log R); non-universal scale.
assumptions (5)
  • standard math The O(n to 0) model maps to self-avoiding walks (de Gennes)
    Standard high-temperature expansion; loop contributions vanish as n to 0.
  • domain assumption Levy-SAW jump distribution P(r) approximately r^{-(1+sigma)} captures the LR O(n) interaction
    The long-range spin coupling J_ij approximately |r_i-r_j|^{-(d+sigma)} maps to jump probability with the same sigma; the power-law tail is the relevant part.
  • domain assumption Hyperscaling gamma=(2-eta)nu holds for Levy-SAW
    Assumed to derive Eq. (5) and to convert gamma/nu into eta; standard for SR critical phenomena but not derived here.
  • ad hoc to paper fRG flow equations (S10)-(S12) for O(n) in d greater than or equal to 2 remain valid for n=0 and d=1
    SM Section 3: 'Assuming that Eqs. (S10)-(S12) hold for n=0 and d=1'; used to derive correction exponent Delta=sigma-sigma*.
  • ad hoc to paper Finite-size correction ansatz xi(N)=N^{nu_infinity}(1+a N^{-Delta}+...)
    SM Eqs. (S15)-(S16); assumed to describe all observables in both LR and SR regimes, without a dedicated derivation for Levy-SAW.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Long range to short range crossover in one dimension." pith.science (2026). https://pith.science/paper/T6DIV4IR

@misc{pith2026250708092,
  author       = {Pith},
  title        = {Pith review of: Long range to short range crossover in one dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6DIV4IR}},
  note         = {Machine review of arXiv:2507.08092}
}
read the original abstract

This work investigates the critical behavior of one-dimensional systems with long-range (LR) interactions, focusing on the crossover to short-range (SR) universality. Through large-scale Monte Carlo simulations of self-avoiding L\'evy flights on a 1D lattice, we compute the anomalous dimension \eta, the correlation length exponent \nu, and the susceptibility exponent \gamma across a wide range of LR decay parameters \sigma. Our results provide strong numerical evidence that supports Sak's scenario. They identify the crossover at \sigma^* = 1 and demonstrate the continuity of critical exponents across this point, with strong corrections to scaling. The study also reveals deviations from Flory-type scaling predictions and discusses the limitations of effective dimension approaches in general. These findings clarify the nature of the LR-SR crossover in low-dimensional systems and open avenues for exploring criticality in disordered and complex networks.

Figures

Figures reproduced from arXiv: 2507.08092 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Critical exponent [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Critical exponent [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

55 extracted references · 48 canonical work pages

  1. [1]

    This trend is more evident atσ= 1.3and1.4, and byσ= 1.5,ηaligns with ηSR, which supports the Sak scenario over Fisher’s

    Atσ= 1.1and1.2, large error bars include both Fisher and Sak predictions, although atσ= 1.2, our MC estimate clearly deviates from Fisher toward Sak. This trend is more evident atσ= 1.3and1.4, and byσ= 1.5,ηaligns with ηSR, which supports the Sak scenario over Fisher’s. Large un- certainties appear atσ≳1, where our maximum walk lengths are beyond LR scali...

    work page

  2. [2]

    Campa, T

    A. Campa, T. Dauxois, and S. Ruffo, Physics Reports480, 57 (2009)

    work page 2009

  3. [3]

    Defenu, T

    N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo, and A. Trombettoni, Reviews of Modern Physics95, 035002 (2023)

    work page 2023

  4. [4]

    Häffner, C

    H. Häffner, C. F. Roos, and R. Blatt, Physics Reports469, 155 (2008)

    work page 2008

  5. [5]

    Saffman, T

    M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys.82, 2313 (2010)

    2010

  6. [6]

    Ritsch, P

    H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Reviews of Modern Physics85, 553 (2013)

    2013

  7. [7]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner,et al., Nature551, 579 (2017)

    2017

  8. [8]

    Monroe, W

    C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V . Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano,et al., Reviews of Modern Physics93, 025001 (2021)

    work page 2021

Show all 55 references
  1. [9]

    Defenu, A

    N. Defenu, A. Lerose, and S. Pappalardi, Physics Reports1074, 1–92 (2024)

    2024

  2. [10]

    F. J. Dyson, Communications in Mathematical Physics12, 91 (1969)

    1969

  3. [11]

    Thouless, Physical Review187, 732 (1969)

    D. Thouless, Physical Review187, 732 (1969)

    1969

  4. [12]

    J. M. Kosterlitz, Physical Review Letters37, 1577 (1976)

    1976

  5. [13]

    C. Chen, G. Bornet, M. Bintz, G. Emperauger, L. Leclerc, V . S. Liu, P. Scholl, D. Barredo, J. Hauschild, S. Chatterjee, M. Schuler, A. M. Läuchli, M. P. Zaletel, T. Lahaye, N. Y . Yao, and A. Browaeys, Nature616, 691–695 (2023)

    2023

  6. [14]

    L. Feng, O. Katz, C. Haack, M. Maghrebi, A. V . Gorshkov, Z. Gong, M. Cetina, and C. Monroe, Nature623, 713–717 (2023)

    2023

  7. [15]

    Defenu, A

    N. Defenu, A. Codello, S. Ruffo, and A. Trombettoni, Journal of Physics A: Mathematical and Theoretical53, 143001 (2020)

    2020

  8. [16]

    M. E. Fisher, S.-k. Ma, and B. Nickel, Physical Review Letters 29, 917 (1972)

    1972

  9. [17]

    Sak, Physical Review B8, 281 (1973)

    J. Sak, Physical Review B8, 281 (1973)

    1973

  10. [18]

    A. C. van Enter, Physical Review B26, 1336 (1982)

    1982

  11. [19]

    Yamazaki, Physics Letters A61, 207 (1977)

    Y . Yamazaki, Physics Letters A61, 207 (1977)

    1977

  12. [20]

    Yamazaki, Physica A: Statistical Mechanics and its Applica- tions92, 446 (1978)

    Y . Yamazaki, Physica A: Statistical Mechanics and its Applica- tions92, 446 (1978)

    1978

  13. [21]

    Gusmao and W

    M. Gusmao and W. Theumann, Physical Review B28, 6545 (1983)

    1983

  14. [22]

    Luijten and H

    E. Luijten and H. W. Blöte, Physical Review Letters89, 025703 (2002)

    2002

  15. [23]

    M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, Physical Re- view E89, 062120 (2014)

    2014

  16. [24]

    Brezin, G

    E. Brezin, G. Parisi, and F. Ricci-Tersenghi, Journal of Statisti- cal Physics157, 855 (2014)

    2014

  17. [25]

    Defenu, A

    N. Defenu, A. Trombettoni, and A. Codello, Physical Review E 92, 052113 (2015)

    2015

  18. [26]

    Picco, arXiv preprint arXiv:1207.1018 (2012)

    M. Picco, arXiv preprint arXiv:1207.1018 (2012)

    2012 arXiv

  19. [27]

    Blanchard, M

    T. Blanchard, M. Picco, and M. Rajabpour, Europhysics Letters 101, 56003 (2013)

    2013

  20. [28]

    Behan, L

    C. Behan, L. Rastelli, S. Rychkov, and B. Zan, Physical Review Letters118, 241601 (2017)

    2017

  21. [29]

    Behan, L

    C. Behan, L. Rastelli, S. Rychkov, and B. Zan, Journal of Physics A: Mathematical and Theoretical50, 354002 (2017)

    2017

  22. [30]

    Benedetti, E

    D. Benedetti, E. Lauria, D. Mazá ˇc, and P. van Vliet, Physical Review Letters134, 201602 (2025)

    2025

  23. [31]

    Giachetti, N

    G. Giachetti, N. Defenu, S. Ruffo, and A. Trombettoni, Physical Review Letters127, 156801 (2021)

    2021

  24. [32]

    T. Xiao, D. Yao, C. Zhang, Z. Fan, and Y . Deng, arXiv preprint arXiv:2404.08498 (2024)

    2024

  25. [33]

    G. Gori, M. Michelangeli, N. Defenu, and A. Trombettoni, Physical Review E96, 012108 (2017)

    2017

  26. [34]

    De Gennes,Scaling concepts in polymer physics(Cornell University Press, 1979)

    P.-G. De Gennes,Scaling concepts in polymer physics(Cornell University Press, 1979)

    1979

  27. [35]

    R. J. Baxter,Exactly solved models in statistical mechanics(El- sevier, 2016)

    2016

  28. [36]

    Slade, Proceedings of the Royal Society A475, 20180549 (2019)

    G. Slade, Proceedings of the Royal Society A475, 20180549 (2019)

    2019

  29. [37]

    de Gennes, Physics Letters A38, 339 (1972)

    P.-G. de Gennes, Physics Letters A38, 339 (1972)

    1972

  30. [38]

    Slade, Communications in Mathematical Physics358, 343 (2018)

    G. Slade, Communications in Mathematical Physics358, 343 (2018)

    2018

  31. [39]

    Grassberger, Journal of Physics A: Mathematical and General 18, L463 (1985)

    P. Grassberger, Journal of Physics A: Mathematical and General 18, L463 (1985)

    1985

  32. [40]

    However, since the jump(length) distribution in Lévy-SAW follows a heavy- tailed power-law,⟨R 2 N ⟩becomes ill-defined

    Note that in short-range SAWs (typically defined on a lattice with nearest-neighbor jumps), the mean squared end-to-end dis- tance is expected to scale as⟨R 2 N ⟩ ∼N2νSR. However, since the jump(length) distribution in Lévy-SAW follows a heavy- tailed power-law,⟨R 2 N ⟩becomes...

  33. [41]

    Madras and G

    N. Madras and G. Slade,The self-avoiding walk(Springer Sci- ence & Business Media, 2013)

    2013

  34. [42]

    Halley and H

    J. Halley and H. Nakanishi, Phys. Rev. Lett.55, 551 (1985)

    1985

  35. [43]

    A. D. Sokal, arXiv preprint hep-lat/9405016 (1994)

    1994 arXiv

  36. [44]

    D. P. Landau and K. Binder,A Guide to Monte Carlo Simula- tions in Statistical Physics(Cambridge University Press, 2009)

    2009

  37. [45]

    See Supplemental Material at THIS URL for details on the ex- traction of critical exponentsη,γandν, finite-size corrections, and results forν

  38. [46]

    Solfanelli and N

    A. Solfanelli and N. Defenu, Phys. Rev. E110, 044121 (2024)

    2024

  39. [47]

    A. P. Millán, G. Gori, F. Battiston, T. Enss, and N. Defenu, Physical Review Research3, 023015 (2021)

    2021

  40. [48]

    Sarkar, T

    M. Sarkar, T. Enss, and N. Defenu, Physical Review B110, 014208 (2024)

    2024

  41. [49]

    Bighin, T

    G. Bighin, T. Enss, and N. Defenu, Nature Communications15, 4207 (2024)

    2024

  42. [50]

    E. C. King, M. Linnebacher, P. P. Orth, M. Rizzi, and G. Morigi, arXiv preprint arXiv:2501.08148 (2025)

    2025

  43. [51]

    Long range to short range crossover in one dimension

    B. D. Hughes,Random walks and random environments(Ox- ford University Press, 1996). 6 Supplemental Material for “Long range to short range crossover in one dimension” Mrinal Sarkar1, Nicolò Defenu2, and Tilman Enss1 1Institut für Theoretische Physik, Universität Heidelberg, 69...

    1996

  44. [52]

    Extraction of the anomalous dimensionη. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

  45. [53]

    Extraction of SAW critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

  46. [54]

    Finite size correction to SAW critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

  47. [55]

    Behavior of the SAW critical exponentν. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Extraction of the anomalous dimensionη To extract the exponentηfor a givenσ, we proceed as foll...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.