Pith. sign in

REVIEW 4 major objections 6 minor 79 references

The Role of Integration Cycles in Complex Langevin Simulations

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read In quartic toy models, boundary-term-free complex Langevin averages are numerically confirmed to be observable-independent linear combinations of integration-cycle averages, with the kernel phase controlling which cycles contribute.

desk verdict Solid 1D confirmation of the cycle decomposition, a genuinely new but explicitly conditional 2D test, and a useful cycle-counting algorithm; worth a serious referee. read the letter →

arxiv 2412.17137 v2 pith:FFPOIPP4 submitted 2024-12-22 hep-lat hep-th

Michael W. Hansen , Michael Mandl , Erhard Seiler , Dénes Sexty This is my paper · ORCID
classification hep-lathep-th PACS 02.50.Ey05.10.Gg
keywords complexLangevinsignproblemintegrationcycleskernelwrongconvergenceboundarytermsDyson-Schwingerequationstwo-dimensionaltoymodel
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

Complex Langevin simulations are a candidate cure for the sign problem, but they can converge to wrong answers even when the standard boundary-term test looks clean. This paper's central claim is that in such clean cases the simulation average is still a linear combination of integration-cycle averages, with the same coefficients for every observable, and that the kernel in the Langevin equation decides which combination is sampled. The authors verify the decomposition numerically in a one-dimensional quartic model, where it is a theorem, and in a two-dimensional extension, where it is only a conjecture. The practical punchline is that vanishing boundary terms are necessary but not sufficient for correctness, and that kernels used to cure the sign problem can silently redirect the simulation onto the wrong cycles.

What carries the argument

The central object is the integration cycle: an equivalence class of integration contours in the complexified field space along which $e^{-S}$ vanishes at infinity; for polynomial actions, cycles connect different angular good regions where the Boltzmann factor decays. The load-bearing identity is the decomposition (10) of the paper, which says every boundary-term-free complex Langevin average is a linear combination of cycle averages with observable-independent coefficients. The second piece of machinery is the kernel $K$ in the Langevin equation $dz = -K S'(z)\,d\tau + \sqrt{K}\,dw$; the argument of $K$ rotates the sampled distribution in the complex plane and thereby selects which cycle or mixture of cycles is sampled. To extract the coefficients, the paper fits measured monomial observables against numerically computed cycle averages by generalized least squares, using whitened residuals to judge whether the fit is trustworthy. A geometric counting algorithm based on decomposing cuboids in the angular torus supplies the number $N_\gamma$ of independent cycles, including the drop from nine to two cycles at $|a|=2$.

What would settle it

Re-measure the even monomial observables in the model (34) at $a=2$ with kernels near $m=10$ and $m=34$, fit the two-cycle coefficients on subsets of observables of increasing order, and check whether the residuals drift with observable order; any order-dependent drift would falsify the claim that one coefficient set describes all observables. Independently, evaluate the integral of $\operatorname{Re} S$ along the ray $r_1 = r_2$ for $a=2$: if no non-integrable divergence appears, the predicted reduction to $N_\gamma = 2$ would need another explanation.

Watch

Extended reading notes

Core claim

The central claim, carried through every simulation, is that in the absence of boundary terms a complex Langevin expectation value obeys the decomposition $\langle O\rangle_{\mathrm{CL}} = \sum_{i=1}^{N_\gamma} a_i \langle O\rangle_{\gamma_i}$, where the $\gamma_i$ are the independent integration cycles of the complexified theory and the complex coefficients $a_i$ are the same for all observables, with $\sum_i a_i = 1$. In one dimension the paper reproduces the known cases and exposes a strong example: for $\lambda = e^{5i\pi/6}$ with trivial kernel the boundary terms vanish yet the result is a 50/50 mixture of the real and imaginary cycles, not the desired real-cycle average. Scanning kernel phases $K = e^{im\pi/24}$, the coefficients move through a series of plateaus: only the real cycle near $m=10$, only the imaginary cycle near $m=34$, and mixed combinations on smaller plateaus near $m=22$ and $46$. In the two-dimensional model $S_a = \frac{\lambda}{4}(z_1^4+z_2^4+a z_1^2 z_2^2)$, the same decomposition is found numerically, including the strong-coupling regime $a \ge 2$ in which an angular counting algorithm predicts only two independent cycles and the fits correspondingly use $N_\gamma = 2$. The authors therefore conclude that the one-dimensional theorem plausibly extends to arbitrary dimension.

Load-bearing premise

The argument assumes the integration-cycle decomposition holds in two dimensions, a conjecture the authors explicitly flag as unproved, and counts $N_\gamma = 2$ for $a \ge 2$ on the strength of an asserted but not derived non-integrable singularity on the line $r_1 = r_2$.

Editorial extensions

If this is right

  • Vanishing boundary terms are confirmed insufficient for correctness: simulations can converge to wrong averages composed of several integration cycles while all measured boundary terms stay consistent with zero.
  • The kernel acts as a dial on cycle coefficients, at least in the quartic model: changing its phase moves the simulation from the real cycle to the imaginary cycle, with intermediate mixtures, and large plateaus of the kernel phase give correct results.
  • Kernel searches must track cycle content, not only boundary terms: there are large kernel regions with clean boundary terms but wrong averages.
  • In the two-dimensional model the number of independent cycles drops from nine to two for strong coupling $|a| \ge 2$, and numerical fits there support the two-cycle decomposition.
  • Because the decomposition held in every boundary-term-free run, the authors expect a proof of the decomposition for arbitrary dimensions to be within reach.

Reading between the lines

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

  • If the two-dimensional conjecture is right, integration-cycle content should be part of the interpretation of every boundary-term-free complex Langevin simulation, not just one-dimensional models.
  • Automated kernel searches (for example by machine learning) could lock onto a harmless-looking kernel on a wrong-convergence plateau; using a family of observables, especially high powers, rather than a single one, would make such plateaus detectable.
  • The counting algorithm's prediction that strong coupling reduces cycle number suggests that some strongly interacting theories might have tractably few cycles, and that the danger of cycle contamination may be largest at weak coupling.
  • The asserted singularity at $r_1=r_2$ for $a=2$ is testable in isolation: a direct numerical integration along that ray would either substantiate or weaken the $N_\gamma=2$ counting.
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

4 major / 6 minor

Summary. The paper studies complex Langevin (CL) simulations and the role of integration cycles. It numerically tests the Salcedo-Seiler theorem (Eq. (10)) that, in the absence of boundary terms, CL expectation values are a linear combination of integration-cycle averages with observable-independent coefficients. In one dimension (action S = λ z^4/4, kernel K = e^{iπm/24}), the authors measure monomial observables and boundary terms, and fit the coefficients a_i via covariance-weighted least squares to cycle averages computed by numerical integration. They find the theorem confirmed whenever boundary terms vanish, with the sum rule (11) satisfied; different kernels rotate the cycle-coefficient vector, and the 'perfect' kernel K = λ^{-1/2} selects the real cycle. In two dimensions (action S = λ/4 (z1^4 + z2^4 + a z1^2 z2^2)), they study a = 2 (O(2)-symmetric, Nγ = 2) and a < 2 (Nγ = 9). For a = 2 they find a1 ≈ 1 and a2 ≈ 0 for all λ with vanishing boundary terms; for a = 0.5 and suitable kernels they find four contributing cycles. They conclude that Eq. (10) holds in all their simulations, providing a first hint of its validity beyond one dimension.

Significance. If correct, the paper provides a useful numerical confirmation of the integration-cycle picture and, more importantly, demonstrates that kernels can be used to control which cycles are sampled, with practical implications for kernel searches in complex Langevin simulations. The one-dimensional results are convincing: the model is nontrivial, the fits are covariance-weighted, the sum rule is checked, alternative observable sets give consistent results, and the data and analysis scripts are publicly available. The two-dimensional results are weaker because Eq. (10) is a conjecture there and the cycle count at a = 2 relies on an asserted singularity that is not derived. Nevertheless, the paper is transparent about these limitations, and the two-dimensional evidence is a plausible first step toward a generalized theorem.

major comments (4)
  1. [Appendix A, Eq. (36)] The count Nγ = 2 for |a| ≥ 2 is load-bearing for the two-dimensional fits, but the argument rests on the unproven assertion in Appendix A that for a = 2 and ϕ = π/4 (the line r1 = r2) 'a non-integrable singularity arises in the integral over certain cycles.' No derivation, estimate, or reference is given. Since the number of columns of the design matrix X in Eq. (23) and hence the a = 2 fits in Sec. V B 1 (Figs. 8 and 11) depend on this count, the authors should provide the missing derivation or an explicit numerical check (e.g., demonstrate divergence of candidate cycle integrals, or show rank deficiency of the matrix Mij = ⟨Oi⟩γj for nine candidate cycles). Without this, the a = 2 fits cannot be taken as a confirmation of Eq. (10).
  2. [Sec. V B 2 and Eq. (40)] For the a < 2 fits of Sec. V B 2, the nine-cycle basis (40) is used at a = 0.5, but the validity of representatives involving the one-dimensional γ3 for a ≠ 0 is only asserted in Appendix A. The numerical contours used to compute ⟨O⟩γi for the design matrix X are not specified. Please specify the contours and provide a convergence check for all nine cycle integrals at the couplings used (at least a = 0.5), or restrict the basis to cycles whose validity is demonstrated. Otherwise the small fitted values of a5...a9 in Fig. 15 may be artifacts of non-convergent integrals.
  3. [Appendix B and Sec. V A 1] The Shapiro-Wilk test used in Appendix B as the goodness-of-fit criterion tests whether the whitened residuals are Gaussian, not whether their mean is zero. A model that is systematically offset by a constant would produce shifted Gaussian residuals and could still pass the p > 0.05 threshold. Since the statement 'we find the fit to be good if and only if it produces this unique answer' (Sec. V A 1) underpins the one- and two-dimensional conclusions, add an explicit zero-mean test on the whitened residuals (e.g., a t-test, or equivalently a χ² per degree of freedom after whitening) and report its outcome for all fits shown.
  4. [Abstract and Sec. V B] The abstract states that the paper 'confirm[s] numerically' Eq. (10), but for d = 2 the equation is a conjecture, as acknowledged in Secs. III A and VI. Because the two-dimensional fits assume the linear model (10) as the regression model, they test compatibility with one particular linear combination rather than the validity of the decomposition. To strengthen the two-dimensional confirmation, perform a hold-out test: fit the coefficients a_i on a subset of observables and verify that the remaining observables are reproduced within uncertainties for the points in Figs. 8 and 11. This would mitigate the concern that a non-cycle distribution matching the first few monomial moments could pass the current test.
minor comments (6)
  1. [Sec. II B] The word 'Heavyside' should be 'Heaviside'.
  2. [Sec. IV D 2] The phrase 'it has do be done online' should read 'it has to be done online'.
  3. [Sec. IV D 3] The covariance matrix Σ is said to be 'normalized by 1/√Nmeas' in the fit, while Appendix B uses the same symbol for the unnormalized covariance; please clarify the two uses.
  4. [Sec. V A 1, Table I] The sentence 'The results are rounded to the first significant digit of the respective statistical uncertainties' is clear, but consider stating explicitly that the imaginary parts in the l = -1 and l = 1 rows are consistent with zero within the quoted errors.
  5. [References] Reference [31] has an incomplete author entry ('T¨or¨ok' without initials); please complete the citation.
  6. [Sec. IV and Data Availability] The FAIR principles are invoked, but the simulation code is only 'available upon request'; posting the code alongside the data would be more consistent with the stated reproducibility goals.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central numerical test of Eq. (10) is independent of the theorem it checks, and the 2D conjecture is explicitly flagged.

full rationale

The paper's claimed result is a numerical confirmation of Eq. (10), not a derivation. In one dimension, Eq. (10) is a proven theorem cited from Salcedo and Seiler [1]; in two dimensions, the paper explicitly states that the validity of (10) has not been established and treats it as a conjecture. The complex Langevin expectation values come from independent simulations, the cycle averages <O>_{gamma_i} are computed by separate numerical integration, and the coefficients a_i are free least-squares outputs. The fit demonstrably fails when boundary terms are present, so the confirmation is falsifiable rather than forced by construction. The main self-citation is [1], on which one author is a co-author, but that work contains a published proof whose stated assumptions do not include the numerical results of the present paper; it is not an unverified premise invoked to forbid alternatives. The Appendix A count N_gamma = 2 at a = 2 relies on an asserted non-integrable singularity, and the two-dimensional extension of (10) is unproven, but the paper states both limitations transparently; these are correctness risks, not circular reductions. The Shapiro-Wilk criterion tests normality of residuals rather than the full linear-form hypothesis, which weakens the 2D evidence but does not make the fitted coefficients inputs to the result. No equation is defined in terms of the quantity it is claimed to predict, and no fitted parameter is renamed as a prediction. Score 1 reflects the presence of a non-load-bearing self-citation in an otherwise independent central test.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper does not introduce new entities. Its central interpretive framework is the prior theorem (10) (proven for d=1, conjectured for d=2), which is assumed when fitting coefficients. The 2D cycle count relies on an unproved analytic statement about the a=2 model. No free parameters are fitted into the derivation; the coefficients a_i are measured outputs.

assumptions (3)
  • domain assumption Salcedo-Seiler theorem (10) in d=1: any linear functional satisfying Dyson-Schwinger equations with no boundary terms is a linear combination of integration cycles.
    Invoked in Sec. III A as the foundation for interpreting ⟨O⟩_CL. It is proven in [1] for one dimension, but the paper assumes the same form when fitting in two dimensions.
  • domain assumption Non-integrability of certain a=2 integrals at r1=r2 (φ=π/4) leading to Nγ=2 for |a|≥2.
    Appendix A: stated without derivation; the conclusion that only two rectangles survive at a=2 depends on this.
  • domain assumption Vanishing of measured boundary terms B_{z^n}(Y) for all measured n implies the simulation is in the regime where (10) applies.
    The paper's fit selection uses the 3σ plateau criterion in Sec. V A 1 as a proxy for the mathematically exact absence of boundary terms.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Role of Integration Cycles in Complex Langevin Simulations." pith.science (2026). https://pith.science/paper/FFPOIPP4

@misc{pith2026241217137,
  author       = {Pith},
  title        = {Pith review of: The Role of Integration Cycles in Complex Langevin Simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FFPOIPP4}},
  note         = {Machine review of arXiv:2412.17137}
}
read the original abstract

Complex Langevin simulations are an attempt to solve the sign (or complex-action) problem encountered in various physical systems of interest. The method is based on a complexification of the underlying degrees of freedom and an evolution in an auxiliary time dimension. The complexification, however, does not come without drawbacks, the most severe of which is the infamous 'wrong convergence' problem, stating that complex Langevin simulations sometimes fail to produce correct answers despite their apparent convergence. It has long been realized that wrong convergence may - in principle - be fixed by the introduction of a suitable kernel into the complex Langevin equation, such that the conventional correctness criteria are met. However, as we discuss in this work, complex Langevin results may - especially in the presence of a kernel - still be affected by unwanted so-called integration cycles of the theory spoiling them. Indeed, we confirm numerically that in the absence of boundary terms the complex Langevin results are given by a linear combination of such integration cycles, as put forward by Salcedo & Seiler. In particular, we shed light on the way different choices of kernel affect which integration cycles are being sampled in a simulation and how this knowledge can be used to ensure correct convergence in simple toy models.

Figures

Figures reproduced from arXiv: 2412.17137 by the authors.

Figure 1
Figure 1. FIG. 1. Topography of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Real and imaginary parts of the difference ∆ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 5
Figure 5. FIG. 5. Real and imaginary parts of the observables [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figures from the paper (13 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Real and imaginary parts of the coefficients [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Real and imaginary parts of the coefficients [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 10
Figure 10. Figure 10: As before, we once again observe large plateaus in the vicinity of m = 10 and m = 34, respectively. In fact, these plateaus appear to be slightly larger than those in [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Histograms of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Analogous to Fig [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Histograms of [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Real and imaginary parts of the observable [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Histograms of [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Real and imaginary parts of the coefficients [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. ‘Sign’ of [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 17
Figure 17. Figure 17: , where the rectangle on the left plot is decom￾posed into atomic rectangles after two iteration steps. It is not hard to see that in total there are nine such atomic rectangles in this theory, from which we conclude that there are nine linearly independent integratio…
Figure 18
Figure 18. Figure 18: FIG. 18. Distribution of the whitened residuals [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p024_19.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

79 extracted references · 34 canonical work pages

  1. [1]

    In order to define the two linearly independent integration cycles for this model, we introduce two variables ξ1 and ξ2 in analogy to (30)

    O(2)-symmetric case We start by discussing the case a = 2, i.e., the model S2(z1, z2) = λ 4 (z2 1 + z2 2)2 , (37) which features an O(2) symmetry. In order to define the two linearly independent integration cycles for this model, we introduce two variables ξ1 and ξ2 in analogy to (30). Then, as before, we define γ1 to be the real in- tegration cycle in th...

    work page

  2. [2]

    However, as is known since [62], the introduction of a nontrivial kernel K ≈ λ−1/2 in (4) can ensure ⟨zn⟩CL = ⟨zn⟩exact for arbitrary λ ̸= 0

    Nontrivial kernel We have seen that for most values of λ, measurements in complex Langevin simulations with K = 1 are either affected by boundary terms (i.e., slowly decaying proba- bility distributions) or by the sampling of unwanted in- tegration cycles. However, as is known since [62], the introduction of a nontrivial kernel K ≈ λ−1/2 in (4) can ensure...

    work page

  3. [3]

    General couplings We have seen before that the class of models (34) behaves in a qualitatively different way depending on whether a is smaller or larger than two, since the number of independent integration cycles jumps from 9 to 2 at that value according to (36). As the final analysis of this work, we shall devote this subsection to a discussion of how t...

    work page

  4. [4]

    L. L. Salcedo and E. Seiler, J. Phys. A 52, 035201 (2019), arXiv:1809.06888 [math-ph]

    arXiv 2019

  5. [5]

    Y. Aoki, G. Endr˝ odi, Z. Fodor, S. D. Katz, and K. K. Szab´ o, Nature443, 675 (2006), arXiv:hep-lat/0611014

    arXiv 2006

  6. [6]

    Troyer and U.-J

    M. Troyer and U.-J. Wiese, Phys. Rev. Lett. 94, 170201 (2005), arXiv:cond-mat/0408370

    arXiv 2005

  7. [7]

    de Forcrand, S

    P. de Forcrand, S. Kim, and T. Takaishi, Nucl. Phys. B – Proc. Suppl. 119, 541 (2003), arXiv:hep-lat/0209126

    arXiv 2003

  8. [8]

    Goksch, Phys

    A. Goksch, Phys. Rev. Lett. 61, 2054 (1988)

    work page 1988

Show all 79 references
  1. [9]

    Chandrasekharan, PoS (LA TTICE 2008) , 003 (2009), arXiv:0810.2419 [hep-lat]

    S. Chandrasekharan, PoS (LA TTICE 2008) , 003 (2009), arXiv:0810.2419 [hep-lat]

    2009 arXiv

  2. [10]

    Parisi, Phys

    G. Parisi, Phys. Lett. B 131, 393 (1983)

    1983

  3. [11]

    J. R. Klauder, J. Phys. A 16, L317 (1983)

    1983

  4. [12]

    Cristoforetti, F

    M. Cristoforetti, F. Di Renzo, and L. Scorzato (Aurora- Science Collaboration), Phys. Rev. D 86, 074506 (2012), arXiv:1205.3996 [hep-lat]

    2012 arXiv

  5. [13]

    M. C. Ba˜ nuls, K. Cichy, J. I. Cirac, K. Jansen, and S. K¨ uhn, PoS (LA TTICE2018), 022 (2019), arXiv:1810.12838 [hep-lat]

    2019 arXiv

  6. [14]

    Alford, A

    M. Alford, A. Kapustin, and F. Wilczek, Phys. Rev. D 59, 054502 (1999), arXiv:hep-lat/9807039v1

    1999 arXiv

  7. [15]

    Allton, S

    C. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, C. Schmidt, and L. Scorzato, Phys. Rev. D 66, 074507 (2002), arXiv:hep-lat/0204010

    2002 arXiv

  8. [16]

    Parisi and Y.-S

    G. Parisi and Y.-S. Wu, Sci. Sin. 24, 483 (1981)

    1981

  9. [17]

    Damgaard and H

    P. Damgaard and H. H¨ uffel, Phys. Rep.152, 227 (1987)

    1987

  10. [18]

    Namiki, Prog

    M. Namiki, Prog. Theor. Phys. Suppl. 111, 1 (1993)

    1993

  11. [19]

    Karsch and H

    F. Karsch and H. W. Wyld, Phys. Rev. Lett. 55, 2242 (1985)

    1985

  12. [20]

    S¨ oderberg, Nucl

    B. S¨ oderberg, Nucl. Phys. B295, 396 (1988)

    1988

  13. [21]

    Gausterer and S

    H. Gausterer and S. Lee, J. Stat. Phys. 73, 147 (1992), arXiv:hep-lat/9211050

    1992 arXiv

  14. [22]

    Okano, L

    K. Okano, L. Sch¨ ulke, and B. Zheng, Prog. Theor. Phys. Suppl. 111, 313 (1993)

    1993

  15. [23]

    Ambjørn and S

    J. Ambjørn and S. K. Yang, Phys. Lett. B 165, 140 (1985)

    1985

  16. [24]

    Ambjørn, M

    J. Ambjørn, M. Flensburg, and C. Peterson, Nucl. Phys. B 275, 375 (1986)

    1986

  17. [25]

    L. L. Salcedo, Phys. Lett. B 304, 125 (1993)

    1993

  18. [26]

    Fujimura, K

    K. Fujimura, K. Okano, L. Sch¨ ulke, K. Yamagishi, and B. Zheng, Nucl. Phys. B 424, 675 (1994), arXiv:hep- th/9311174

    1994

  19. [27]

    Berges and I.-O

    J. Berges and I.-O. Stamatescu, Phys. Rev. Lett. 95, 202003 (2005), arXiv:hep-lat/0508030

    2005 arXiv

  20. [28]

    Berges, S

    J. Berges, S. Bors´ anyi, D. Sexty, and I.-O. Stamatescu, Phys. Rev. D 75, 045007 (2007), arXiv:hep-lat/0609058

    2007 arXiv

  21. [29]

    Aarts, F

    G. Aarts, F. A. James, E. Seiler, and I.-O. Stamatescu, Phys. Lett. B 687, 154 (2010), arXiv:0912.0617 [hep-lat]

    2010 arXiv

  22. [30]

    Seiler, D

    E. Seiler, D. Sexty, and I.-O. Stamatescu, Phys. Lett. B 723, 213 (2013), arXiv:1211.3709 [hep-lat]

    2013 arXiv

  23. [31]

    Sexty, Phys

    D. Sexty, Phys. Lett B. 729, 108 (2014), arXiv:1307.7748 [hep-lat]

    2014 arXiv

  24. [32]

    Sexty, Nucl

    D. Sexty, Nucl. Phys. A 931, 856 (2014), arXiv:1408.6767 [hep-lat]

    2014 arXiv

  25. [33]

    Aarts, E

    G. Aarts, E. Seiler, D. Sexty, and I.-O. Stamatescu, Phys. Rev. D 90, 114505 (2014), arXiv:1408.3770 [hep-lat]

    2014 arXiv

  26. [34]

    Fodor, S

    Z. Fodor, S. D. Katz, D. Sexty, and T¨ or¨ ok, Phys. Rev. D 92, 094516 (2015), arXiv:1508.05260 [hep-lat]

    2015 arXiv

  27. [35]

    Attanasio, B

    F. Attanasio, B. J¨ ager, and F. P. G. Ziegler, arXiv:2203.13144 [hep-lat]

    arXiv

  28. [36]

    Aarts, F

    G. Aarts, F. Attanasio, B. J¨ ager, and D. Sexty, JHEP 2016, 087, arXiv:1606.05561 [hep-lat]

    2016 arXiv

  29. [37]

    C. E. Berger, L. Rammelm¨ uller, A. C. Loheac, F. Ehmann, J. Braun, and J. E. Drut, Phys. Rep. 892, 1 (2021), arXiv:1907.10183 [cond-mat]

    2021 arXiv

  30. [38]

    Aarts, E

    G. Aarts, E. Seiler, and I.-O. Stamatescu, Phys. Rev. D 81, 054508 (2010), arXiv:0912.3360 [hep-lat]

    2010 arXiv

  31. [39]

    Aarts, F

    G. Aarts, F. A. James, E. Seiler, and I.-O. Stamatescu, Eur. Phys. J. C 71, 1756 (2011), arXiv:1101.3270 [hep- lat]. 25

    2011 arXiv

  32. [40]

    L. L. Salcedo, Phys. Rev. D 94, 114505 (2016), arXiv:1611.06390 [hep-lat]

    2016 arXiv

  33. [41]

    Nagata, J

    K. Nagata, J. Nishimura, and S. Shimasaki, Phys. Rev. D 94, 114515 (2016), arXiv:1606.07627 [hep-lat]

    2016 arXiv

  34. [42]

    Scherzer, E

    M. Scherzer, E. Seiler, D. Sexty, and I.-O. Stamatescu, Phys. Rev. D 99, 014512 (2019), arXiv:1808.05187 [hep- lat]

    2019 arXiv

  35. [43]

    Scherzer, E

    M. Scherzer, E. Seiler, D. Sexty, and I.-O. Stamatescu, Phys. Rev. D 101, 014501 (2020), arXiv:1910.09427 [hep- lat]

    2020 arXiv

  36. [44]

    Seiler, D

    E. Seiler, D. Sexty, and I.-O. Stamatescu, Phys. Rev. D 109, 014509 (2024), arXiv:2304.00563 [hep-lat]

    2024 arXiv

  37. [45]

    Witten, AMS/IP Stud

    E. Witten, AMS/IP Stud. Adv. Math. 50, 347 (2011), arXiv:1001.2933 [hep-th]

    2011 arXiv

  38. [46]

    Witten, arXiv:1009.6032 [hep-th]

    E. Witten, arXiv:1009.6032 [hep-th]

    arXiv

  39. [47]

    M. D. Wilkinson et al., Sci. Data 3, 160018 (2016)

    2016

  40. [48]

    L. L. Salcedo, J. Math. Phys. 38, 1170 (1997), arXiv:hep- lat/9607044

    1997

  41. [49]

    Weingarten, Phys

    D. Weingarten, Phys. Rev. Lett. 89, 240201 (2002), arXiv:quant-ph/0210195

    2002 arXiv

  42. [50]

    L. L. Salcedo, J. Phys. A 40, 9399 (2007), arXiv:0706.4359 [hep-lat]

    2007 arXiv

  43. [51]

    N. M. Lampl and D. Sexty, arXiv:2309.06103 [hep-lat]

    arXiv

  44. [52]

    Alvestad, R

    D. Alvestad, R. Larsen, and A. Rothkopf, JHEP 2023, 057, arXiv:2211.15625 [hep-lat]

    2023 arXiv

  45. [53]

    Alvestad, A

    D. Alvestad, A. Rothkopf, and D. Sexty, Phys. Rev. D 109, L031502 (2024), arXiv:2310.08053 [hep-lat]

    2024 arXiv

  46. [54]

    M. W. Hansen and D. Sexty, Phys. Rev. D 111, 054508 (2025), arXiv:2405.20709 [hep-lat]

    2025 arXiv

  47. [55]

    Mandl, M

    M. Mandl, M. W. Hansen, E. Seiler, and D. Sexty, PoS (LA TTICE2024), 059 (2025), arXiv:2410.13423 [hep- lat]

    2025 arXiv

  48. [56]

    Garc ´ ıa, Z

    S. Garc ´ ıa, Z. Guralnik, and G. S. Guralnik, arXiv:hep- th/9612079

  49. [57]

    Guralnik and C

    G. Guralnik and C. Pehlevan, Nucl. Phys. B 811, 519 (2009), arXiv:0710.3756 [hep-th]

    2009 arXiv

  50. [58]

    Guralnik and Z

    G. Guralnik and Z. Guralnik, Ann. Phys. 325, 2486 (2010), arXiv:0710.1256 [hep-th]

    2010 arXiv

  51. [59]

    M. W. Hansen, M. Mandl, E. Seiler, and D. Sexty, Zen- odo 10.5281/zenodo.14517088 (2024)

    2024 doi

  52. [60]

    M. W. Hansen, M. Mandl, E. Seiler, and D. Sexty, Zen- odo 10.5281/zenodo.14517265 (2024)

    2024 doi

  53. [61]

    Ukawa and M

    A. Ukawa and M. Fukugita, Phys. Rev. Lett. 55, 1854 (1985)

    1985

  54. [62]

    L’Ecuyer, R

    P. L’Ecuyer, R. Simard, E. J. Chen, and W. D. Kelton, Oper. Res. 50, 1073 (2002)

    2002

  55. [63]

    G. E. P. Box and M. E. Muller, Ann. Math. Statist. 29, 610 (1958)

    1958

  56. [64]

    Vingelmann, and F

    NVIDIA, P. Vingelmann, and F. H. Fitzek, CUDA, re- lease: 10.2.89 (2020)

    2020

  57. [65]

    Okamoto, K

    H. Okamoto, K. Okano, L. Sch¨ ulke, and S. Tanaka, Nucl. Phys. B 324, 684 (1989)

    1989

  58. [66]

    Aarts, Phys

    G. Aarts, Phys. Rev. D 88, 094501 (2013), arXiv:1308.4811 [hep-lat]

    2013 arXiv

  59. [67]

    Aarts, L

    G. Aarts, L. Bongiovanni, E. Seiler, and D. Sexty, JHEP 2014, 159, arXiv:1407.2090 [hep-lat]

    2014 arXiv

  60. [68]

    Boguslavski, P

    K. Boguslavski, P. Hotzy, and D. I. M¨ uller, SciPost Phys. 18, 092 (2025), arXiv:2412.02396 [hep-lat]

    2025 arXiv

  61. [69]

    Okano, L

    K. Okano, L. Sch¨ ulke, and B. Zheng, Phys. Lett. B258, 421 (1991)

    1991

  62. [70]

    van Rossum and J

    G. van Rossum and J. de Boer, CWI Quarterly 4, 283 (1991)

    1991

  63. [71]

    van der Walt, S

    S. van der Walt, S. C. Colbert, and G. Varoquaux, Com- put. Sci. Eng. 13, 22 (2011)

    2011

  64. [72]

    Virtanen, R

    SciPy 1.0 Contributors: P. Virtanen, R. Gommers et al., Nature Methods 17, 261 (2020)

    2020

  65. [73]

    McKinney, Proceedings of the 9th Python in Science Conference (Scipy 2010) , 51 (2010)

    W. McKinney, Proceedings of the 9th Python in Science Conference (Scipy 2010) , 51 (2010)

    2010

  66. [74]

    J. D. Hunter, Comput. Sci. Eng. 9, 90 (2007)

    2007

  67. [75]

    Crameri, Zenodo 10.5281/zenodo.8409685 (2023)

    F. Crameri, Zenodo 10.5281/zenodo.8409685 (2023)

    2023 doi

  68. [76]

    Wolfram Research, Inc., Mathematica, Version 13.3

  69. [77]

    Andrae, T

    R. Andrae, T. Schulze-Hartung, and P. Melchior, arXiv:1012.3754 [astro-ph]

    arXiv

  70. [78]

    Kessy, A

    A. Kessy, A. Lewin, and K. Strimmer, Am. Stat. 72, 309 (2018), arXiv:1512.00809 [stat]

    2018 arXiv

  71. [79]

    S. S. Shapiro and M. B. Wilk, Biometrika 52, 591 (1965)

    1965

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.