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arxiv: 2603.26083 · v2 · submitted 2026-03-27 · ❄️ cond-mat.stat-mech

Crossover Scaling of Binder Cumulant and its application in Non-reciprocal Sandpiles

Wei Zhong , Youjin Deng This is my paper

Pith reviewed 2026-05-14 23:30 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Binder cumulantnon-reciprocal interactionsManna sandpileuniversality classesmean-field criticalityfinite-size scalingconserved nonequilibrium systems
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0 comments X

The pith

Non-reciprocal interactions drive critical exponents in sandpiles toward mean-field values

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a pre-asymptotic scaling law for the Binder cumulant that holds in both disordered and ordered phases near criticality. This law is applied to the conserved Manna sandpile to test the stability of its universality class when reciprocity is broken. Reciprocal biases only shift the location of the critical point while leaving the class unchanged, but non-reciprocal interactions prove relevant and cause the exponents to flow toward mean-field values. The work identifies non-reciprocity as a general mechanism that can induce mean-field criticality in conserved nonequilibrium systems.

Core claim

We unveil a robust pre-asymptotic scaling regime for the Binder cumulant U_L, demonstrating U_L ∼ N^{-1} |t|^{-dν} in the disordered phase and 2/3 − U_L ∼ N^{-1} |t|^{-dν} in the ordered phase. For the conserved Manna sandpile, reciprocal biases preserve the universality class while non-reciprocal interactions act as relevant perturbations that drive the critical exponents toward their mean-field values. This establishes non-reciprocity as a generic mechanism for inducing mean-field criticality in conserved non-equilibrium systems.

What carries the argument

The pre-asymptotic Binder cumulant scaling U_L ∼ N^{-1} |t|^{-dν}, which enables tracking of universality class flow under perturbations without requiring full asymptotic data.

Load-bearing premise

The pre-asymptotic scaling regime for the Binder cumulant remains robust and can be used to diagnose universality-class flow without requiring full asymptotic data.

What would settle it

A simulation of a non-reciprocal Manna sandpile in which the effective exponents extracted from Binder cumulant data do not approach mean-field values such as ν = 1/2 as system size increases would falsify the claim of flow to mean-field criticality.

Figures

Figures reproduced from arXiv: 2603.26083 by Wei Zhong, Youjin Deng.

Figure 1
Figure 1. Figure 1: FIG. 1. Discovery and validation of the pre-asymptotic [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The log-log plot of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The effective exponents (a) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: confirms the scaling behavior reported in Eq. (2)-(3) of the main tex. 10-5 10-4 10-3 10-2 10-1 100 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows that for the pure 2D conserved Manna sandpile, we extract the exponents from the scaling behavior and we find ν = 0.80(2), β = 0.64(2) and γ ′ = 0.38(2) which are consistent with the results reported in Ref. [3] In this paper, we consider three different bias to the system. They are 1 Reciprocal Bias (RB, Case I): qright = qleft = 0.25+δ, qup = qdown = 0.25−δ. This preserves spatial inversion symmetr… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a)-(c) Critical behavior of [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The critical behavior of the 1D conserved Manna sandpile. The obtained critical exponents are consistence with the [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The critical behavior of the non-reciprocal bias interaction ( [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The critical behavior of the non-reciprocal bias interaction ( [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Critical exponents [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
read the original abstract

In this letter, we unveil a robust, pre-asymptotic scaling regime for the Binder cumulant $U_L$, a central finite-size scaling tool, demonstrating $U_L\sim N^{-1} |t|^{-d\nu}$ (disordered phase) and $\frac{2}{3}-U_L\sim N^{-1} |t|^{-d\nu}$ (ordered phase), with $t$ being the reduced control parameter, and $N$, $d$, $\nu$ represent the total number of sites, the dimensionality, and correlation length exponent, respectively. Leveraging this result, we resolve a fundamental question on the stability of universality classes under the breaking of microscopic reciprocity. For the conserved Manna sandpile, we show that reciprocal biases preserve its universality class, merely shifting the critical point. In striking contrast, any non-reciprocal interaction acts as a relevant perturbation, decisively driving the system's critical exponents to flow from their non-mean-field values towards the mean-field related ones. This flow establishes non-reciprocity as a generic mechanism inducing mean-field criticality in conserved, non-equilibrium systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims a robust pre-asymptotic scaling regime for the Binder cumulant: U_L ∼ N^{-1} |t|^{-dν} (disordered phase) and 2/3−U_L ∼ N^{-1} |t|^{-dν} (ordered phase). It applies this form to the conserved Manna sandpile to conclude that reciprocal biases only shift the critical point while any non-reciprocal interaction is relevant and drives the exponents from non-mean-field toward mean-field values.

Significance. If the scaling derivation and its robustness under non-reciprocity hold, the result supplies a practical finite-size diagnostic for universality-class flow and identifies non-reciprocity as a generic mechanism for mean-field criticality in conserved non-equilibrium systems. The approach could be useful for other driven lattice models where full asymptotic data are inaccessible.

major comments (3)
  1. [Abstract and scaling regime section] The scaling form U_L ∼ N^{-1} |t|^{-dν} (and its ordered-phase counterpart) is presented as a discovery but without derivation details, explicit assumptions on the control parameter t, or checks that the leading finite-size correction remains unchanged when non-reciprocal terms are added. This form is load-bearing for the claim that non-reciprocity induces exponent flow.
  2. [Application to non-reciprocal sandpiles] The diagnosis of universality-class flow rests on fitting the same finite-size data both to locate the critical point (via t) and to extract effective exponents from the proposed scaling; no independent benchmark or cross-validation is shown, creating a circularity risk that directly affects the central claim.
  3. [Universality class stability discussion] The weakest assumption—that the pre-asymptotic regime remains valid and directly yields flowing exponents when non-reciprocal perturbations are present—is not tested against possible modifications to leading corrections or additional relevant operators. Without such a test the conclusion that any non-reciprocal interaction drives mean-field flow cannot be considered established.
minor comments (2)
  1. [Methods and figures] Specify the precise numerical procedure used to determine the reduced control parameter t and the fitting window in N and t; include error bars on all reported exponents and scaling collapses.
  2. [Discussion] Clarify whether the scaling form is expected to hold only for the Manna class or more generally, and add a brief comparison with at least one other known model where the Binder cumulant scaling is already established.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points regarding the derivation, potential circularity in analysis, and robustness of the pre-asymptotic regime. We have revised the manuscript to address these concerns by adding explicit derivations, independent benchmarks, and additional numerical tests. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract and scaling regime section] The scaling form U_L ∼ N^{-1} |t|^{-dν} (and its ordered-phase counterpart) is presented as a discovery but without derivation details, explicit assumptions on the control parameter t, or checks that the leading finite-size correction remains unchanged when non-reciprocal terms are added. This form is load-bearing for the claim that non-reciprocity induces exponent flow.

    Authors: We have expanded the scaling regime section with a step-by-step derivation starting from the definition of the Binder cumulant U_L = 1 - <m^4>/(3<m^2>^2) and the finite-size scaling ansatz for the order parameter distribution. The reduced control parameter t is explicitly defined as (p - p_c)/p_c, with p the driving probability. Additional simulations for non-reciprocal variants confirm that the leading correction term remains N^{-1} |t|^{-dν} without modification to its functional form. These details are now included in the revised main text. revision: yes

  2. Referee: [Application to non-reciprocal sandpiles] The diagnosis of universality-class flow rests on fitting the same finite-size data both to locate the critical point (via t) and to extract effective exponents from the proposed scaling; no independent benchmark or cross-validation is shown, creating a circularity risk that directly affects the central claim.

    Authors: To eliminate circularity, we now determine p_c independently via the crossing point of U_L curves for different system sizes L, which relies only on the standard finite-size scaling property that U_L is size-independent at criticality and does not invoke the proposed N^{-1} |t|^{-dν} form. Effective exponents are subsequently extracted from data collapse using this fixed p_c. We added a cross-validation by comparing the flow with the exactly known mean-field Binder cumulant value (U=2/3) in the strong non-reciprocity limit. revision: yes

  3. Referee: [Universality class stability discussion] The weakest assumption—that the pre-asymptotic regime remains valid and directly yields flowing exponents when non-reciprocal perturbations are present—is not tested against possible modifications to leading corrections or additional relevant operators. Without such a test the conclusion that any non-reciprocal interaction drives mean-field flow cannot be considered established.

    Authors: We performed new simulations varying the non-reciprocal bias strength over two orders of magnitude and explicitly checked that the leading correction exponent remains unchanged while the effective ν flows continuously toward its mean-field value. No additional relevant operators altering the N^{-1} prefactor were observed within the accessible scaling window. While a full renormalization-group treatment lies outside the scope of this letter, the numerical evidence supports the relevance of non-reciprocity; we have clarified this limitation in the revised discussion. revision: partial

Circularity Check

1 steps flagged

Pre-asymptotic Binder cumulant scaling presented as discovery then used to extract flowing exponents from same finite-size data

specific steps
  1. fitted input called prediction [Abstract]
    "Leveraging this result, we resolve a fundamental question on the stability of universality classes under the breaking of microscopic reciprocity. ... any non-reciprocal interaction acts as a relevant perturbation, decisively driving the system's critical exponents to flow from their non-mean-field values towards the mean-field related ones."

    The scaling regime U_L ∼ N^{-1}|t|^{-dν} is unveiled from finite-size data; the same data (with t defined relative to the fitted critical point) is then used to extract effective exponents that are reported to flow under non-reciprocity. The flow is therefore a direct readout of the scaling fit rather than an independent test of relevance.

full rationale

The paper derives or fits the pre-asymptotic form U_L ~ N^{-1}|t|^{-dν} (and ordered-phase counterpart) from simulations, then leverages it directly to conclude that non-reciprocal terms drive effective exponents toward mean-field values. Because t and the critical point are themselves located from the same data sets, the reported flow is statistically forced by the scaling assumption rather than independently verified. No self-citation chain or ansatz smuggling is evident; the central claim retains some independent content from the numerical evidence but the diagnostic step reduces to a fitted-input prediction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the pre-asymptotic scaling relation being exact in the stated regime and on the assumption that non-reciprocal terms are relevant operators whose strength can be tuned independently of the control parameter t.

free parameters (1)
  • effective scaling window for t
    The range of reduced control parameter t over which the N^{-1} |t|^{-d nu} form is claimed to hold is not derived from first principles and must be chosen or fitted.
axioms (1)
  • domain assumption Standard finite-size scaling hypothesis for the Binder cumulant near criticality
    Invoked to extend the usual U_L scaling into the pre-asymptotic window.

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