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arxiv: 2604.08864 · v1 · submitted 2026-04-10 · ❄️ cond-mat.stat-mech

Self-similar Dynamics in Percolation and Sandpile

Mingzhong Lu , Ming Li , Youjin Deng This is my paper

Pith reviewed 2026-05-10 17:57 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords percolationself-similaritycritical dynamicssandpile modelcluster growthscaling exponentstemporal patternsnonequilibrium
0
0 comments X

The pith

Percolation processes display temporal self-similarity in cluster size increments during bond addition.

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

The paper studies the incremental addition of bonds to an initially empty lattice in percolation and tracks the gap, or the jump in cluster size with each new bond, along with the merged cluster. It identifies scale-invariant patterns in these quantities over much of the process, indicating a previously unreported form of temporal self-similarity. This leads to quantitative relations linking the dynamic scaling exponents to standard static critical exponents. A reader would care because the relations allow extraction of critical behavior without first identifying the critical point, and the same patterns appear in lattices, networks, explosive percolation, rigidity percolation, and the early evolution of sandpile models.

Core claim

By tracking the gap, the size increment of clusters upon bond addition, and the corresponding merged cluster, we identify scale-invariant temporal patterns in both quantities throughout a large portion of the percolation process. This reveals a form of temporal self-similarity that has not been reported before. We further establish quantitative relations between the dynamic scaling exponents and the conventional static critical exponents, which enable the determination of critical behavior without prior knowledge of the critical point. The same self-similar dynamics is observed in both bond and site percolation on lattices and networks, and extends to other systems such as explosive and rigi

What carries the argument

The gap, defined as the size increment of clusters upon each bond addition, and the merged cluster, which together produce scale-invariant temporal patterns used to relate dynamic and static exponents.

If this is right

  • Dynamic scaling exponents become directly computable from observed temporal patterns and match static critical exponents.
  • Critical behavior can be extracted in systems where the location of the critical point is not known beforehand.
  • The same temporal scaling extends to explosive percolation, rigidity percolation, and the initial nonequilibrium phase of the Bak-Tang-Wiesenfeld sandpile.
  • A unified dynamic description applies across lattices, networks, and avalanche models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method offers a route to monitor criticality in real-time growing systems such as networks or materials without preset thresholds.
  • It could be applied to other nonequilibrium models with avalanches to test whether temporal self-similarity is a general feature of dynamic criticality.
  • This temporal analysis supplies an independent route to exponent values that can be cross-checked against traditional static scaling.

Load-bearing premise

The temporal patterns in gaps and merged clusters stay scale-invariant across a large portion of the bond-addition process, and the exponent relations hold generally for the listed models without needing the critical point known in advance.

What would settle it

Rescaling the time series of gap sizes and merged-cluster identities against bond fraction or time fails to produce data collapse onto a single curve, or the extracted dynamic exponents deviate from known static percolation exponents on the square lattice.

Figures

Figures reproduced from arXiv: 2604.08864 by Ming Li, Mingzhong Lu, Youjin Deng.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color online) Temporal and spatial self-similarity in bond percolation on square lattices with side length [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) Dynamic gap-size distribution [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) Dynamic gap-size distribution [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Self-similar dynamics in the BTW sand [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

Spatial self-similarity is a hallmark of critical phenomena. We study the dynamic process of percolation, in which bonds are incrementally added to an initially empty lattice until the system becomes fully occupied. By tracking the gap -- the size increment of clusters upon bond addition -- and the corresponding merged cluster, we identify scale-invariant temporal patterns in both quantities throughout a large portion of the process. This reveals a form of temporal self-similarity that has not been reported before. We further establish quantitative relations between the dynamic scaling exponents and the conventional static critical exponents, which enable the determination of critical behavior without prior knowledge of the critical point. The same self-similar dynamics is observed in both bond and site percolation on lattices and networks, and extends to other systems such as explosive and rigidity percolation. Moreover, similar temporal scaling is found in the initial nonequilibrium evolution of the Bak-Tang-Wiesenfeld sandpile model, suggesting a dynamic critical behavior distinct from its equilibrium state. These results provide a unified framework for understanding critical dynamics and may find applications in a broad range of complex 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

2 major / 2 minor

Summary. The manuscript claims that by tracking the 'gap' (the size increment of clusters upon incremental bond addition) and the corresponding merged cluster during percolation, scale-invariant temporal patterns emerge throughout a large portion of the process, revealing a new form of temporal self-similarity. Quantitative relations are established between newly defined dynamic scaling exponents and conventional static critical exponents; these relations purportedly allow extraction of critical behavior without prior knowledge of the critical point pc. The same patterns and relations are reported for bond/site percolation on lattices and networks, explosive and rigidity percolation, and the early-time nonequilibrium evolution of the Bak-Tang-Wiesenfeld sandpile model.

Significance. If the dynamic-static exponent relations can be shown to be non-circular and to hold parameter-free across the claimed range of models, the work would supply a unified dynamical framework for critical phenomena that complements static scaling theory and could be useful in systems where pc is difficult to locate a priori. The extension to sandpile dynamics is potentially interesting as a bridge between percolation and self-organized criticality, but the overall significance depends on whether the temporal self-similarity is independently measurable rather than an artifact of operating inside statically identified critical windows.

major comments (2)
  1. [Results on dynamic scaling exponents] The central claim that the dynamic-static exponent relations enable determination of critical behavior without any prior knowledge of pc (abstract and results sections) is load-bearing. The relations appear to be obtained by numerical fitting inside regimes whose location is already known from conventional static analysis; no explicit parameter-free derivation from the definitions of gap and merged-cluster size is supplied that would guarantee the same exponents emerge when pc is truly unknown a priori. This raises the possibility that the observed scale invariance is an artifact of the critical window rather than a self-contained method.
  2. [§5] §5 (extensions to explosive/rigidity percolation and sandpile): the adaptation of the gap concept to these models is not derived from first principles, and the exponent-matching tests are performed on systems whose critical points or driving rates are already known from prior literature. A concrete test on a lattice or network with deliberately unknown pc (or on sandpile with unknown driving rate) is needed to substantiate the generality asserted in the abstract.
minor comments (2)
  1. [Abstract] The abstract asserts 'quantitative relations' but does not state their explicit functional form or name the dynamic exponents; adding one or two equations would improve clarity.
  2. [Figures and Methods] Figure captions and methods should specify the fitting procedure for the temporal scaling (e.g., range of bond-addition steps used, error estimation on exponents) and confirm that the same scaling window is used for both dynamic and static quantities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, providing clarifications and indicating the revisions we will make to strengthen the presentation and address the concerns about independence from prior knowledge of pc and the generality of the extensions.

read point-by-point responses

  1. Referee: [Results on dynamic scaling exponents] The central claim that the dynamic-static exponent relations enable determination of critical behavior without any prior knowledge of pc (abstract and results sections) is load-bearing. The relations appear to be obtained by numerical fitting inside regimes whose location is already known from conventional static analysis; no explicit parameter-free derivation from the definitions of gap and merged-cluster size is supplied that would guarantee the same exponents emerge when pc is truly unknown a priori. This raises the possibility that the observed scale invariance is an artifact of the critical window rather than a self-contained method.

    Authors: We acknowledge that demonstrating independence from prior knowledge of pc is essential for the central claim. The gap is defined directly as the size increment of the merged cluster upon each incremental bond addition, and the temporal self-similarity manifests as power-law scaling in the distributions of these quantities over extended intervals of the process. The dynamic exponents are extracted from these observed scalings, after which the relations to static exponents are applied. While known pc values are used for validation and to highlight the critical regime, the scale-invariant patterns can be identified by searching for the broadest power-law regimes in the gap time series without presupposing pc. We will add a new subsection and example in the revised manuscript that details a parameter-free procedure: scan possible temporal windows to maximize the scaling range and fit quality for the gap and merged-cluster quantities, then apply the exponent relations to obtain static exponents. This will be illustrated with a figure showing the extraction process and subsequent verification against known values, clarifying that the method is self-contained once the dynamic patterns are detected. revision: partial

  2. Referee: [§5] §5 (extensions to explosive/rigidity percolation and sandpile): the adaptation of the gap concept to these models is not derived from first principles, and the exponent-matching tests are performed on systems whose critical points or driving rates are already known from prior literature. A concrete test on a lattice or network with deliberately unknown pc (or on sandpile with unknown driving rate) is needed to substantiate the generality asserted in the abstract.

    Authors: We agree that the adaptations in §5 would benefit from clearer motivation and a direct test with unknown parameters. The gap is generalized by tracking the relevant size increment during the incremental process: for explosive percolation, the change in the largest cluster under the product rule; for rigidity percolation, the merger of rigid clusters upon constraint addition; and for the sandpile, the increments in avalanche sizes during early-time driving. In the revision, we will expand the text to derive these definitions more explicitly from the shared incremental dynamics. To provide the requested concrete test, we will include new results for a random network where pc is not used a priori: the dynamic exponents are extracted solely from the observed temporal scaling of the gap, the relations are applied to predict static exponents, and these are then compared to independently determined literature values. For the sandpile, we will add an analysis treating the driving rate as unknown and showing its inference from the temporal scaling exponents. These additions will be placed in an updated §5 to support the generality. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's core contribution is the numerical identification of scale-invariant temporal patterns in the gap (cluster size increment) and merged-cluster size during bond/site addition, followed by empirical observation of quantitative links between the resulting dynamic scaling exponents and known static percolation exponents. These links are presented as observed relations rather than a closed-form derivation or first-principles prediction; the method is shown to operate across a broad temporal window without explicit dependence on pre-known pc in the reported simulations. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text. The derivation remains self-contained as an empirical unification of dynamic and static scaling, with the generality to other models (explosive percolation, sandpiles) asserted via direct observation rather than reduction to prior inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard percolation theory assumptions about incremental cluster growth and the existence of static critical exponents; no explicit free parameters or new invented entities are described in the abstract.

axioms (2)
  • domain assumption Percolation proceeds by incremental addition of bonds or sites, producing clusters whose size increments (gaps) and merges can be tracked over time.
    This is the procedural foundation for defining the gap and merged-cluster quantities whose temporal scaling is analyzed.
  • domain assumption Static critical exponents of percolation are well-defined and conventional.
    The paper invokes these as the reference quantities to which dynamic exponents are quantitatively related.

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Reference graph

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