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arxiv: 2604.06040 · v1 · submitted 2026-04-07 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· nlin.AO

Dynamical phase diagram of synchronization in one dimension: universal behavior from Edwards-Wilkinson to random deposition through Kardar-Parisi-Zhang

Ricardo Gutierrez , Rodolfo Cuerno This is my paper

Pith reviewed 2026-05-10 18:26 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nnnlin.AO
keywords synchronizationKardar-Parisi-ZhangEdwards-Wilkinsonphase oscillatorsone dimensiondynamical phase diagramuniversality classesphase slips
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0 comments X

The pith

Synchronization of one-dimensional phase oscillators crosses over from Edwards-Wilkinson to Kardar-Parisi-Zhang scaling as randomness strength or coupling nonoddity increases.

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

Numerical simulations map the full dynamical phase diagram for chains of phase oscillators subject to either columnar disorder or time-dependent noise. Outside the synchronized region the growth of phase differences follows random deposition or linear growth. Inside the synchronized region a crossover occurs from Edwards-Wilkinson to Kardar-Parisi-Zhang scaling, with the effective KPZ coupling parameter capturing part of the combined influence of randomness strength and nonodd coupling. Saturation times, saturation values, and the gradual emergence of synchronous dynamics are all characterized. Near the desynchronization boundary, frequent phase slips distort the observed scaling.

Core claim

The paper shows that one-dimensional synchronization displays generic scale invariance whose universality class depends on parameters: random deposition or linear growth appears in the absence of synchronization, while a crossover from Edwards-Wilkinson to Kardar-Parisi-Zhang behavior with the matching type of noise occurs inside the synchronous region when randomness strength or the nonoddity of the coupling is increased, their joint effect being partly encoded in the KPZ coupling.

What carries the argument

The KPZ coupling, which partially encodes the joint effect of randomness strength and coupling nonoddity and drives the crossover between universality classes.

If this is right

  • In the absence of synchronization the phase-difference dynamics belong to random deposition for time-dependent noise and to linear growth for columnar disorder.
  • Inside the synchronous region the effective KPZ coupling controls the gradual replacement of Edwards-Wilkinson by Kardar-Parisi-Zhang scaling.
  • Phase slips near the desynchronization boundary produce measurable distortions of the scaling behavior.
  • Saturation times and saturated roughness values are mapped across the entire parameter space for both types of randomness.

Where Pith is reading between the lines

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

  • The same crossover mechanism may persist in higher dimensions provided the mapping between oscillator coupling and surface-growth nonlinearity remains valid.
  • Experimental realizations with electronic or chemical oscillators could test the predicted distortion of scaling caused by phase slips if the system is tuned close to the synchronization boundary.
  • The phase diagram supplies concrete parameter ranges where clean Edwards-Wilkinson or Kardar-Parisi-Zhang exponents should be observable in large-scale simulations or experiments.

Load-bearing premise

Finite-size numerical simulations of finite oscillator chains correctly identify the asymptotic universality class without residual finite-size corrections or insufficient ensemble averaging, especially near the desynchronization boundary.

What would settle it

Systematic deviation of measured roughness and growth exponents from the expected Edwards-Wilkinson or Kardar-Parisi-Zhang values when system size is increased by an order of magnitude or ensemble averaging is substantially extended.

Figures

Figures reproduced from arXiv: 2604.06040 by Ricardo Gutierrez, Rodolfo Cuerno.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
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Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
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Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
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Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
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Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 8
Figure 8. Figure 8: With our numerical accuracy, it is very hard to distin [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
read the original abstract

Synchronization in one dimension displays generic scale invariance with universal properties previously observed in surface kinetic roughening and the wider context of the Kardar-Parisi-Zhang (KPZ) universality class. This has been established for phase oscillators and also for some limit-cycle oscillators, both in the presence of columnar (quenched) disorder and of time-dependent noise, by extensive numerical simulations, and has been analytically motivated by continuum approximations in the strong oscillator coupling limit. The robustness and the precise boundaries in parameter space for such critical behavior remain unclear, however, which may preclude further developments, including the extension of these results to higher dimensions and the experimental observation of nonequilibrium criticality in synchronizing (e.g.~electronic or chemical) oscillators. We here present complete numerical phase diagrams of one-dimensional synchronization, including saturation times and values, but, most importantly, also dynamical features giving insight into the gradual emergence of synchronous dynamics, based on systems of phase oscillators with either type of randomness. In the absence of synchronization, the dynamics evolves as expected for random deposition (for time-dependent noise) or linear growth (for columnar disorder), while a crossover from Edwards-Wilkinson to Kardar-Parisi-Zhang behavior (with the corresponding type of randomness) is observed as the randomness strength, or the nonoddity of the coupling among oscillators, is increased in the synchronous region -- their combined effect being partially captured by the so-called KPZ coupling. The distortion of scaling due to phase slips near the desynchronization boundary, a feature that is likely to play a role in experimental contexts, is also discussed.

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 numerically constructs dynamical phase diagrams for one-dimensional synchronization of phase oscillators driven by either time-dependent noise or columnar (quenched) disorder. It reports that the unsynchronized regime reproduces random-deposition or linear-growth scaling, while the synchronized regime exhibits a crossover from Edwards-Wilkinson to Kardar-Parisi-Zhang universality (with matching roughness and growth exponents) as randomness strength or coupling non-oddity is increased; this crossover is said to be partially captured by an effective KPZ coupling. Saturation times, dynamical features, and the distorting role of phase slips near the desynchronization boundary are also presented.

Significance. If the reported exponents and scaling functions are shown to be asymptotic, the work supplies a complete 1D phase diagram that clarifies the boundaries and gradual emergence of KPZ-like behavior in synchronization, extending earlier continuum approximations and numerical observations. The explicit treatment of both noise types, saturation dynamics, and phase-slip effects adds concrete value for potential experimental realizations in oscillator arrays and for future higher-dimensional extensions.

major comments (2)
  1. [Abstract and results on the synchronous region] Abstract and results on the synchronous region: the central claim of an asymptotic EW-to-KPZ crossover (with α=1/2, β=1/3) as randomness or non-oddity increases rests on finite-length oscillator chains, yet no system sizes, number of realizations, or multi-size extrapolations are referenced to demonstrate convergence without residual corrections, especially near the desynchronization line where phase slips are stated to distort scaling.
  2. [Discussion of phase slips near the desynchronization boundary] Discussion of phase slips near the desynchronization boundary: the manuscript notes that phase slips distort scaling in this region, but provides no quantitative test (e.g., comparison of effective exponents or saturation times across chain lengths L=100, 200, … or data-collapse metrics) to confirm that the reported universality-class boundaries remain reliable rather than being shifted by these finite-size or slip-induced transients.
minor comments (2)
  1. [Abstract] The abstract refers to 'extensive numerical simulations' without summarizing key numerical parameters (chain lengths, ensemble sizes, integration times); adding a short methods paragraph or table would improve reproducibility.
  2. [Figures] Figure captions would benefit from explicit listing of the oscillator parameters and randomness strengths used in each panel to allow direct comparison with the phase-diagram boundaries.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments provided. We respond to each major comment below and indicate the revisions we will make to address the concerns.

read point-by-point responses

  1. Referee: Abstract and results on the synchronous region: the central claim of an asymptotic EW-to-KPZ crossover (with α=1/2, β=1/3) as randomness or non-oddity increases rests on finite-length oscillator chains, yet no system sizes, number of realizations, or multi-size extrapolations are referenced to demonstrate convergence without residual corrections, especially near the desynchronization line where phase slips are stated to distort scaling.

    Authors: We thank the referee for highlighting this important point. The central claim is supported by extensive numerical simulations on finite chains, with convergence checked through varying system sizes and ensemble averages. However, we acknowledge that the abstract and main text do not sufficiently reference the specific system sizes, number of realizations, or details of the extrapolations. In the revised manuscript, we will include a clear description of these numerical aspects, including how we ensured the absence of significant residual corrections, with particular attention to the region near the desynchronization line. This will strengthen the presentation of the asymptotic EW-to-KPZ crossover. revision: yes

  2. Referee: Discussion of phase slips near the desynchronization boundary: the manuscript notes that phase slips distort scaling in this region, but provides no quantitative test (e.g., comparison of effective exponents or saturation times across chain lengths L=100, 200, … or data-collapse metrics) to confirm that the reported universality-class boundaries remain reliable rather than being shifted by these finite-size or slip-induced transients.

    Authors: We agree that a quantitative test would enhance the discussion of phase slips. The manuscript currently provides a qualitative description of their distorting effects near the desynchronization boundary. We will revise the manuscript to incorporate quantitative analyses, including comparisons of effective exponents and saturation times across different chain lengths, as well as data-collapse metrics. These additions will help confirm that the reported universality-class boundaries are reliable and not unduly shifted by finite-size or slip-induced effects. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct numerical simulation compared to external literature benchmarks

full rationale

The paper establishes its phase diagram and EW-to-KPZ crossover through direct numerical integration of the oscillator equations on finite chains, extraction of saturation times, roughness exponents, and growth exponents, and comparison of these measured quantities to independently known values from the surface-growth literature. No equation or claim reduces a derived quantity to a parameter fitted from the same data, nor does any load-bearing step rely on a self-citation whose content is itself unverified or defined circularly within the present work. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The phase-diagram claims rest on numerical integration of standard phase-oscillator equations whose scaling is then matched to independently tabulated universality classes; no new entities are postulated and the only free parameters are the physical coupling and randomness strengths that are scanned rather than fitted to a target result.

axioms (1)
  • domain assumption Measured growth exponents and scaling functions in the synchronous regime can be unambiguously assigned to the Edwards-Wilkinson or Kardar-Parisi-Zhang classes by comparison with known values.
    The paper identifies crossovers by matching numerical data to the established exponents and scaling forms of surface-roughening models.

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discussion (0)

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