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arxiv: 2512.15366 · v3 · submitted 2025-12-17 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn

Exponents and front fluctuations in the quenched Kardar-Parisi-Zhang universality class of one- and two- dimensional interfaces

\'Angela Tajuelo-Valbuena , Jara Trujillo-Mulero , Juan J. Mel\'endez , Rodolfo Cuerno , Juan J. Ruiz-Lorenzo This is my paper

Pith reviewed 2026-05-16 21:49 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nn
keywords quenched KPZdepinning transitiondirected percolationinterface rougheningscaling exponentsfront fluctuationsprobability density functionuniversality class
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The pith

Quenched KPZ interfaces in one and two dimensions show scaling exponents matching the directed percolation depinning class.

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

The paper simulates an automaton model of the quenched Kardar-Parisi-Zhang equation to examine interface growth at the depinning transition. It measures the roughness exponent, growth exponent, and dynamic exponent along with related quantities, finding the full set largely matches the directed percolation depinning universality class. The authors also extract the probability density function of front height fluctuations in the growth regime and show it is strongly non-Gaussian while differing from the distribution known for the standard KPZ equation. A reader would care because the result supplies a concrete mapping that lets predictions for disordered interfaces be made from percolation concepts rather than solving the full stochastic equation.

Core claim

Simulations of an automaton discretization of the quenched KPZ equation in one and two dimensions at the depinning transition directly yield the exponents α, β, θ, δ, and z. These values are compatible with the directed percolation depinning class. The probability density function of the front fluctuations is computed in the growth regime and found to be asymptotically non-Gaussian with skewness and kurtosis values that also differ from the KPZ equation with time-dependent noise in both the central region and the tails.

What carries the argument

The automaton discretization of the quenched KPZ equation, together with direct measurement of the height-difference correlation function to obtain the dynamic exponent z.

If this is right

  • Interface roughening and depinning in quenched media obey the same scaling relations as directed percolation in one and two dimensions.
  • The probability density function of front fluctuations supplies an independent signature that distinguishes quenched noise from time-dependent noise.
  • Universality allows direct transfer of known directed percolation results to predict correlation lengths and fluctuation statistics in physical interfaces.
  • The measured dynamic exponent z obtained from real-space height correlations closes the set of scaling relations for the universality class.

Where Pith is reading between the lines

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

  • Real experiments on fluid invasion or magnetic domain walls in two dimensions could test the predicted non-Gaussian front statistics without needing to resolve microscopic disorder details.
  • If the compatibility persists when the same automaton is run in three dimensions, the directed percolation mapping would extend beyond the physical substrate dimensions studied here.
  • The difference between the quenched and time-dependent PDFs suggests that annealed versus quenched noise may belong to distinct universality classes even when the deterministic part of the equation is identical.

Load-bearing premise

The automaton discretization faithfully reproduces the continuum quenched KPZ dynamics and the simulated system sizes and times are large enough to reach the asymptotic scaling regime without crossover effects.

What would settle it

A larger-scale simulation or laboratory experiment on a quenched KPZ interface in one or two dimensions that returns an exponent set, for example the roughness exponent α, clearly outside the range reported for the directed percolation depinning class.

Figures

Figures reproduced from arXiv: 2512.15366 by \'Angela Tajuelo-Valbuena, Jara Trujillo-Mulero, Juan J. Mel\'endez, Juan J. Ruiz-Lorenzo, Rodolfo Cuerno.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Average front vs. time for different values of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Logarithmic plot of the saturation squared roughness [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Squared roughness vs. time for different values of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Correlation length computed from Eq. (19) with [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. PDF of the front fluctuations for the 1D model. (a) PDF com [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. PDF of the front fluctuations for the 2D model. (a) PDF com [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: shows two sample unscaled C2(r, t) functions, arbitrarily taken as representatives for both the 1D (a) and 2D (b) models, for several times, and fixed L. As previously mentioned, no signs of anomalous scaling (typically, a time￾dependent shift of the C2(r, t) vs. r curves for increasing t, see e.g. Ref. [50]) are observed, which is a sound demonstra￾tion of the validity of the FV scaling for both dimensio… view at source ↗
read the original abstract

We have simulated an automaton version of the quenched Kardar-Parisi-Zhang (qKPZ) equation in one and two dimensions in order to study the scaling properties of the interface at the depinning transition. Specifically, the $\alpha$, $\beta$, $\theta$, and $\delta$ critical exponents characterizing the surface kinetic roughening and depinning behaviors have been directly computed from the simulations. In addition, by studying the height-difference correlation function in real space, we have also been able to directly compute the dynamic correlation length and its associated dynamic critical exponent $z$. The full sets of scaling exponents are largely compatible with those of the Directed Percolation Depinning universality class for one and two dimensional interfaces. Furthermore, we have computed numerically the probability density function (PDF) of the front fluctuations in the growth regime, finding its asymptotic form in one and two dimensions. While the PDF features strongly non-Gaussian skewness and kurtosis values, it also differs from the PDF of the KPZ equation with time-dependent noise for physical substrate dimensions, both in the central part and at the tails of the distribution.

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 manuscript reports simulations of an automaton discretization of the quenched Kardar-Parisi-Zhang (qKPZ) equation in one and two dimensions at the depinning transition. The authors directly measure the roughness exponent α, growth exponent β, and the additional exponents θ, δ, and z from the height-difference correlation function, concluding that the full set is largely compatible with the Directed Percolation Depinning (DPD) universality class. They further compute the probability density function of front fluctuations in the growth regime and characterize its non-Gaussian features, which differ from those of the KPZ equation with time-dependent noise.

Significance. If the simulations are shown to reach the true asymptotic regime, the work would strengthen the numerical case for qKPZ depinning belonging to the DPD class in low dimensions and supply a concrete characterization of fluctuation statistics. The direct extraction of z from real-space correlations and the PDF analysis are useful additions to the literature on interface depinning.

major comments (3)
  1. [Numerical methods and scaling analysis] The central claim of compatibility with the DPD class rests on the assumption that the automaton faithfully reproduces the continuum qKPZ dynamics and that the measured exponents are asymptotic. No explicit tests of exponent stability versus system size L or time are presented, nor is there a comparison against alternative discretizations or continuum integrations; this is especially critical in 2D where depinning crossovers are known to be slow.
  2. [Results on critical exponents] The statement that the exponents are 'largely compatible' with DPD values is not accompanied by quantitative error bars on the differences or by data-collapse quality metrics; without these, it is difficult to assess whether the agreement is within numerical uncertainty or merely pre-asymptotic.
  3. [Dynamic correlation length] For the dynamic exponent z extracted from the height-difference correlation function, the manuscript does not discuss possible corrections-to-scaling or the fitting window used; this leaves open the possibility that the reported z reflects a transient rather than the true asymptotic value.
minor comments (2)
  1. [Front fluctuation statistics] The PDF figures would benefit from explicit reporting of skewness and kurtosis values together with their uncertainties.
  2. [Introduction] Notation for the exponents (α, β, θ, δ, z) should be defined once in a dedicated table or section to avoid ambiguity when comparing to DPD literature.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. We address each major comment below and have revised the manuscript to strengthen the presentation of our numerical results and scaling analysis.

read point-by-point responses

  1. Referee: [Numerical methods and scaling analysis] The central claim of compatibility with the DPD class rests on the assumption that the automaton faithfully reproduces the continuum qKPZ dynamics and that the measured exponents are asymptotic. No explicit tests of exponent stability versus system size L or time are presented, nor is there a comparison against alternative discretizations or continuum integrations; this is especially critical in 2D where depinning crossovers are known to be slow.

    Authors: We agree that explicit checks of asymptotic convergence are important, particularly in 2D. In the revised manuscript we add supplementary figures that plot the effective exponents versus L and versus time for the largest accessible scales; these show clear stabilization of α, β, θ, δ and z within the statistical uncertainties of our runs. We also expand the methods section to recall that the automaton is a standard, previously validated discretization of qKPZ whose exponents have been shown to match continuum results in the literature. While we have not performed new continuum integrations (owing to their substantially higher computational cost), the added convergence data directly address the concern about pre-asymptotic behavior. revision: partial

  2. Referee: [Results on critical exponents] The statement that the exponents are 'largely compatible' with DPD values is not accompanied by quantitative error bars on the differences or by data-collapse quality metrics; without these, it is difficult to assess whether the agreement is within numerical uncertainty or merely pre-asymptotic.

    Authors: We have revised the results section to report statistical error bars on every exponent, obtained from both ensemble fluctuations across independent realizations and from the covariance of the fitting procedures. We further include data-collapse plots for the height-difference correlation function together with a quantitative goodness-of-collapse metric (mean-squared deviation from the master curve). These additions allow the reader to judge that the observed agreement with DPD values lies within the estimated numerical uncertainties. revision: yes

  3. Referee: [Dynamic correlation length] For the dynamic exponent z extracted from the height-difference correlation function, the manuscript does not discuss possible corrections-to-scaling or the fitting window used; this leaves open the possibility that the reported z reflects a transient rather than the true asymptotic value.

    Authors: The revised text now explicitly states the fitting windows (in both real-space and time) used to extract z and shows the effective exponent versus scale to demonstrate the absence of strong drift. We also include a brief analysis of leading corrections-to-scaling by fitting an extended form that incorporates a sub-leading term; the leading exponent remains stable and consistent with the DPD value once the fitting window is restricted to the largest scales. revision: yes

Circularity Check

0 steps flagged

No circularity: all results are direct numerical measurements

full rationale

The paper reports direct numerical simulations of an automaton discretization of the qKPZ equation in 1D and 2D. Scaling exponents (α, β, θ, δ, z) and the PDF of front fluctuations are computed from the raw simulation trajectories without any analytic derivation, parameter fitting that is then re-labeled as a prediction, or load-bearing self-citation of a uniqueness theorem. The compatibility statement with the Directed Percolation Depinning class is a post-hoc numerical comparison, not a reduction of the reported values to the paper's own inputs. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the automaton belongs to the qKPZ class and that finite-size simulations have reached the scaling limit; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption The discrete automaton version belongs to the quenched KPZ universality class
    Invoked when the authors equate their measured exponents to the directed percolation depinning class.

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