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REVIEW 2 major objections 4 minor 64 references

Geometry dependence in linear interface growth

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper establishes that the linear Edwards-Wilkinson and Mullins-Herring universality classes split into flat and radial subclasses, giving universal height-fluctuation variances and covariances that depend on geometry.

desk verdict A solid, useful paper: the new exact 1D radial MH variance is real and the geometry splitting for EW/MH holds up well, but the 2D radial values are torus values until a spherical-mode computation confirms them. read the letter →

arxiv 1908.10892 v1 pith:QT6APIT2 submitted 2019-08-28 cond-mat.stat-mech

I. S. S. Carrasco , T. J. Oliveira This is my paper · ORCID
classification cond-mat.stat-mech
keywords Edwards-WilkinsonclassMullins-Herringinterfacegrowthradialgeometryflatheightdistributionvariancespatialcovariancetemporal
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

Interface growth universality classes are usually grouped by exponents, but this paper shows the grouping is finer: even the linear Edwards-Wilkinson (EW) and Mullins-Herring (MH) classes separate into flat and radial subclasses in one and two dimensions. In both geometries the one-point height distribution is Gaussian with zero mean, so the geometry dependence is carried entirely by the variance $\langle\chi^2\rangle_c$ of the rescaled fluctuation variable $\chi$. The paper derives a new exact 1D radial MH variance, $\approx 0.94573$ against $\approx 0.64697$ flat, confirms the 1D EW pair ($\approx 1.25331$ radial against $\approx 0.79788$ flat), and gives numerical 2D values from several discrete models and integrations of the growth equations. The rescaled spatial and temporal covariances also collapse onto universal but geometry-dependent curves, and the 2D radial EW temporal covariance appears to decay logarithmically. This matters because geometry dependence is thus a general property of linear classes, not a peculiarity of nonlinear growth.

What carries the argument

The load-bearing object is the rescaled height fluctuation $\chi$ defined by the KPZ ansatz $h\simeq v_\infty t+(\Theta t)^\beta \chi$, whose variance $\langle\chi^2\rangle_c$ is the single non-null cumulant of the Gaussian height distribution. For the radial case in 1D the argument runs through the linearized growth equation in polar coordinates, Eq. (A1), with noise amplitude $2D/r$; Fourier decomposition of the radial fluctuations yields the closed formula $\langle\chi^2\rangle_c = c_z(z-1)^{1/z}I_z/(2^{1/z}\pi)$, with $z=2$ for EW and $z=4$ for MH, giving the new radial MH value. The flat-geometry variances come from the standard exact solutions of the EW and MH equations on fixed substrates. Numerically, the radial geometry is realized by stochastically duplicating columns so the substrate size grows as $L(t)=L_0+\omega t$, and universality is checked by comparing discrete models with direct integrations of the growth equations.

What would settle it

Directly integrate the 2D radial EW and MH equations in coordinates adapted to a growing circular or spherical domain and compare the extrapolated $\langle\chi^2\rangle_c$ values with the expanding-substrate numbers in Table III; a mismatch would show the column-duplication mapping fails in 2D.

Watch

Extended reading notes

Core claim

The central claim is that the EW and MH universality classes split into flat and radial subclasses, with universal, geometry-dependent one-point fluctuations and two-point correlations. Using the KPZ ansatz $h\simeq v_\infty t+(\Theta t)^\beta \chi$, the authors show that $P(\chi)$ is Gaussian for every case studied, with $\langle\chi\rangle=0$ and all higher cumulants zero; the only distinguishing cumulant is $\langle\chi^2\rangle_c$. For 1D, the exact values are $\langle\chi^2\rangle_c\approx 0.79788$ flat and $\approx 1.25331$ radial for EW, and $\approx 0.64697$ flat and $\approx 0.94573$ radial for MH, the last derived here from the radial linear equation. In 2D, where analytic mode sums are not available, extrapolations of integrations and of discrete models give EW variances $\approx 0.076$ flat versus $\approx 0.160$ radial, and MH variances $\approx 0.199$ flat versus $\approx 0.415$ radial. The rescaled spatial covariance $F(r/\xi)$ is universal for each geometry but different between them; in particular the oscillatory decay of flat MH becomes a monotonic decay in the radial case, and the 1D flat EW curve agrees with the Airy1 curve of flat 1D KPZ interfaces. The temporal covariances match the known analytical expressions in 1D and follow the predicted asymptotic exponents in 2D.

Load-bearing premise

The load-bearing premise is that growth on a substrate whose size increases stochastically and linearly in time, implemented by column duplication, faithfully reproduces true radial growth in two dimensions, since the equivalence is verified against exact results only in 1D.

Editorial extensions

If this is right

  • The variance $\langle\chi^2\rangle_c$ can serve as a class indicator for Gaussian interfaces, since EW and MH values differ substantially in the same geometry and dimension.
  • Geometry must be reported and controlled when height distributions or covariances are used to identify universality classes in experiments such as thin-film growth or electrodeposition.
  • The 1D flat EW spatial covariance coinciding with the Airy1 curve suggests that flat 1D EW interfaces share a deeper statistical description with flat 1D KPZ interfaces.
  • In 2D radial EW the temporal covariance does not follow a power-law decay because $\beta=0$, so its logarithmically corrected decay must be accounted for in scaling analyses.
  • The column-duplication method for expanding substrates, previously used for nonlinear classes, also produces the radial statistics of linear classes.

Reading between the lines

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

  • I would expect the same flat-versus-radial variance split to appear for the MH class in three dimensions, where $\beta=1/8$ and the growth regime is non-logarithmic; the paper does not test this.
  • The closed formula for $\langle\chi^2\rangle_c$ in Appendix A is stated for general even $z$, so a direct check for $z=6$ would test whether the identity extends beyond EW and MH.
  • The observed 1D flat EW covariance matching Airy1 hints that exact covariance identities for linear interfaces may be derivable from the same machinery as the KPZ class; the paper only reports the numerical coincidence.
  • Since the paper's radial variances come from an expanding-substrate mapping rather than from genuinely curved coordinates in 2D, a direct integration of the linear equations on an expanding sphere would be a decisive independent check.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript studies whether the linear Edwards-Wilkinson (EW) and Mullins-Herring (MH) interface growth universality classes split into distinct flat and radial subclasses, analogous to the known splitting in nonlinear KPZ and VLDS classes. The authors derive exact one-dimensional variances for radial EW and, newly for radial MH (Eq. A12), and confirm these against numerical integrations of the growth equations and simulations of several discrete models. For 2D systems, they report variances extracted from integrations and discrete models on substrates whose size grows linearly in time (Tables II and III), and they analyze spatial and temporal covariances in both geometries and dimensions, finding universal but geometry-dependent scaling functions. The central claim is that the splitting is general to linear classes, including at the 2D EW upper critical dimension.

Significance. If the 2D results are accepted, the paper establishes a new general phenomenon: geometry-induced subclass splitting exists in linear as well as nonlinear universality classes. The 1D exact results, particularly the new radial MH variance, are clean and provide a useful anchor for the numerical method. The identification of universal, geometry-dependent amplitudes is also of practical interest for distinguishing universality classes in thin-film growth experiments. The paper is generally well organized, and the 1D analysis is convincing; the main risk is the unverified identification of the expanding square substrate with the true 2D radial (spherical) geometry, on which the central 2D claim depends.

major comments (2)
  1. [II, III.B, Table III] The identification of the expanding square lattice (a flat torus) with the 2D radial (spherical) geometry is not validated. In 1D, a circle and a periodic line have the same mode spectrum, so the exact circular EW and MH variances from Appendix A anchor the expanding-substrate data. In 2D, the sphere and the expanding flat torus have different Laplace-Beltrami spectra (spherical harmonics with degeneracy 2l+1 versus plane waves on a torus), and no exact or independent numerical result for the 2D spherical EW/MH variance is provided. The consistency across two parameter sets and two discrete models in Table III shows only that the duplication algorithm is robust, not that it reproduces spherical radial geometry. Because the central claim that 2D EW and MH classes split into flat and radial subclasses rests on the values <chi^2> = 0.160(1) (EW radial) and 0.415(3) (MH radial) in Table III and on the radial covariance curves in Figs. 3(c-d) and 5(c-d), the authors should either compute the variance from the spherical equations using the approach of Ref. [52], or provide an independent numerical test of the expanding-substrate equivalence in 2D, or explicitly reframe the claim as applying to expanding flat substrates and justify why this is the relevant radial subclass.
  2. [V, Fig. 5(c)] The 2D radial EW temporal covariance does not collapse onto a single scaling curve, and the authors introduce a three-parameter fit A(y) = a + b/(c + ln y) to describe the average data. This is a stated failure for one of the paper's main two-point quantities, and it weakens the universality claim for the temporal covariances in 2D radial geometry. The authors should either provide a theoretical argument for the logarithmic decay form, or explicitly present this case as an exception where universality is not yet established, rather than relying on an ad hoc fit.
minor comments (4)
  1. [Abstract] The phrase 'university classes' in the abstract should be 'universality classes'.
  2. [Appendix A, Eq. A9] The Poisson summation is truncated to the k=0 term with the statement that the main contribution comes from k=0. Since the z=4 result is new and used for calibration, it would be helpful to justify this truncation by estimating the k != 0 terms in the large-time limit, or to state explicitly that the result is asymptotic rather than exact.
  3. [III.B, Eq. 7] For the 2D EW class, the logarithmic ansatz in Eq. (7) is introduced separately from the general KPZ ansatz in Eq. (1). The authors should clarify the relationship between the two, since a reader may wonder whether the logarithmic form follows by a limiting procedure from Eq. (1) when beta = 0.
  4. [IV, Fig. 3] The fitted values of the exponent gamma for the 2D radial cases (gamma = 0.39 for EW and gamma = 0.61 for MH) are reported without error bars. Including uncertainties would make the claimed universality of the asymptotic decay more persuasive.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central variances are exact solutions or cross-validated numerical integrations, and the expanding-substrate method is independently checked in 1D.

full rationale

The paper's central 1D results are exact solutions of the linear EW and MH equations: the flat variances come from standard results, the radial EW variance from Singha's solution, and the radial MH variance is derived in Appendix A from the polar-coordinate equation. These are parameter-free mathematical consequences, not fitted inputs. In 2D, the variances are obtained by direct numerical integration of the same linear equations on fixed and expanding substrates, with two independent parameter sets agreeing with each other; the discrete-model g2 values provide an independent cross-check because they compare separately simulated roughness amplitudes against the equation-derived variances. The expanding-substrate protocol is motivated by prior works including the authors' own, but it is validated in 1D against exact circular solutions, and in 2D it is an unverified geometric extrapolation (expanding square versus true spherical geometry) that constitutes a correctness risk, not a circular reduction. No fitted parameter is renamed as a prediction, and no load-bearing claim reduces to a self-citation or to a definitional identity. Therefore no circular step is identified.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

The central results rest on the standard linear stochastic growth equations, on the ansatz relating height to a rescaled universal variable, and on the identification of expanding substrates with radial geometry. Fitted coefficients enter only for the discrete-model comparison, not for the exact 1D predictions.

free parameters (10)
  • nu_z (Family model, EW) = 0.78(2)
    Obtained via inverse method (Sec. III.A and Appendix B); used with D=1/2 to compute Theta and to test universality.
  • nu_z (LC1 model, MH) = 0.61(2)
    Inverse method (Sec. III.A); first estimate for LC1, used to compute Theta.
  • nu_z (LC2 model, MH) = 0.142(6)
    Inverse method (Sec. III.A); consistent with prior nu_4 approx 0.14 estimate from Ref. [14].
  • 2D EW <chi^2>_c (flat) = 0.077(2) (EWI), 0.075(3) (EWII)
    Extrapolated from w^2/(Theta ln t) for integrations of the 2D EW equation (Sec. III.B, Fig. 2a); used to compute Theta for discrete models.
  • 2D EW <chi^2>_c (radial) = 0.160(1) (EWI), 0.161(2) (EWII)
    Same extrapolation on expanding substrates (Fig. 2a); central evidence for geometry dependence in 2D.
  • 2D MH <chi^2>_c (flat) = 0.200(2) (MHI), 0.198(2) (MHII)
    Extrapolated from w^2/(Theta t)^{2 beta} for the 2D MH equation (Fig. 2b).
  • 2D MH <chi^2>_c (radial) = 0.415(3) (MHI), 0.416(3) (MHII)
    Extrapolated on expanding substrates (Fig. 2b); supports geometry-dependent variance in 2D.
  • gamma (spatial covariance exponent, 2D EW) = approx 0.67 (flat), approx 0.39 (radial)
    Fitted in Eq. 8 for large s (Sec. IV, Fig. 3c); descriptive, not load-bearing.
  • gamma (spatial covariance exponent, 2D MH) = approx 0 (flat), approx 0.61 (radial)
    Fitted in Eq. 8 (Sec. IV, Fig. 3d); descriptive, not load-bearing.
  • a, b, c (temporal covariance fit, 2D radial EW) = not reported
    Fit A(y) = a + b/(c + ln y) to average data (Sec. V, Fig. 5c); acknowledged as not a clean collapse.
assumptions (6)
  • domain assumption Growth of the discrete models is described in the continuum by the linear EW or MH equation (Eqs. 3-4) with Gaussian white noise; the models belong to these universality classes.
    Central to extracting Theta and comparing with theory; standard in the field and supported by known mappings (e.g., SSS to the kinetic Ising model).
  • domain assumption An interface growing on a substrate whose size increases linearly in time, L(t) = L0 + omega t, realizes the radial (curved) geometry for linear growth equations.
    Section II; validated in 1D against exact radial solutions, assumed in 2D.
  • domain assumption For radial geometry the linear equation can be linearized by replacing r^z with <r>^z = (r0 + F t)^z, and the noise correlator scaled by 1/(r0 + F t).
    Appendix A, Eq. A2; standard small-fluctuation approximation.
  • standard math The k=0 term dominates the Poisson summation in Eq. A9 (S_z approx c_z sigma).
    Asymptotically valid for large sigma; used to derive the exact variance in Eq. A12.
  • domain assumption The KPZ-type ansatz (Eq. 1), h approx v_inf t + (Theta t)^beta chi, applies to linear classes, with Gaussian chi and zero mean.
    Section III; from linearity the single-point height is Gaussian; the ansatz defines the rescaled variable.
  • domain assumption For 2D EW (beta=0), the width grows as (D/nu_2)<chi^2> ln(nu_2 t/a), so the rescaled ansatz becomes h approx v_inf t + sqrt(Theta ln t) chi (Eq. 7).
    Standard logarithmic growth at the upper critical dimension.

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Cite this review

Pith. "Pith review of Geometry dependence in linear interface growth." pith.science (2026). https://pith.science/paper/QT6APIT2

@misc{pith2026190810892,
  author       = {Pith},
  title        = {Pith review of: Geometry dependence in linear interface growth},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QT6APIT2}},
  note         = {Machine review of arXiv:1908.10892}
}
abstract

The effect of geometry in the statistics of \textit{nonlinear} universality classes for interface growth has been widely investigated in recent years and it is well known to yield a split of them into subclasses. In this work, we investigate this for the \textit{linear} classes of Edwards-Wilkinson (EW) and of Mullins-Herring (MH) in one- and two-dimensions. From comparison of analytical results with extensive numerical simulations of several discrete models belonging to these classes, as well as numerical integrations of the growth equations on substrates of fixed size (flat geometry) or expanding linearly in time (radial geometry), we verify that the height distributions (HDs), the spatial and the temporal covariances are universal, but geometry-dependent. In fact, the HDs are always Gaussian and, when defined in terms of the so-called "KPZ ansatz" $[h \simeq v_{\infty} t + (\Gamma t)^{\beta} \chi]$, their probability density functions $P(\chi)$ have mean null, so that all their cumulants are null, except by their variances, which assume different values in the flat and radial cases. The shape of the (rescaled) covariance curves is analyzed in detail and compared with some existing analytical results for them. Overall, these results demonstrate that the splitting of such university classes is quite general, being not restricted to the nonlinear ones.

Figures

Figures reproduced from arXiv: 1908.10892 by the authors.

Figure 1
Figure 1. FIG. 1. Extrapolations to the long time limit of the rescaled [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Extrapolations to the long time limit of rescaled [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Rescaled spatial covariances for EW (left) and MH [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of rescaled spatial covariances for EW [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Calculation of the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.