{"id":"c9cea176-a1c3-49a9-ac20-a4674de5c4f0","arxiv_id":"1908.10892","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":10,"one_line_summary":"For Edwards-Wilkinson and Mullins-Herring interfaces in 1D and 2D, the rescaled height fluctuations remain Gaussian but their variance and the two-point covariance functions become geometry-dependent, confirming that flat/radial splitting is a general feature of interface universality classes.","lead":"This paper shows that the two classic linear classes of growing interfaces, Edwards-Wilkinson and Mullins-Herring, split into flat and circular subclasses, just as the nonlinear KPZ and VLDS classes do. The result matters because it gives experimenters a simple Gaussian-width test to determine both the universality class and the growth geometry of a measured surface.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 2D radial subclass values in Table III depend on an unverified equivalence between expanding flat-square substrates and true spherical radial geometry; a direct spherical-mode computation should settle it.","rationale":"The strongest part of the paper is 1D: the exact EW circular variance and the newly derived MH variance (Eq. A12) are consistent with simulations, and universality across models is demonstrated. My concern is not about the analytic derivation, which appears sound in leading order, nor about the existence of a geometry effect in 1D. It is specifically that the 2D extension, which is essential to the abstract's claim that the splitting is general, uses an expanding flat square as a proxy for radial geometry. In 1D the proxy is exact because S^1 and the periodic line share the same spectrum; in 2D the sphere and the flat torus do not. The linearity of EW/MH makes the exact spherical calculation straightforward, so the absence of this check is the weakest load-bearing point. If the check reproduces Table III, the verdict should remain ACCEPT; if not, the 2D central claim is not supported.","tokens_in":16406,"tokens_out":23009,"duration_ms":258095,"concrete_test":"Compute the width variance for the linear EW (z=2) and MH (z=4) equations on an expanding 2-sphere, R(t) = F t, using the exact spherical-harmonic mode equations d v_l/dt = -2ν [l(l+1)]^{z/2}/(F^z t^z) v_l + (2D/(F^2 t^2)) (with the correct spherical-harmonic noise normalization), sum over l with degeneracy 2l+1, and extract the asymptotic coefficients of w^2 (for EW, the coefficient of ln t; for MH, the coefficient of t^{2β}). Compare these coefficients with the <chi^2>_c entries in Table III for EWI/EWII and MHI/MHII. Agreement within error bars confirms the expanding-square mapping; disagreement means the 2D radial subclasses are artifacts of the duplication algorithm.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central 2D claim (EW and MH split into flat/radial subclasses, with universal variances in Table III) rests on the identification of a substrate whose size grows as L(t) = L0 + ωt with column/row duplication (Sec. II) with radial geometry. In 1D this identification is calibrated: the circle and the periodic line have the same mode spectrum, so exact circular variances for EW and the new MH result (Eq. A12) anchor the expanding-substrate data. In 2D the expanding square is a flat torus, not a sphere; the spherical EW/MH equations cited in Sec. IV have a different spectrum (spherical harmonics, degeneracy 2l+1). No exact or independent numerical result for the 2D spherical variance is given, so the values <chi^2> = 0.160(1) (EW) and 0.415(3) (MH) in Table III, and the radial covariance curves, could depend on the duplication algorithm rather than on the radial geometry. The paper's appendix solves only the 1D case, leaving this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16683,"tokens_out":10633,"duration_ms":115633,"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":[{"comment":"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.","section":"II, III.B, Table III"},{"comment":"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.","section":"V, Fig. 5(c)"}],"minor_comments":[{"comment":"The phrase 'university classes' in the abstract should be 'universality classes'.","section":"Abstract"},{"comment":"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.","section":"Appendix A, Eq. A9"},{"comment":"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.","section":"III.B, Eq. 7"},{"comment":"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.","section":"IV, Fig. 3"}],"recommendation":"major_revision","confidential_remarks":"The 1D part of the paper is solid and the new MH radial variance is a useful contribution. My main concern is the 2D radial equivalence: because the paper's prior work on KPZ (Ref. [33]) used the same expanding-substrate method, the authors may believe the equivalence is already established, but for linear classes the mode spectrum directly sets the variance amplitude, so the flat-torus versus sphere distinction is material. I would recommend asking the authors to obtain an independent 2D spherical result, even numerically, or to explicitly restrict the 2D claim to expanding flat substrates. If they can address this, I would support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuinely useful paper that fills a real gap. The geometry splitting established for the nonlinear classes (KPZ, VLDS) is now systematically demonstrated for the linear EW and MH classes, and it comes with a new exact result — the 1D radial MH variance, <chi^2>_c approximately 0.94573, derived in Appendix A — that checks out against numerics. I would send this to a serious referee.\n\nWhat is actually new and good: the Appendix A derivation is parameter-free and the simulations converge to it within 2%. The 2D survey — four discrete models (SSS, Family, LC1, LC2) plus direct integration of the continuum equations with two parameter sets each, in both geometries — is the kind of cross-validation that gives this subfield confidence. The covariance results, especially the MH spatial covariance changing from oscillatory (flat) to monotonic (radial) decay, parallel the VLDS picture in a satisfying way. I trust the 1D results completely.\n\nSoft spots, in proportion:\n\n1. The 2D radial geometry is the real issue. An expanding square with periodic boundaries is a flat torus, not a sphere, and the two have different mode spectra. The 1D validation against exact circular results is reassuring, but 1D is the special case where the expanding line literally is the circle. No exact or independent 2D spherical result is reported, so the radial values in Table III (0.160 for EW, 0.415 for MH) could depend on the duplication algorithm rather than on the radial geometry. This is checkable — a spherical-harmonics computation of the 2D EW variance would settle it — and I would ask for that, or for direct spherical simulations, before treating those numbers as radial universality. Not a fatal flaw; a review request.\n\n2. The Airy1 suggestion (1D flat EW spatial covariance coincides with Airy1) is an overreach. A curve collapse with one fitted rescaling is suggestive, not conclusive, and the claim is not needed for the paper's message. Soften or support it.\n\n3. Minor: the 2D radial EW temporal covariance collapses poorly and is fit with an ad hoc three-parameter form; the appendix's printed intermediate constants (c4 and I4) do not quite reproduce the quoted final value, though the final number is right; and no code or data is shipped. All fixable.\n\nWho this is for: people working on interface-growth universality classes, and experimentalists using height-distribution diagnostics in thin-film growth. The Gaussian-plus-geometry-dependent-variance diagnostic is practically useful. The paper deserves peer review; the spherical check should be the main request.","headline":"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.","tokens_in":17271,"tokens_out":17778,"would_cite":true,"duration_ms":159939,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Edwards-Wilkinson class","Mullins-Herring class","interface growth","radial geometry","flat geometry","height distribution variance","spatial covariance","temporal covariance"],"falsifier":"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.","tokens_in":16162,"feed_emoji":"📈","tokens_out":10209,"duration_ms":92163,"temperature":0.7,"pith_summary":"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.","feed_headline":"Flat vs radial geometry splits linear growth universality","feed_subtitle":"Universal height and covariance statistics differ by geometry, even in linear growth classes.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the exact flat one-dimensional EW and MH squared-width amplitudes and scaling exponents, setting the flat variance references.","marker":"[14]"},{"why":"States the height-fluctuation ansatz $h\\simeq v_\\infty t+(\\Theta t)^\\beta \\chi$ that defines the fluctuation variable whose variance is the central quantity.","marker":"[15]"},{"why":"Gives exact radial temporal covariances for 1D EW and MH and the asymptotic decay exponent, providing the analytical curves matched by simulations.","marker":"[35]"},{"why":"Reports the 1D radial EW variance used as the radial reference for the expanding-substrate method.","marker":"[36]"},{"why":"Derives the asymptotic spatial-covariance form $F(s)\\sim s^{-\\gamma}e^{-cs^\\delta}$ used to fit the flat and radial covariance tails.","marker":"[37]"},{"why":"Provides the flat-geometry temporal covariance scaling function $A_f(y)$ used as the analytical comparison for the flat cases.","marker":"[38]"},{"why":"Shows the symmetric single-step model maps exactly to the EW equation with $\\nu_2=D=1$, fixing model parameters for one of the discrete tests.","marker":"[47]"},{"why":"Establishes the linearly expanding substrate method for realizing radial geometry in simulations, which the paper extends to linear classes.","marker":"[28]"}],"fun_headline_variants":["Linear growth universality splits by flat vs radial geometry","Geometry splits Edwards-Wilkinson and Mullins-Herring classes","Flat vs radial: new subclasses for linear interface growth","Linear growth classes: geometry changes variance, not Gaussianity","Universality beyond nonlinear: geometry splits linear growth too"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Linear growth universality splits by flat vs radial geometry","Geometry splits Edwards-Wilkinson and Mullins-Herring classes","Flat vs radial: new subclasses for linear interface growth","Linear growth classes: geometry changes variance, not Gaussianity","Universality beyond nonlinear: geometry splits linear growth too"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1520,"prompt_tokens":1111,"completion_tokens":409,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":727,"completion_tokens_details":{"reasoning_tokens":338}},"tokens_in":727,"tokens_out":409,"duration_ms":4584,"temperature":1.0,"reasoning_tokens":338,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:31:37.782646+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives exact radial temporal covariances for 1D EW and MH and the asymptotic decay exponent, providing the analytical curves matched by simulations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the 1D radial EW variance used as the radial reference for the expanding-substrate method."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the asymptotic spatial-covariance form $F(s)\\sim s^{-\\gamma}e^{-cs^\\delta}$ used to fit the flat and radial covariance tails."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the flat-geometry temporal covariance scaling function $A_f(y)$ used as the analytical comparison for the flat cases."},{"cited_title":"KPZ ansatz","cited_arxiv_id":null,"evidence_quote":"Shows the symmetric single-step model maps exactly to the EW equation with $\\nu_2=D=1$, fixing model parameters for one of the discrete tests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the linearly expanding substrate method for realizing radial geometry in simulations, which the paper extends to linear classes."}],"review_version":1}