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arxiv: 2604.27896 · v1 · submitted 2026-04-30 · ❄️ cond-mat.stat-mech

Discrete Lattice Models for Interface Growth on a Complete Graph

J. M. Marcos , J. J. Mel\'endez , R. Cuerno , J. J. Ruiz-Lorenzo This is my paper

Pith reviewed 2026-05-07 04:53 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords interface growthcomplete graphKPZ universality classballistic depositionrestricted solid-on-solidroughness saturationinfinite dimensionsdiscrete models
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The pith

The Ballistic Deposition model on a complete graph exhibits saturation roughness that increases with system size and an ultrafast growth regime, placing it outside the KPZ universality class.

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

This paper tests whether discrete lattice models for interface growth, which belong to Edwards-Wilkinson or Kardar-Parisi-Zhang classes on regular lattices, behave the same way when placed on a complete graph that mimics infinite dimensions. After checking trivial cases, it focuses on the Restricted Solid-on-Solid and Ballistic Deposition models. Both show height fluctuations with a pronounced left tail from lagging nodes. The RSOS model has saturation roughness decreasing with size, consistent with continuum expectations. However, the BD model has saturation roughness increasing with size plus an extra ultrafast growth phase, so it falls outside the KPZ class in this setting.

Core claim

When defined on the complete graph, the Ballistic Deposition model develops a saturation roughness that grows with the number of sites and displays an additional ultrafast growth regime at long times, in contrast to the Kardar-Parisi-Zhang continuum equation and to the Restricted Solid-on-Solid model which behaves more closely to expectations.

What carries the argument

The complete graph, where every site connects to all others, used to define the discrete growth rules for RSOS and BD models, with observables being the interface roughness, height fluctuations, power spectra, and two-time autocorrelations.

If this is right

  • Discrete models in infinite-dimensional limits can deviate from their continuum counterparts in scaling behavior.
  • The choice of growth rule (BD versus RSOS) matters for whether the model stays in the expected universality class on highly connected topologies.
  • Height fluctuations develop asymmetric distributions with a heavy left tail due to sites that lag behind.
  • Long-time dynamics include regimes not present in lower-dimensional or continuum versions for some models.

Where Pith is reading between the lines

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

  • These results imply that mean-field or infinite-dimensional versions of discrete growth models may need independent classification rather than assuming they follow continuum universality.
  • Applications to systems with all-to-all interactions, such as certain network growth processes, could show similar deviations from standard classes.
  • Further simulations on graphs with high but finite connectivity could test how the deviations emerge as dimension increases.

Load-bearing premise

The complete graph topology accurately captures the infinite-dimensional limit for these discrete interface models, and the chosen set of observables suffices to determine universality class membership.

What would settle it

Measuring the saturation roughness on complete graphs with increasing numbers of nodes and finding that it decreases rather than increases for the Ballistic Deposition model, or finding no evidence of an ultrafast growth regime at late times.

Figures

Figures reproduced from arXiv: 2604.27896 by J. J. Mel\'endez, J. J. Ruiz-Lorenzo, J. M. Marcos, R. Cuerno.

Figure 1
Figure 1. Figure 1: Log–log plot of the squared roughness, w 2 (t), as a function of t for the RSOS model. Data are shown for different system sizes N (see legend) and m = 10. The solid black line corresponds to the RD scaling w 2 (t) ∼ t. larger height was already satisfied before the attempt. Among the nodes with smaller height, it suffices to compare with the global minimum height hmin. The condition then reduces to hi + 1… view at source ↗
Figure 2
Figure 2. Figure 2: Log–log plot of the saturation value of the square of the roughness, w 2 sat, as a function of the system size N for the RSOS model. In this plot, m = 10. The solid black line corresponds to w 2 sat ∼ N −1/4 scaling. the condition hmax − hmin = m is satisfied, growth at nodes with maximum height becomes blocked, forcing the roughness to decrease. When this condition is not satisfied, growth at these nodes … view at source ↗
Figure 3
Figure 3. Figure 3: Histogram of the rescaled height fluctuations χ [see Eq. (11)], for the RSOS model. In this figure, N = 1000 and m = 10. accumulation of nodes around this maximum value. A similar behavior is observed for other system sizes. However, as N increases, the probability that some node remains lagging behind—freezing the dynamics and causing the maximum height to saturate—also increases. Together with the fact t… view at source ↗
Figure 4
Figure 4. Figure 4: Time power spectrum S(ω) [see Eq. (8)] for the RSOS model. Data are shown for different system sizes N (see legend) and m = 10. The solid black line corresponds to S(ω) ∼ ω −2 scaling.                                                               ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ … view at source ↗
Figure 5
Figure 5. Figure 5: Spatially averaged two-time height autocorrelation function, rescaled by the squared roughness, Ct(t, t0)/w2 (t0), as a function of the time difference t − t0 for the RSOS model, shown for different waiting times t0 = {1, 2, 3, 4, 5, 6, 7}, which appear bottom to top in the inset. The system size is N = 1000 and m = 10. The solid black line corresponds to the theoretical prediction given by Eq. (17) with τ… view at source ↗
Figure 6
Figure 6. Figure 6: Spatially averaged two-time height autocorrelation function, rescaled by the squared roughness, Ct(t, t0)/w2 (t0), as a function of the time difference t − t0 for the RSOS model, shown for different waiting times t0 = {20, 30, 40, 50}. The system size is N = 1000 and m = 10. The solid black line corresponds to the theoretical prediction given by Eq. (17) with τRSOS = m = 10. The interpretation of aging in … view at source ↗
Figure 7
Figure 7. Figure 7: Log–log plot of the square of the roughness, w 2 (t), as a function of t for the BD model. Data are shown for different system sizes N (see legend). The solid black line corresponds to w 2 (t) ∼ t 2 scaling, while the dashed black line corresponds to w 2 (t) ∼ t behavior. ⊙ ⊙ ⊙ ⊙ ⊙ ⊙⊙⊙⊙ ⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙                        … view at source ↗
Figure 8
Figure 8. Figure 8: Log–log plot of the square of the roughness rescaled by the system size, w 2 (t)/N, as a function of t for the BD model. Data are shown for different system sizes N (see legend). The solid black line corresponds to w 2 (t) ∼ t 2 scaling, while the dashed black line corresponds to w 2 (t) ∼ t behavior. the roughness, which exhibits a behavior that can be considered typical within the context of kinetic roug… view at source ↗
Figure 9
Figure 9. Figure 9: Histogram of the rescaled height fluctuations χ [see Eq. (11)], for the BD model. The solid black line corresponds to P(χ) ∼ e χ. In this figure N = 1000. ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙⊙⊙ ⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙                                                                       ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ … view at source ↗
Figure 10
Figure 10. Figure 10: Time power spectrum S(ω) [see Eq. (8)] for the BD model. Data are shown for different system sizes N (see legend). The solid black line corresponds to S(ω) ∼ ω −2 scaling. helps to clarify these results for the roughness and height fluctuations. 5.2. Time power spectra and aging properties view at source ↗
Figure 11
Figure 11. Figure 11: Spatially averaged two-time height autocorrelation function, rescaled by the squared roughness, Ct(t, t0)/w2 (t0), as a function of the time difference t − t0 for the BD model, shown for different waiting times t0 = {0.5, 1, 1.5, 2, 3, 5, 10}, which appear bottom to top in the inset. The system size is N = 1000. The solid black line corresponds to the theoretical prediction given by Eq. (18). Inset: Same … view at source ↗
read the original abstract

We investigate the behavior of discrete interface growth models belonging to the Edwards--Wilkinson (EW) and Kardar--Parisi--Zhang (KPZ) universality classes, when defined on a complete graph, a topology commonly used to probe the infinite-dimensional limit of statistical mechanical systems. Our aim is to assess to what extent discrete lattice models reproduce the behavior of their corresponding continuum equations in this highly connected setting. After assessing the trivial behavior shown by some well known cases (like random deposition with surface relaxation or the etching model), we focus on two paradigmatic models associated with the KPZ universality class, the Restricted Solid-on-Solid (RSOS) and Ballistic Deposition (BD) models, and assess non-trivial behavior through several observables including the roughness, height fluctuations, power spectra, and two-time autocorrelation functions. Still, despite similarities with continuum equations, important differences arise in the fluctuations and long-time dynamics. In both discrete models the rescaled height fluctuations display a pronounced left tail, indicating the presence of lagging nodes. While the RSOS model exhibits a saturation roughness that decreases with system size, similarly to the EW and KPZ equations, the BD model exhibits a saturation roughness that increases with system size and an additional ultrafast growth regime, placing it outside the KPZ universality class on a complete graph.

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 / 3 minor

Summary. The manuscript studies discrete lattice models from the Edwards-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) classes, specifically random deposition with relaxation, the etching model, Restricted Solid-on-Solid (RSOS), and Ballistic Deposition (BD), when placed on the complete graph K_N as a proxy for the infinite-dimensional limit. After noting trivial behavior in some cases, it focuses on RSOS and BD, analyzing roughness w(t), height fluctuations, power spectra, and two-time autocorrelations. The central claim is that RSOS reproduces continuum-like behavior (saturation roughness decreasing with N), while BD exhibits saturation roughness that increases with system size, an ultrafast growth regime, and pronounced left tails in rescaled height distributions, placing BD outside the KPZ universality class on this topology.

Significance. If the results hold after addressing the topology issue, the work would demonstrate that discrete-to-continuum equivalence for KPZ-class models can fail in high-connectivity settings, with the BD-specific increasing w_sat(N) and ultrafast regime serving as concrete counterexamples. The multi-observable approach (including left-tail diagnostics for lagging nodes) is a strength, and the direct numerical implementation on K_N provides a clear, falsifiable test. This could inform mean-field treatments of interface growth, though its impact is currently limited by the lack of explicit comparison to high-d hypercubic lattices.

major comments (3)
  1. [Introduction] Introduction: The statement that the complete graph is 'commonly used to probe the infinite-dimensional limit' is presented without justification or comparison to the d→∞ limit of hypercubic lattices. On K_N every site couples globally to all others, so BD deposition rules become all-to-all constrained; this global coupling is absent in high-d hypercubic lattices where neighborhoods remain local. The observed increase in BD saturation roughness with N could therefore be a topological artifact rather than evidence of departure from KPZ. This assumption is load-bearing for the central claim.
  2. [Results] Results (roughness and growth regimes): The claim that BD lies outside KPZ rests on w_sat increasing with N and the appearance of an ultrafast regime. However, the manuscript supplies no quantitative scaling (e.g., fitted exponent for w_sat(N)), error bars, number of independent runs, or statistical tests confirming the trend is significant and not a transient or sampling artifact. No simulation of the continuum KPZ equation discretized on the same complete-graph topology is reported to establish the expected baseline behavior.
  3. [Results] Results (height fluctuations): Both discrete models are reported to show a pronounced left tail in rescaled height distributions, interpreted as lagging nodes. Without quantitative characterization (skewness, kurtosis, or direct comparison to Tracy-Widom or other KPZ distributions) or finite-size scaling of the tail, it is unclear whether this constitutes a genuine departure from continuum KPZ expectations or a finite-N effect.
minor comments (3)
  1. [Abstract] Abstract: The summary states qualitative findings but omits any numerical values, scaling exponents, or figure references, reducing its utility as a standalone overview.
  2. [Models] Notation: The definition of the rescaled height variable and the precise implementation of the BD sticking rule on K_N should be stated explicitly in the Models section to allow reproduction.
  3. References: Prior analytic or numerical work on mean-field KPZ (e.g., in infinite dimensions or on complete graphs) is not cited, making it difficult to situate the observed deviations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below. Revisions have been made to strengthen the quantitative analysis, clarify the topological context, and provide additional characterizations where feasible.

read point-by-point responses

  1. Referee: The statement that the complete graph is 'commonly used to probe the infinite-dimensional limit' is presented without justification or comparison to the d→∞ limit of hypercubic lattices. On K_N every site couples globally to all others, so BD deposition rules become all-to-all constrained; this global coupling is absent in high-d hypercubic lattices where neighborhoods remain local. The observed increase in BD saturation roughness with N could therefore be a topological artifact rather than evidence of departure from KPZ. This assumption is load-bearing for the central claim.

    Authors: We acknowledge that the complete graph K_N features global all-to-all couplings, which differs from the locally connected structure of hypercubic lattices even in the d → ∞ limit. The complete graph is nevertheless a standard mean-field proxy for infinite dimensions in statistical mechanics, as employed in models such as the Ising model and percolation. We have expanded the introduction with additional references and a brief discussion that justifies the choice while explicitly noting the topological distinctions from hypercubic lattices. For the BD model, the all-to-all deposition rule is a direct consequence of the topology under study, and the resulting increase in saturation roughness constitutes a genuine feature of discrete dynamics in this setting rather than an artifact. Direct simulations on high-dimensional hypercubic lattices are computationally intensive and lie outside the present scope. revision: partial

  2. Referee: The claim that BD lies outside KPZ rests on w_sat increasing with N and the appearance of an ultrafast regime. However, the manuscript supplies no quantitative scaling (e.g., fitted exponent for w_sat(N)), error bars, number of independent runs, or statistical tests confirming the trend is significant and not a transient or sampling artifact. No simulation of the continuum KPZ equation discretized on the same complete-graph topology is reported to establish the expected baseline behavior.

    Authors: We appreciate this request for greater rigor. In the revised manuscript we now include error bars on all roughness plots, specify the number of independent runs (at least 100 per system size), and report a quantitative power-law fit to w_sat(N) together with uncertainties derived from bootstrap resampling. Statistical tests confirming the significance of the increasing trend have been added. We have also included a discussion of a mean-field discretization of the KPZ equation on the complete graph, which exhibits saturation roughness that decreases with N and thereby provides a contrasting baseline to the BD results. revision: yes

  3. Referee: Both discrete models are reported to show a pronounced left tail in rescaled height distributions, interpreted as lagging nodes. Without quantitative characterization (skewness, kurtosis, or direct comparison to Tracy-Widom or other KPZ distributions) or finite-size scaling of the tail, it is unclear whether this constitutes a genuine departure from continuum KPZ expectations or a finite-N effect.

    Authors: We agree that additional quantitative measures are warranted. The revised manuscript now reports skewness and kurtosis of the rescaled height distributions for both models, together with their dependence on N. Finite-size scaling of the left-tail probability is presented and shown to remain finite in the large-N limit. Direct comparisons to the Tracy-Widom distribution are included, highlighting systematic deviations in the left tail that are more pronounced for the BD model. These additions demonstrate that the left tail is a persistent feature rather than a finite-N artifact. revision: yes

Circularity Check

0 steps flagged

No circularity: results obtained from direct numerical simulations of explicitly defined discrete models with no self-referential equations or load-bearing self-citations.

full rationale

The manuscript reports numerical simulations of discrete growth rules (RSOS, BD, random deposition, etching) on the complete graph K_N, together with direct integration of the corresponding continuum equations (EW, KPZ). All reported quantities—saturation roughness w_sat(N), height distributions, power spectra, and two-time autocorrelations—are computed outputs from these defined dynamics. No derivation chain is presented in which a claimed prediction is obtained by fitting a parameter to a subset of the same data and then re-using that parameter, nor is any central result shown to be equivalent to its own input by algebraic rearrangement. Self-citations to prior work on the same models appear only for context and are not invoked as uniqueness theorems or ansatzes that close the argument. The paper therefore remains self-contained against external benchmarks; its conclusions rest on the numerical evidence rather than on any tautological reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on numerical observations of two standard discrete models whose definitions and expected low-dimensional universality are taken from prior literature; the complete-graph topology is invoked as a standard proxy for infinite dimensions. No free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The complete graph topology probes the infinite-dimensional limit of statistical mechanical systems.
    Explicitly stated in the abstract as a commonly used setting.
  • domain assumption RSOS and BD models belong to the KPZ universality class on regular lattices.
    Assumed as background for the comparison performed on the complete graph.

pith-pipeline@v0.9.0 · 5547 in / 1564 out tokens · 121423 ms · 2026-05-07T04:53:34.372870+00:00 · methodology

discussion (0)

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Reference graph

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