arxiv: 2604.10082 · v1 · submitted 2026-04-11 · 🌊 nlin.CG

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Coarsening and Bifurcations in Wide-Range Two-Dimensional Totalistic Cellular Automata

Franco Bagnoli , Luca Mencarelli

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:43 UTC · model grok-4.3

classification 🌊 nlin.CG

keywords cellular automatatotalistic rulesmajority votefrustrated majoritycoarseningbifurcationtwo-dimensional latticeasymptotic density

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The pith

Wide-range frustrated majority vote cellular automata exhibit a bifurcation in asymptotic density above a critical interaction radius.

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

This paper examines Boolean totalistic cellular automata on a two-dimensional lattice with either a majority vote or frustrated majority vote update rule, where the interaction range can be varied. For the majority rule, the system reaches absorbing states with a bifurcation depending on the starting density, though at half density it shows coarsening that stabilizes at clusters of specific curvature. The frustrated version produces persistent active patterns with fixed density, unlike the chaotic behavior predicted by mean-field theory, and displays a bifurcation in final density versus initial density once the range exceeds a threshold. A reader might care because these findings highlight how extended local interactions can generate stable structures and phase-like transitions not captured by simpler approximations.

Core claim

In the majority vote model, absorbing states appear with a bifurcation according to initial density, consistent with mean-field, but at initial density 0.5 the dynamics coarsen until clusters of definite curvature radius form. In the frustrated majority vote model, active patterns with stable density are observed instead of mean-field chaos or limit cycles, and above a critical interacting radius a bifurcation of the asymptotic density occurs as a function of the initial density.

What carries the argument

Variable-span totalistic cellular automata using majority or frustrated majority vote rules, which allow simulation of wide-range interactions on the lattice to reveal deviations from mean-field predictions.

If this is right

The majority vote model develops stable clusters with fixed curvature after coarsening at half density.
The frustrated model maintains active patterns rather than oscillating chaotically.
A density bifurcation emerges in the frustrated model when the interaction radius surpasses a critical value.
These behaviors differ from mean-field approximations due to the spatial structure in two dimensions.

Where Pith is reading between the lines

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

This could imply that similar bifurcations might appear in other extended-interaction models like opinion dynamics or Ising systems with long-range coupling.
The coarsening process suggests analogies to phase separation in physical systems where interface curvature drives evolution.
Testing on larger lattices might confirm if the curvature radii and stable densities persist without finite-size effects.

Load-bearing premise

That the finite-size simulations with selected interaction ranges faithfully represent the behavior in infinite systems without artifacts from boundaries or synchronous updating.

What would settle it

Running simulations on significantly larger grids or with varied boundary conditions and observing whether the density bifurcation and stable curvature radii remain unchanged.

Figures

Figures reproduced from arXiv: 2604.10082 by Franco Bagnoli, Luca Mencarelli.

**Figure 1.** Figure 1: The difference K(R) − C(R) between the number of point K(R) within distance R from the origin in a regular two-dimensional grid of unitary spacing, and the approximation C(R) = ⌊πR2 ⌋, for 0 ≤ R ≤ 40. The set of cells that are interacting with a given one is called its neighborhood vi vi = {sj | aij = 1}. We consider periodic boundary conditions, so all operations on the x and y coordinates are, respectiv… view at source ↗

**Figure 2.** Figure 2: The mean-field return map of the majority model, Eq. (2), and a few iterations of the map. Left: R = 1 (K = 5), center: R = 2 (K = 13), right: R = 3 (K = 29). The observable we consider is the average value of the state of cells (the “density”), ρ(s) = 1 N X i si . The mean-field approximation of the evolution is obtained by neglecting all correlations (i.e., shuffling the configuration or choosing random … view at source ↗

**Figure 3.** Figure 3: The mean-field approximation function ρ ′ = F(ρ) for the frustrated majority model with (left) R = 1 (K = 5), (center) R = 2 (K = 13), (right) R = 3 (K = 29), with some iterations of the map. The mean-field equation for the voter model is ρ ′ = X K v=0 v K K v ρ v (1 − ρ) K−v = (performing the summation) = ρ, so that in average the density does not change, disregarding fluctuations. This is consistent … view at source ↗

**Figure 4.** Figure 4: Serial majority model for various values of R. Left column: the asymptotic configuration for a lattice 100 × 100 cells and ρ0 = 0.5. Center and right columns: average over 10 runs for a 100 × 100 and 200 × 200 lattices. Center column: the asymptotic density ρ(T) vs initial density ρ(0) (the values for the different lattice sizes are overlapped). Right column: the relaxation time T versus the initial dens… view at source ↗

**Figure 5.** Figure 5: Left: The procedure to numerical fit the curvature radius r for a given interacting radius R. Right: the numerical fit of r and the approximations for some values of R. Circles: The curvature radius r numerically fitted for various interacting radius R. Red continuous line: the empirical fit r = 3(R − 1.5)2 . Dotted curves: analytical approximations for different values of the parameter σ. horizontal and … view at source ↗

**Figure 6.** Figure 6: Right: The point c keeps the state of other sites in the green radius if the area A is larger than the area B. Left: The variation of the number of grid points in a circle of radius r = 12.9 after increasing the interacting radius R = 3.8 by the minimal amount σ = 0.1 The problem is that the correct value of σ depends on r, since its location on the circle is not easily determined. This problem is related … view at source ↗

**Figure 7.** Figure 7: Typical patters of the frustrated majority problem of Eq. (3)for a 100 × 100 lattice. Top-left: R = 1, top-right: R − 2; bottom: R = 3 starting with (left) ρ0 = 0.1 and (right) ρ0 = 0.9. 5 Conclusions We have investigated Boolean, totalistic cellular automata with a majority rule and extended interaction range R. The mean-field approximation of this model, which is analogous to the voter one, gives for the… view at source ↗

**Figure 8.** Figure 8: The bifurcation diagram of the probability distribution of the asymptotic density ρ vs. ρ0 for various values of the interacting radius R. Lattocie size 200×200 cells, 32 samples with ρ0 varing from 1/33 to 32/33, averaged over 103 time steps after a transient of 4 · 103 steps. We think that these two problems deserve further studies. References [1] Stephen Wolfram. “Statistical mechanics of cellular auto… view at source ↗

read the original abstract

We investigate Boolean, totalistic cellular automata with a majority or frustrated majority vote rule, and an interaction range of variable span. These two models show a behavior which differs from the mean-field one. The majority vote model is characterized by the presence of absorbing states, and there is a related bifurcation according to the initial density, in agreement with the mean-field approximation. For initial density equal to $0.5$, however, the dynamics is dominated by a coarsening process, which stops when clusters with a definite curvature radius are established. For the frustrated majority vote model, the mean-field approximation gives chaotic oscillations or a limit cycle. Instead, we observe active patterns, with stable density. Above a certain critical value for the interacting radius there is a bifurcation of the asymptotic density as a function of the initial one.

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.

Desk Editor's Note private letter to a colleague

Simulations of wide-range totalistic CA show curvature-limited coarsening in majority vote and a density bifurcation in the frustrated case above some radius, but without finite-size scaling the infinite-limit claims stay provisional.

read the letter

These simulations of wide-range totalistic cellular automata turn up two things worth noting: coarsening that stops at a fixed curvature in the majority vote model, and a bifurcation in asymptotic density for the frustrated version once the range exceeds some value. The work extends earlier majority-vote studies by allowing the interaction range to vary. They iterate the rules on finite lattices and observe how the patterns settle. In the standard majority case, domains grow but then stabilize with a characteristic curvature, which differs from the mean-field expectation of absorbing states or full consensus. For the frustrated majority, mean-field predicts oscillations or cycles, but here they find persistent active patterns with a fixed density. When the range is large enough, that density splits into two branches depending on the starting density. This combination of range dependence and the specific spatial outcomes is not in the prior mean-field literature they cite. The approach is straightforward: they apply the Boolean totalistic rules directly without extra parameters or approximations. The results come straight from the dynamics, so they are reproducible if the rules and initial conditions are matched. The main limitation is that everything is shown for particular finite sizes. The bifurcation and the critical radius for it are presented without checks on how they shift as the lattice gets bigger. No plots of density versus radius for increasing system sizes appear, which leaves open whether the split persists or the critical value converges in the large-system limit. The way they measure the curvature radius or identify the stable densities also lacks detail, though the full text might clarify that. These are not fatal, but they mean the central claims rest on unverified extrapolation from the simulated grids. This paper is aimed at people who study cellular automata and discrete models of opinion dynamics or voting. Someone interested in how interaction range changes the long-term behavior in totalistic rules would find the numerical examples useful. It does not claim to solve broader problems in complex systems but documents these specific deviations from mean-field. I would send it to peer review. The observations are concrete and the models are simple to check, so referees can assess the numerics and ask for the missing scaling if needed. It is worth a look from someone in the field.

Referee Report

2 major / 2 minor

Summary. The manuscript examines wide-range two-dimensional totalistic cellular automata using majority and frustrated majority vote rules. It reports that the majority vote model has absorbing states with a bifurcation in behavior based on initial density, aligning with mean-field predictions, but at initial density 0.5 exhibits coarsening that ceases at clusters of definite curvature radius. For the frustrated majority vote model, stable active patterns with constant density are observed instead of mean-field chaos or cycles, and above a critical interaction radius, the asymptotic density bifurcates depending on the initial density.

Significance. If substantiated, these findings demonstrate significant departures from mean-field approximations in cellular automata with extended neighborhoods, particularly the stabilization of densities in frustrated systems and the bifurcation phenomenon. This could have implications for understanding non-equilibrium dynamics, pattern formation, and phase transitions in discrete lattice models. The numerical exploration of variable range interactions adds to the literature on CA beyond small neighborhoods.

major comments (2)

[Abstract and results section] Abstract and numerical results on the frustrated majority vote model: the central claim that above a critical interaction radius there is a bifurcation of the asymptotic density as a function of the initial density rests on finite-lattice simulations. No finite-size scaling analysis (e.g., density-vs-initial-density curves or critical-radius values for multiple L such as 64, 128, 256) is reported to demonstrate convergence to the infinite-system limit, leaving the bifurcation vulnerable to boundary or size artifacts as noted in the weakest assumption.
[Abstract and majority vote results] Abstract and coarsening discussion for majority vote model at initial density 0.5: the claim that coarsening stops when 'clusters with a definite curvature radius are established' lacks a precise operational definition, measurement protocol, or quantitative criteria (including error estimates) for identifying this radius from post-simulation configurations, which is load-bearing for distinguishing the observed behavior from mean-field.

minor comments (2)

[Abstract] The abstract refers to 'Boolean, totalistic cellular automata' but does not specify the exact totalistic threshold functions or neighborhood definitions used for the majority and frustrated rules.
[Methods or numerical section] Simulation details such as lattice sizes, boundary conditions, update order, and number of runs should be stated explicitly in the methods to allow reproducibility of the reported stable densities and patterns.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive major comments. We address each point below and commit to revisions that will strengthen the claims.

read point-by-point responses

Referee: [Abstract and results section] Abstract and numerical results on the frustrated majority vote model: the central claim that above a critical interaction radius there is a bifurcation of the asymptotic density as a function of the initial density rests on finite-lattice simulations. No finite-size scaling analysis (e.g., density-vs-initial-density curves or critical-radius values for multiple L such as 64, 128, 256) is reported to demonstrate convergence to the infinite-system limit, leaving the bifurcation vulnerable to boundary or size artifacts as noted in the weakest assumption.

Authors: We agree that finite-size effects must be addressed to confirm the bifurcation is not an artifact. In the revised manuscript we will add simulations for L = 64, 128, 256 (and 512 where feasible), together with plots of asymptotic density versus initial density for each size. These will demonstrate convergence of both the bifurcation location and the critical radius with increasing L, including error bars obtained from multiple independent realizations. revision: yes
Referee: [Abstract and majority vote results] Abstract and coarsening discussion for majority vote model at initial density 0.5: the claim that coarsening stops when 'clusters with a definite curvature radius are established' lacks a precise operational definition, measurement protocol, or quantitative criteria (including error estimates) for identifying this radius from post-simulation configurations, which is load-bearing for distinguishing the observed behavior from mean-field.

Authors: We acknowledge that the current description is qualitative and requires a rigorous operational definition. We will revise the text to define the curvature radius explicitly: cluster boundaries are extracted via connected-component labeling and edge tracing; circular arcs are fitted to boundary segments by least-squares minimization; the reported radius is the ensemble-averaged value of these fits over the largest clusters, with standard-error estimates from independent runs. A new methods subsection and supporting figures will document the full protocol. revision: yes

Circularity Check

0 steps flagged

No circularity: claims are direct outputs of rule iteration on finite lattices

full rationale

The manuscript reports numerical observations from iterating totalistic cellular automaton rules (majority and frustrated majority vote) with variable interaction range. The central claim of a bifurcation in asymptotic density for the frustrated model above a critical radius is presented as an empirical outcome of the dynamics, not as an analytic derivation or fitted prediction. No equations are solved that reduce to their own inputs by construction, no parameters are tuned on a data subset and then called predictions, and no load-bearing steps rely on self-citations or imported uniqueness theorems. The results are therefore self-contained against external benchmarks (the rule definitions themselves) and receive the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on direct iteration of standard Boolean totalistic rules on a 2D lattice; no additional free parameters, ad-hoc axioms, or invented entities are introduced beyond the definition of the majority and frustrated rules themselves.

axioms (2)

standard math Cells update synchronously according to a totalistic Boolean function of the neighbor sum.
Standard definition of totalistic cellular automata invoked throughout the abstract.
domain assumption The lattice is two-dimensional with periodic or open boundaries (unspecified).
Implicit in all 2D CA simulations; boundary choice can affect coarsening.

pith-pipeline@v0.9.0 · 5437 in / 1286 out tokens · 44518 ms · 2026-05-10T16:43:21.104836+00:00 · methodology

discussion (0)

Reference graph

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7 extracted references · 4 canonical work pages

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