bibkey: fuksfates2026diploid
authors: Henryk Fukś, Nazim Fatès
year: 2026
title: "Analysis of the asymptotic density of endogamous diploid cellular automata"
doi: 10.1140/epjs/s11734-026-02447-z
url: https://arxiv.org/abs/2607.01917v1
claim: "Diploid elementary cellular automata mix two rules cell by cell with probability λ. For rule 10-245, third-order local structure approximation gives the asymptotic density 1/4 + λ/2 from the symmetric Bernoulli start, and §5 asks whether this expression can be formally demonstrated to be exact."
strata_touched:
  - D5/S3/StatisticalMechanics/CellularAutomata/Diploid10245Density
license: citation-only
triage: anchor

Fukś–Fatès, asymptotic density of endogamous diploid cellular automata

H. Fukś and N. Fatès, Analysis of the asymptotic density of endogamous diploid cellular automata, arXiv:2607.01917v1 [nlin.CG] (2 July 2026); Eur. Phys. J. Spec. Top. (2026), DOI 10.1140/epjs/s11734-026-02447-z.

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DOI: 10.1140/epjs/s11734-026-02447-z (Eur. Phys. J. Spec. Top., published 2026; the journal full text was not read). Primary version read: https://arxiv.org/abs/2607.01917v1 (the only arXiv version). Source: https://arxiv.org/pdf/2607.01917v1 (text read from the PDF):

• §2, Eqs. (1)–(3) and the definitions of d_t and d∞;
• §3.2, the transition table of rule 10-245 and its local structure approximations;
• §4.1, the solvable family;
• §5, the closing open problems.

Source statements

§2, Eq. (1): “Pr(s_i(t+1) = a) = w(a | s_{i−1}(t), s_i(t), s_{i+1}(t)).”

§2, Eq. (2): “w(1|x₁x₂x₃) = (1−λ) f(x₁,x₂,x₃) + λ g(x₁,x₂,x₃).”

§2: “Suppose that we start with a symmetric Bernoulli distribution … d∞ = lim_{t→∞} d_t. Note that d_t and d∞ as defined above are theoretical expected values of s_i(t), for infinite lattice and assuming initial symmetric Bernoulli measure.”

§3.2: “The diploid 10-245 has the following transition probabilities: w(1|x,y,z) = λ, 1−λ, λ, 1−λ, λ, λ, λ, λ” for xyz = 000, …, 111. The order-3 local structure approximation “yields the expression d = 1/4 + λ/2.”

§5: “for rule 10-245, the expression for the asymptotic density, d = 1/4 + λ/2, appears to be exact. Could this be formally demonstrated?”