Abstract
Consider an infinite graph with nodes initially labeled by independent Bernoulli random variables of parameter p. We want to find a (probabilistic or deterministic) cellular automaton or a finite-range interacting particle system that decides if p is smaller or larger than 1/2. Precisely, the trajectories should converge to the uniform configuration with only 0’s if p < 1/2, and only 1’s if p > 1/2. We present solutions to that problem on ℤd, for any d ≥ 2, and on the regular infinite trees. For ℤ, we propose some candidates that we back up with numerical simulations.
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References
Cox, J.T., Durrett, R.: Nonlinear voter models. In: Random Walks, Brownian Motion, and Interacting Particle Systems. Progr. Probab., vol. 28, pp. 189–201. Birkhäuser Boston, Boston (1991)
Dobrushin, R., Kryukov, V., Toom, A.: Stochastic cellular systems: ergodicity, memory, morphogenesis. Nonlinear Science. Manchester University Press (1990)
Fatès, N.: Stochastic Cellular Automata Solve the Density Classification Problem with an Arbitrary Precision. In: STACS 2011, vol. 9, pp. 284–295 (2011)
Fontes, L.R., Schonmann, R.H., Sidoravicius, V.: Stretched exponential fixation in stochastic Ising models at zero temperature. Comm. Math. Phys. 228(3), 495–518 (2002)
Fukś, H.: Nondeterministic density classification with diffusive probabilistic cellular automata. Phys. Rev. E 66 (2002)
Gács, P.: A Toom rule that increases the thickness of sets. J. Statist. Phys. 59(1-2), 171–193 (1990)
Gács, P.: Reliable cellular automata with self-organization. J. Statist. Phys. 103(1-2), 45–267 (2001)
Gonzaga de Sá, P., Maes, C.: The Gacs-Kurdyumov-Levin automaton revisited. J. Statist. Phys. 67(3-4), 507–522 (1992)
Grimmett, G.: Percolation, Grundlehren der Mathematischen Wissenschaften, 2nd edn., vol. 321. Springer, Berlin (1999)
Howard, C.D.: Zero-temperature Ising spin dynamics on the homogeneous tree of degree three. J. Appl. Probab. 37(3), 736–747 (2000)
Kanoria, Y., Montanari, A.: Majority dynamics on trees and the dynamic cavity method. Ann. Appl. Probab. 21(5), 1694–1748 (2011)
Kurka, P.: Cellular automata with vanishing particles. Fund. Inform. 58(3-4), 203–221 (2003)
Land, M., Belew, R.K.: No perfect two-state cellular automata for density classification exists. Phys. Rev. Lett. 74, 5148–5150 (1995)
Le Gloannec, B.: Around Kari’s traffic cellular automaton for the density classification. Master’s thesis (2009)
Liggett, T.M.: Interacting particle systems. Classics in Mathematics. Springer, Berlin (2005); reprint of the 1985 original
Packard, N.H.: Dynamic Patterns in Complex Systems. World Scientific, Singapore (1988)
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Bušić, A., Fatès, N., Mairesse, J., Marcovici, I. (2012). Density Classification on Infinite Lattices and Trees. In: Fernández-Baca, D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29344-3_10
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DOI: https://doi.org/10.1007/978-3-642-29344-3_10
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