×

zbMATH — the first resource for mathematics

A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit. (English) Zbl 1315.82005
Summary: We provide an example of a discrete-time Markov process on the three-dimensional infinite integer lattice with \(\mathbb{Z}_q\)-invariant Bernoulli-increments which has as local state space the cyclic group \(\mathbb{Z}_q\). We show that the system has a unique invariant measure, but remarkably possesses an invariant set of measures on which the dynamics is conjugate to an irrational rotation on the continuous sphere \(S^1\). The update mechanism we construct is exponentially well localized on the lattice.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
68Q80 Cellular automata (computational aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Acebrón, J. A.; Bonilla, L. L.; Pérez Vicente, C. J.; Ritort, F.; Spigler, R., The Kuramoto model: a simple paradigm for synchronization phenomena, Rev. Modern Phys., 137-185, (2005)
[2] Aizenman, M.; Lebowitz, J. L., Metastability effects in bootstrap percolation, J. Phys. A, 21, 19, 3801-3813, (1988) · Zbl 0656.60106
[3] Balogh, J.; Bollobás, B.; Morris, R., Bootstrap percolation in three dimensions, Ann. Probab., 37, 4, 1329-1380, (2009) · Zbl 1187.60082
[4] Barron, A. R., The strong ergodic theorem for densities: generalized Shannon-mcmillan-breiman theorem, Ann. Probab., 13, 1292-1303, (1985) · Zbl 0608.94001
[5] B. Bollobás, H. Duminil-Copin, R. Morris, P. Smith, Universality of two-dimensional critical cellular automata, 2014. arXiv:1406.6680.
[6] Chassaing, P.; Mairesse, J., A non-ergodic probabilistic cellular automaton with a unique invariant measure, Stochastic Process. Appl., 121, 11, 2474-2487, (2011) · Zbl 1237.60076
[7] Chen, M. F., From Markov chains to non-equilibrium particle systems, (2004), World Scientific
[8] Chen, M. F., Eigenvalues, inequalities and ergodic theory, (2005), Springer-Verlag London, Ltd. London · Zbl 1079.60005
[9] Dai Pra, P.; den Hollander, F., Mckean-Vlasov limit for interacting random processes in random media, J. Stat. Phys., 84, 3-4, 735-772, (1996) · Zbl 1081.60554
[10] Dai Pra, P.; Louis, P.; Roelly, S., Stationary measures and phase transition for a class of probabilistic cellular automata, ESAIM Probab. Stat., 6, 89-104, (2002) · Zbl 1003.60090
[11] De Masi, A.; Galves, A.; Löcherbach, E.; Presutti, E., Hydrodynamic limit for interacting neurons, J. Stat. Phys., (2014) · Zbl 1315.35222
[12] Duminil-Copin, H.; van Enter, A. C.D., Sharp metastability threshold for an anisotropic bootstrap percolation model, Ann. Probab., 41, 3A, 1218-1242, (2013) · Zbl 1356.60166
[13] Ermolaev, V. N.; Külske, C., Low-temperature dynamics of the Curie-Weiss model: periodic orbits, multiple histories and loss of Gibbsianness, J. Stat. Phys., 141, 5, 727-756, (2010) · Zbl 1208.82042
[14] Fernández, R., Gibbsianness and non-Gibbsianness in lattice random fields, (Les Houches. Vol. LXXXIII, (2005)) · Zbl 05723808
[15] Fernández, R.; den Hollander, F.; Martínez, J., Variational description of Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model, Comm. Math. Phys., 319, 3, 703-730, (2013) · Zbl 1273.82010
[16] Fröhlich, J.; Pfister, C.-E., Spin waves, vortices, and the structure of equilibrium states in the classical XY model, Comm. Math. Phys., 89, 303-327, (1983)
[17] Fröhlich, J.; Simon, B.; Spencer, T., Infrared bounds, phase transitions and continuous symmetry breaking, Comm. Math. Phys., 50, 79-95, (1976)
[18] Fröhlich, J.; Spencer, T., Massless phases and symmetry restoration in abelian gauge symmetries and spin systems, Comm. Math. Phys., 83, 411-454, (1982)
[19] Gács, P., Reliable cellular automata with self-organization, J. Stat. Phys., 103, 1-2, 45-267, (2001) · Zbl 0973.68158
[20] Georgii, H.-O., Gibbs measures and phase transitions, (2011), De Gruyter New York · Zbl 1225.60001
[21] Giacomin, G.; Pakdaman, K.; Pellegrin, X., Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators, Nonlinearity, 25, 1247-1273, (2012) · Zbl 1244.37048
[22] Giacomin, G.; Pakdaman, K.; Pellegrin, X.; Poquet, C., Transitions in active rotator systems: invariant hyperbolic manifold approach, SIAM J. Math. Anal., 44, 6, 4165-4194, (2012) · Zbl 1261.37036
[23] Holley, R., Free energy in a Markovian model of a lattice spin system, Comm. Math. Phys., 23, 87-99, (1971) · Zbl 0241.60096
[24] Jahnel, B.; Külske, C., A class of non-ergodic interacting particle systems with unique invariant measure, Ann. Appl. Probab., 24, 6, 2595-2643, (2014) · Zbl 1304.82014
[25] Jahnel, B.; Külske, C., Synchronization for discrete mean-field rotators, Electron. J. Probab., 19, (2014), Art. 14 · Zbl 1288.82042
[26] Jahnel, B.; Külske, C.; Rudelli, E.; Wegener, J., Gibbsian and non-gibbsian properties of the generalized mean-field fuzzy Potts-model, Markov Process. Related Fields, 20, 601-632, (2014) · Zbl 1318.82012
[27] Kozlov, O. K., Gibbs description of a system of random variables, Probl. Inf. Transm., 10, 258-265, (1974)
[28] Külske, C., Weakly Gibbsian representations for joint measures of quenched lattice spin models, Probab. Theory Related Fields, 119, 1-30, (2001) · Zbl 1052.82016
[29] Külske, C.; Le Ny, A., Spin-flip dynamics of the Curie-Weiss model: loss of Gibbsianness with possibly broken symmetry, Comm. Math. Phys., 271, 431-454, (2007) · Zbl 1138.82012
[30] Külske, C.; Le Ny, A.; Redig, F., Relative entropy and variational properties of generalized Gibbsian measures, Ann. Probab., 32, 2, 1691-1726, (2004) · Zbl 1052.60042
[31] Külske, C.; Opoku, A. A., The posterior metric and the goodness of Gibbsianness for transforms of Gibbs measures, Electron. J. Probab., 1307-1344, (2008) · Zbl 1190.60096
[32] Külske, C.; Redig, F., Loss without recovery of Gibbsianness during diffusion of continuous spins, Probab. Theory Related Fields, 135, 428-456, (2006) · Zbl 1095.60027
[33] Künsch, H., Time reversal and stationary Gibbs measures, Stochastic Process. Appl., 17, 159-166, (1984) · Zbl 0536.60096
[34] Lebowitz, J. L.; Maes, C.; Speer, E., Statistical mechanics of probabilistic cellular automata, J. Stat. Phys., 59, 1-2, 117-170, (1990) · Zbl 1083.82522
[35] Liggett, T., Interacting particle systems, (1985), Springer-Verlag New York · Zbl 0559.60078
[36] Maes, C.; Shlosman, S. B., Rotating states in driven clock- and XY-models, J. Stat. Phys., 144, 1238-1246, (2011) · Zbl 1246.82069
[37] Mountford, T., A coupling of infinite particle systems, J. Math. Kyoto Univ., 35, 1, 43-52, (1995) · Zbl 0840.60097
[38] Pfister, C.-E., Translation invariant equilibrium states of ferromagnetic abelian lattice systems, Comm. Math. Phys., 86, 375-390, (1982)
[39] Ramirez, A. F.; Varadhan, S. R.S., Relative entropy and mixing properties of interacting particle systems, J. Math. Kyoto Univ., 36, 4, 869-875, (1996) · Zbl 0884.60094
[40] Sullivan, W. G., Potentials for almost Markovian random fields, Comm. Math. Phys., 33, 61-74, (1973) · Zbl 0267.60108
[41] Toom, A., Contours, convex sets, and cellular automata, (IMPA Mathematical Publications, (2001), IMPA Rio de Janeiro), Available in english on the web page: http://www.de.ufpe.br/ toom/articles/engmat/index.htm
[42] Toom, A.; Vasilyev, N.; Stavskaya, O.; Mityushin, L.; Kurdyumov, G.; Pirogov, S., Discrete local Markov systems, (Dobrushin, R.; Kryukov, V.; Toom, A., Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, (1990), Manchester University Press Manchester)
[43] van Enter, A. C.D.; Fernández, R.; den Hollander, F.; Redig, F., Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures, Comm. Math. Phys., 226, 101-130, (2002) · Zbl 0990.82018
[44] van Enter, A. C.D.; Fernández, R.; den Hollander, F.; Redig, F., A large-deviation view on dynamical Gibbs-non-Gibbs transitions, Mosc. Math. J., 10, 687-711, (2010) · Zbl 1221.82046
[45] van Enter, A. C.D.; Fernández, R.; Sokal, A. D., Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory, J. Stat. Phys., 72, 879-1167, (1993) · Zbl 1101.82314
[46] van Enter, A. C.D.; Külske, C.; Opoku, A. A., Discrete approximations to vector spin models, J. Phys. A: Math. Theor., 44, (2011) · Zbl 1252.82037
[47] van Enter, A. C.D.; Külske, C.; Opoku, A. A.; Ruszel, W. M., Gibbs-non-Gibbs properties for \(n\)-vector lattice and mean-field models, Braz. J. Probab. Stat., 24, 226-255, (2010) · Zbl 1200.82015
[48] van Enter, A. C.D.; Ruszel, W. M., Gibbsianness vs. non-Gibbsianness of time-evolved planar rotor models, Stochastic Process. Appl., 119, 1866-1888, (2009) · Zbl 1173.82007
[49] van Enter, A. C.D.; Verbitskiy, E. A., On the variational principle for generalized Gibbs measures, Markov Process. Related Fields, 10, 3, 411-434, (2004) · Zbl 1058.60036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.