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Phase Transition and Correlation Decay in Coupled Map Lattices

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Abstract

For a Coupled Map Lattice with a specific strong coupling emulating Stavskaya’s probabilistic cellular automata, we prove the existence of a phase transition using a Peierls argument, and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoupling originally developed for weak coupling. This implies the exponential decay, in space and in time, of the correlation functions of the invariant measures.

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References

  1. Kaneko K.: Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice. Prog. Theor. Phys. 72(3), 480–486 (1984)

    Article  MATH  ADS  Google Scholar 

  2. Kaneko K.: Theory and applications of Coupled Map Lattices. Wiley & Sons, New York (1993)

    MATH  Google Scholar 

  3. Bunimovich L., Sinai Y.: Spacetime chaos in coupled map lattices. Nonlinearity 1(4), 491–516 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Pesin Y.B., Sinai Y.: Space-time chaos in chains of weakly interacting hyperbolic mappings. Adv. in Sov. Math. 3, 165–198 (1991)

    MathSciNet  Google Scholar 

  5. Bricmont J., Kupiainen A.: Coupled analytic maps. Nonlinearity 8(3), 379–396 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Bricmont J., Kupiainen A.: High temperature expansions and dynamical systems. Commun. Math. Phys. 178(3), 703–732 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Bricmont J., Kupiainen A.: Infinite-dimensional SRB measures. Physica D 103, 18–33 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  8. Maes C., Van Moffaert A.: Stochastic stability of weakly coupled lattice maps. Nonlinearity 10(3), 715–730 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Baladi V., Degli Esposti M., Isola S., Järvenpää E., Kupiainen A.: The spectrum of weakly coupled map lattices. J. Math. Pures et appl. 77(6), 539–584 (1998)

    MATH  Google Scholar 

  10. Jiang M., Pesin Y.B.: Equilibrium Measures for Coupled Map Lattices: Existence, Uniqueness and Finite-Dimensional Approximations. Commun. Math. Phys. 193(3), 675–711 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Keller G.: An ergodic theoretic approach to mean field coupled maps, fractal geometry and stochastics. Progr. Probab. 46, 183–208 (1998)

    Google Scholar 

  12. Järvenpää E., Järvenpää M.: On the definition of SRB-measures for coupled map lattices. Commun. Math. Phys. 220(1), 1–12 (2001)

    Article  MATH  Google Scholar 

  13. Keller G., Liverani C.: Coupled map lattices without cluster expansion. Disc. Cont. Dyn. Syst. 11(2–3), 325–335 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Keller, G., Liverani, C.: A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps. Volume 671 Lecture Notes in Physics, Berlin-Heidelberg-New York: Springer, 2005, pp. 115–151

  15. Järvenpää, E.: SRB-measures for coupled map lattices. Volume 671 of Lecture notes in physics, Berlin: Springer, 2005, pp. 95–114

  16. Keller G., Liverani C.: Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Commun. Math. Phys. 262(1), 33–50 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Miller J., Huse D.: Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice. Phys. Rev. E 48(4), 2528–2535 (1993)

    Article  ADS  Google Scholar 

  18. Boldrighini C., Bunimovich L., Cosimi G., Frigio S., Pellegrinotti A.: Ising-type transitions in coupled map lattices. J. Stat. Phys. 80(5-6), 1185–1205 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Boldrighini C., Bunimovich L., Cosimi G., Frigio S., Pellegrinotti A.: Ising-Type and Other Transitions in One-Dimensional Coupled Map Lattices with Sign Symmetry. J. Stat. Phys. 102(5), 1271–1283 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Schmüser F., Just W.: Non-equilibrium behaviour in unidirectionally coupled map lattices. J. Stat. Phys. 105(3–4), 525–559 (2001)

    Article  MATH  Google Scholar 

  21. Gielis G., MacKay R.: Coupled map lattices with phase transition. Nonlinearity 13(3), 867–888 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Just W.: Equilibrium phase transitions in coupled map lattices: A pedestrian approach. J. Stat. Phys. 105(1–2), 133–142 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Blank M., Bunimovich L.A.: Multicomponent dynamical systems: SRB measures and phase transitions. Nonlinearity 16(1), 387–401 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Bardet J.-B., Keller G.: Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling. Nonlinearity 19, 2193 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Vitali G.: Sui gruppi di punti e sulle funzioni di variabili reali. Atti Accad. sci. Torino 43, 229–246 (1908)

    MATH  Google Scholar 

  26. Stavskaya O.: Invariant gibbs measures for markov chains on finite lattices with local interaction. Sbornik: Mathematics 21(3), 395–411 (1973)

    Article  ADS  Google Scholar 

  27. Dobrushin, R., Kryukov, V., Toom, A. (eds): Stochastic cellular systems: ergodicity, memory, morphogenesis. Manchester University Press, Manchester (1990)

    Google Scholar 

  28. Lasota A., Yorke J.: On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186, 481–488 (1973)

    Article  MathSciNet  Google Scholar 

  29. Ionescu Tulcea C., Marinescu G.: Théorie ergodique pour des classes d’opérations non completement continues. Ann. Math. 52(1), 140–147 (1950)

    Article  MathSciNet  Google Scholar 

  30. Hennion H.: Sur un théorème spectral et son application aux noyaux lipschitziens. Proc. Am. Math. Soc. 118(2), 627–634 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Liverani C.: Decay of correlations for piecewise expanding maps. J. Stat. Phys. 78(3–4), 1111–1129 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  32. Keller G.: Interval maps with strictly contracting Perron-Frobenius operators. Int. J. Bifurcation Chaos 9(9), 1777–1783 (1999)

    Article  MATH  Google Scholar 

  33. Baladi, V.: Positive Transfer Operators and Decay of Correlations. Volume 16 of Advanced Series in Nonlinear Dynamics. Singapore: World Scientific Publishing, 2000

  34. Giusti E.: Minimal surfaces and functions of bounded variations. Birkhäuser, Boston (1984)

    Google Scholar 

  35. Ziemer W.: Weakly Differentiable Functions. Springer-Verlag, New York (1989)

    MATH  Google Scholar 

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Authors and Affiliations

  1. UCL, FYMA, Chemin du Cyclotron 2, B-1348, Louvain-la-Neuve, Belgium

    A. de Maere

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  1. A. de Maere
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Correspondence to A. de Maere.

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Communicated by A. Kupiainen

Partially supported by the Belgian IAP program P6/02.

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de Maere, A. Phase Transition and Correlation Decay in Coupled Map Lattices. Commun. Math. Phys. 297, 229–264 (2010). https://doi.org/10.1007/s00220-010-1041-8

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  • DOI: https://doi.org/10.1007/s00220-010-1041-8

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