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Multiscale Analysis in Disordered Systems: Percolation and contact process in a Random Environment

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Probability and Phase Transition

Part of the book series: NATO ASI Series ((ASIC,volume 420))

Abstract

Multiscale analysis is a technique used in the study of disordered systems in the presence of phenomena similar to Griffiths singularities. In this article we illustrate the use of a multiscale analysis by applying it to a very simple model: percolation in a random environment. We also describe the application of this technique to continuous time percolation and contact processes in random environments.

Partially supported by the NSF under grant DMS 92-08029.

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References

  • Aizenman, M., Klein, A., and Newman, C. M. (1993). Percolation methods for disordered quantum Ising models. To appear in Phase Transitions: Mathematics, Physics, Biology, ...(R. Kotecky ed.), World Scientific.

    Google Scholar 

  • Andjel, E. (1993). Survival of multidimensional contact process in random environments. Boletim da Sociedade Brasileira de Matemática 23, 109–119.

    Article  MathSciNet  Google Scholar 

  • Bezuidenhout, C. and Grimmett, G. (1991). Exponential decay for subcritical contact and percolation processes. Annals of Probability 19, 984–1009.

    Article  MathSciNet  MATH  Google Scholar 

  • Bramson, M., Durrett, R., and Schonmann R. H. (1991). The contact process in a random environment. Annals of Probability 19, 960–983.

    Article  MathSciNet  MATH  Google Scholar 

  • Campanino, M. and Klein, A. (1991). Decay of two-point functions for (d + 1)-dimensional percolation, Ising and Potts model with d-dimensional disorder. Communications in Mathematical Physics 135, 483–497.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Campanino, M., Klein, A., and Perez, J. F. (1991). Localization in the ground state of the Ising model with a random transverse field. Communications in Mathematical Physics 135, 499–515.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Dreifus, H. von (1987). On the effects of randomness in ferromagnetic models and Schrödinger operators. Ph.D. Thesis, New York University.

    Google Scholar 

  • Dreifus, H. von and Klein, A. (1989). A new proof of localization in the Anderson tight binding model. Communications in Mathematical Physics 124, 285–299.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Dreifus, H. von and Klein, A. (1991). Localization for random Schrödinger operators with correlated potentials. Communications in Mathematical Physics 140, 133–147.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Fröhlich, J. and Spencer, T. (1983). Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Communications in Mathematical Physics 88, 151–184.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Fröhlich, J., Martinelli, F., Scoppola, E., and Spencer, T. (1985). Constructive proof of localization in the Anderson tight binding model. Communications in Mathematical Physics 101, 21–46.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Griffiths, R. (1969). Non-analytic behavior above the critical point in a random Ising ferromagnet. Physical Review Letters 23, 17–19.

    Article  ADS  Google Scholar 

  • Grimmett, G. (1989). Percolation. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Jitomirskaya, S. and Klein, A. (1993). Ising model in a quasi-periodic transverse field, percolation and contact processes in quasi-periodic environments. Journal of Statistical Physics, to appear.

    Google Scholar 

  • Klein, A. (1992). Extinction of contact and percolation processes in a random environment. Annals of Probability, to appear.

    Google Scholar 

  • Klein, A. (1993). Disordered quantum spin processes, percolation and contact processes. To appear in Phase Transitions: Mathematics, Physics, Biology, (R. Kotecky, ed.), World Scientific.

    Google Scholar 

  • Klein, A. and Perez, J. P. (1992). Localization in the ground state of a disordered array of quantum rotators. Communications in Mathematical Physics 147, 241–252.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Liggett, T. M. (1985). Interacting Particle Systems. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Liggett, T. M. (1991). Spatially inhomogeneous contact processes. In Spatial Stochastic Processes. A Festschrift in honor of the Seventieth Birthday of Ted Harris, Birkhäuser, Boston, 105–140.

    Google Scholar 

  • Liggett, T. M. (1992). The survival of one dimensional contact processes in random environments. Annals of Probability 20, 696–723.

    Article  MathSciNet  MATH  Google Scholar 

  • Spencer, T. (1988). Localization for random and quasi-periodic potentials. Journal of Statistical Physics 51, 1009–1019.

    Article  MathSciNet  ADS  MATH  Google Scholar 

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© 1994 Springer Science+Business Media Dordrecht

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Klein, A. (1994). Multiscale Analysis in Disordered Systems: Percolation and contact process in a Random Environment. In: Grimmett, G. (eds) Probability and Phase Transition. NATO ASI Series, vol 420. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8326-8_8

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  • DOI: https://doi.org/10.1007/978-94-015-8326-8_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4370-2

  • Online ISBN: 978-94-015-8326-8

  • eBook Packages: Springer Book Archive

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