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On a variational problem for an infinite particle system in a random medium Part II: The local growth rate
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  • Published: September 1994

On a variational problem for an infinite particle system in a random medium Part II: The local growth rate

  • A. Greven1 &
  • F. den Hollander2 

Probability Theory and Related Fields volume 100, pages 301–328 (1994)Cite this article

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  • 4 Citations

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Summary

This paper solves the second of two variational problems arising in the study of an infinite system of particles that branch and migrate in a random medium. This variational problem involves a non-linear functional on a subset of the stationary probability measures on [ℕ×ℝ+]ℤ, describing the interplay between particles and medium. It is shown that the variational problem can be solved in terms of the Lyapunov exponent of a product of random ℕ×ℕ matrices. This Lyapunov exponent is calculated via a random continued fraction. By analyzing the latter we are able to compute the maximum and the maximizer in the variational problem. It is found that these quantities exhibit interesting non-analyticities and changes of sign as a function of model parameters, which correspond to phase transitions in the infinite particle system. By combining with results from Part I we obtain a complete picture of the phase diagram.

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References

  • [BB] Batty, C.J.K., Bollmann, H.W.: Generalised Holley-Preston inequalities on measure spaces and their products. Z. Wahrscheinlichkeitstheor. Verw. Geb.53, 157–173 (1980)

    Google Scholar 

  • [BCGH1] Baillon, J.-B., Clément, Ph., Greven, A., den Hollander, F.: A variational approach to branching random walk in random environment. Ann. Probab.21, 290–317 (1993)

    Google Scholar 

  • [BCGH2] Baillon, J.-B., Clément, Ph., Greven, A., den Hollander, F.: On a variational problem for an infinite particle system in a random medium. Part I: The global growth rate. J. Reine Angew. Math. (to appear)

  • [Ge] Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter Studies in Mathematics 9. Berlin: De Gruyter 1988

    Google Scholar 

  • [GH1] Greven, A., den Hollander, F.: Branching random walk in random environment: phase transitions for local and global growth rates. Probab. Theory Relat. Fields91, 195–249 (1992)

    Google Scholar 

  • [GH2] Greven, A., den Hollander, F.: Population growth in random media., I. Variational formula and phase diagram. J. Stat. Phys.65, 1123–1146 (1991)

    Google Scholar 

  • [GH3] Greven, A., den Hollander, F.: Population growth in random media. II. Wave front propagation. J. Stat. Phys.65, 1147–1154 (1991)

    Google Scholar 

  • [GH4] Greven, A., den Hollander, F.: Large deviations for a random walk in random environment. Ann. Probab. (to appear)

  • [Pe] Perron, O.: Die Lehre von den Kettenbrüchen. Leipzig: Teubner 1913

    Google Scholar 

  • [Pr] Preston, C.J.: A generalization of the FKG inequalities. Commun. math. Phys.36, 233–241 (1974)

    Google Scholar 

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Authors and Affiliations

  1. Institut für Stochastik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany

    A. Greven

  2. Mathematisch Instituut, Universiteit Utrecht, P.O. Box 80.010, 3508 TA, Utrecht, The Netherlands

    F. den Hollander

Authors
  1. A. Greven
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  2. F. den Hollander
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Greven, A., den Hollander, F. On a variational problem for an infinite particle system in a random medium Part II: The local growth rate. Probab. Th. Rel. Fields 100, 301–328 (1994). https://doi.org/10.1007/BF01193703

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  • Received: 03 March 1993

  • Revised: 04 April 1994

  • Issue Date: September 1994

  • DOI: https://doi.org/10.1007/BF01193703

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Mathematics Subject Classification 1991

  • 60F10
  • 60J15
  • 82B26
  • 82B44
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