On a variational problem for an infinite particle system in a random medium Part II: The local growth rate |
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Authors: | A. Greven F. den Hollander |
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Affiliation: | (1) Institut für Stochastik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany;(2) Mathematisch Instituut, Universiteit Utrecht, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands |
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Abstract: | 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 [×+], 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 × 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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Keywords: | 60F10 60J15 82B26 82B44 |
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