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January, 1993 A Variational Approach to Branching Random Walk in Random Environment
J.-B. Baillon, Ph. Clement, A. Greven, F. Den Hollander
Ann. Probab. 21(1): 290-317 (January, 1993). DOI: 10.1214/aop/1176989405


This paper considers an infinite system of particles on the integers $\mathbb{Z}$ that: (1) step to the right with a random delay, and (2) split or die along the way according to a random law depending on their position. The exponential growth rate of the particle density is computed in the long time limit in the form of a variational formula that can be solved explicitly. The result reveals two phase transitions associated with localization vs. delocalization and survival vs. extinction. In addition, the system exhibits an intermittency effect. Greven and den Hollander considered the more difficult situation where the particles may step both to the left and right, but the analysis of the phase diagram was less complete.


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J.-B. Baillon. Ph. Clement. A. Greven. F. Den Hollander. "A Variational Approach to Branching Random Walk in Random Environment." Ann. Probab. 21 (1) 290 - 317, January, 1993.


Published: January, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0770.60088
MathSciNet: MR1207227
Digital Object Identifier: 10.1214/aop/1176989405

Primary: 60K35
Secondary: 82C22

Keywords: Branching random walk , large deviations , Phase transitions , Random media , variational problems

Rights: Copyright © 1993 Institute of Mathematical Statistics

Vol.21 • No. 1 • January, 1993
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