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On a variational problem for an infinite particle system in a random medium. (English) Zbl 0814.49033

The authors consider a model of an infinite system of particles resident on a linear lattice labelled by \(\mathbb{Z}\) and subject to two random processes: particles branch according to site-dependent offspring distributions and particles migrate by jumping to nearest neighbour sites with site-independent probabilities. The migration has a drift. In an earlier work, the third and the fourth authors [Prob. Theor. Rel. Fields 91, No. 2, 195-249 (1992; Zbl 0744.60079)] obtained two functions whose maxima provided respectively the exponential growth rates of the global and the local population densities and whose maximizers indicated the path of descent of a typical particle in the global and the local populations, respectively. In the present paper the global functional is studied in considerable detail: in particular, it is found that the maximum and the maximizers show certain phase transitions as the drift varies due to the competition between the branching and the migration.

MSC:

49R50 Variational methods for eigenvalues of operators (MSC2000)
15A45 Miscellaneous inequalities involving matrices
60D05 Geometric probability and stochastic geometry
60K35 Interacting random processes; statistical mechanics type models; percolation theory

Citations:

Zbl 0744.60079