Baillon, J.-B.; Clément, Ph.; Greven, A.; den Hollander, F. On a variational problem for an infinite particle system in a random medium. (English) Zbl 0814.49033 J. Reine Angew. Math. 454, 181-217 (1994). 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. Reviewer: C.S.Sharma (London) Cited in 2 ReviewsCited in 5 Documents 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 Keywords:variational problem; random medium; eigenvalues; infinite system of particles Citations:Zbl 0744.60079 × Cite Format Result Cite Review PDF Full Text: DOI Crelle EuDML Link