Summary
Let (η n ) be the infinite particle system on ℤ whose evolution is as follows. At each unit of time each particle independently is replaced by a new generation. The size of a new generation descending from a particle at sitex has distributionF x and each of its members independently jumps to sitex±1 with probability (1±h)/2,h∈[0, 1]. The sequence {F x } is i.i.d. with uniformly bounded second moment and is kept fixed during the evolution. The initial configurationη 0 is shift invariant and ergodic.
Two quantities are considered:
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(1)
the global particle densityD n (=large volume limit of number of particles per site at timen);
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(2)
the local particle densityd n > (=average number of particles at site 0 at timen).
We calculate the limits ϱ and λ ofn −1 log(D n ) andn −1 log(d n ) explicitly in the form of two variational formulas. Both limits (and variational formulas) do not depend on the realization of {F x } a.s. By analyzing the variational formulas we extract how ϱ and λ depend on the drifth for fixed distribution ofF x . It turns out that the system behaves in a way that is drastically different from what happens in a spatially homogeneous medium:
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(i)
Both \g9(h) and \gl(h) exhibit a phase transition associated with localization vs. delocalization at two respective critical valuesh 1 andh 3 in (0,1). Here the behavior of the path of descent of a typical particle in the whole population resp. in the population at 0 changes from moving on scaleo(n) to moving on scalen. We extract variational expressions forh 1 andh 3.
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(ii)
Both \g9(h) and \gl(h) change sign at two respective critical valuesh 2 andh 4 in (0,1) (for suitable distribution ofF x . That is, the system changes from survival to extinction on a global resp. on a local scale.
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(iii)
\g9(h)\>=\gl(h) for allh; \g9(h)=\gl(h) forh sufficiently small and \g9(h)\s>\gl(h) forh sufficiently large. This means that the system develops a clustering phenomenon ash increases: the population has large peaks on a thin set.
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(iv)
\g9(h)\s>0\s>\gl(h) for a range ofh. (extreme clustering of the system)
We formulate certain technical properties of the variational formulas that are needed in order to derive the qualitative picture of the phase diagram in its full glory. The proof of these properties is deferred to a forthcoming paper dealing exclusively with functional analytic aspects.
The variational formulas reveal a selection mechanism: the typical particle has a path of descent that is best adapted to the given {F x } and that is atypical under the law of the underlying random walk. The random medium induces “selection of the fittest”.
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(i)
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Greven, A., den Hollander, F. Branching random walk in random environment: phase transitions for local and global growth rates. Probab. Th. Rel. Fields 91, 195–249 (1992). https://doi.org/10.1007/BF01291424
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DOI: https://doi.org/10.1007/BF01291424