BRANCHING RANDOM-WALK IN RANDOM ENVIRONMENT - PHASE-TRANSITIONS FOR LOCAL AND GLOBAL GROWTH-RATES

• 成果类型: Article
• 署名作者: GREVEN, A; DENHOLLANDER, F
• 署名单位: Utrecht University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01291424
• 发表日期: 1992
• 页码: 195-249
• 关键词: markov process expectations; asymptotic evaluation; large time; MODEL

摘要

Let (eta(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 site x has distribution F(x) and each of its members independently jumps to site x +/- 1 with probability (1 +/- h)/2, h is-an-element-of [0, 1]. The sequence {F(x)} is i.i.d. with uniformly bounded second moment and is kept fixed during the evolution. The initial configuration eta(o) is shift invariant and ergodic. Two quantities are considered: (1) the global particle density D(n) (= large volume limit of number of particles per site at time n); (2) the local particle density d(n) (= average number of particles at site 0 at time n). We calculate the limits rho and lambda of n-1 log(D(n)) and n-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 rho and lambda-depend on the drift h for fixed distribution of F(x). It turns out that the system behaves in a way that is drastically different from what happens in a spatially homogeneous medium: (i) Both rho(h) and lambda(h) exhibit a phase transition associated with localization vs. delocalization at two respective critical values h1 and h3 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 scale o(n) to moving on scale n. We extract variational expressions for h1 and h3. (ii) Both rho(h) and lambda(h) change sign at two respective critical values h2 and h4 in (0, 1) (for suitable distribution of F(x)). That is, the system changes from survival to extinction on a global resp. on a local scale. (iii) rho(h) greater-than-or-equal-to lambda(h) for all h; rho(h) = lambda(h) for h sufficiently small and rho(h) > lambda(h) for h sufficiently large. This means that the system develops a clustering phenomenon as h increases: the population has large peaks on a thin set. (iv) rho(h) > 0 > lambda(h) for a range of h. (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.