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Growth and saturation in random media. (English) Zbl 0972.60100
Gesztesy, Fritz (ed.) et al., Stochastic processes, physics and geometry: New interplays. I. A volume in honor of Sergio Albeverio. Proceedings of the conference on infinite dimensional (stochastic) analysis and quantum physics, Leipzig, Germany, January 18-22, 1999. Providence, RI: American Mathematical Society (AMS). CMS Conf. Proc. 28, 41-54 (2000).
A model of growth, diffusion and trapping in random environment is considered. This model has three main features: a random environment, an injection pattern and a two way coupling between random walks and random environment (i.e. the random environment acts on the particles by trapping, and the particles act on the environment by saturation of the traps). The three ingredients of the model are the following: 1) The random environment is given by a collection of i.i.d. integer-valued random variables \(\xi(x)\) at each site \(x\) of the lattice \({\mathbb Z}^{d}.\) \(\xi(x)\) is the initial depth (or capacity) of the trap at site \(x,\) with the convention that the site \(x\) is not a trap if the depth \(\xi(x)\) is zero. 2) The injection pattern: At the orign of the lattice \({\mathbb Z}^{d}\) particles are injected (or born) at a time dependent rate. Deterministic injection patterns are mainly studied but most of the results are true with Poissonian injections. 3) The intersection between the medium and the random walks.
The asymptotic behaviour of the probability of survival of a particle born at some given time is computed, both in the annealed and quenched cases, and it is shown that three different situations occur depending on the injection rate. Some related results are discussed. This includes a shape theorem for the high injection regime (\(N(t)\gg t^{d/2}\)) for \(d=1\) and an almost sure hydrodynamic limit result for critical injection (\(N(t)=Ct^{d/2}\), for some constant \(C\)). Here, \(N(t)\) is the number of particles that have born at time \(t.\)
For the entire collection see [Zbl 0953.00048].
MSC:
60K37 Processes in random environments
60F10 Large deviations
60G50 Sums of independent random variables; random walks
60G60 Random fields
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