Source: Ann. Probab. Volume 28, Number 4
(2000), 1470-1527.
We consider a model of diffusion in random media with a two-way
coupling (i.e., a model in which the randomness of the medium influences the
diffusing particles and where the diffusing particles change the medium). In
this particular model, particles are injected at the origin with a
time-dependent rate and diffuse among random traps. Each trap has a finite
(random) depth, so that when it has absorbed a finite (random) number of
particles it is “saturated,” and it no longer acts as a trap.
This model comes from a problem of nuclear waste management. However, a very
similar model has been studied recently by Gravner and Quastel with different
tools (hydrodynamic limits). We compute the asymptotic behavior of the
probability of survival of a particle born at some given time, both in the
annealed and quenched cases, and show that three different situations occur
depending on the injection rate. For weak injection, the typical survival
strategy of the particle is as in Sznitman and the asymptotic behavior of this
survival probability behaves as if there was no saturation effect. For medium
injection rate, the picture is closer to that of internal DLA, as given by
Lawler, Bramson and Griffeath. For large injection rates, the picture is less
understood except in dimension one.
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