Quenched asymptotics for survival probabilities in the random saturation process

We report on the recent work [3]. There, the asymptotics of the survival probabilities of particles in a random environment of obstacles, are computed. The model is the following: particles are injected at a time dependent rate at the origin of the lattice ℤ^d. Once born, they diffuse among sites which are free of traps. Each trap has a random depth, which decreases by one each time a particle is absorbed. The logarithmic asymptotic decay of the probability that a particle born at some fixed time survives at some later time t is computed, showing the presence of three injection regimes. Here we report on the quenched version of these results. A key tool for proving this result is the method of enlargement of obstacles developed by Sznitman [9].