×

zbMATH — the first resource for mathematics

Asymptotic survival probabilities in the random saturation process. (English) Zbl 1108.60315
Summary: 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Antal, P. (1994). Trapping problems for the simple random walk. Thesis, ETH, Z ürich.
[2] Antal, P. (1995). Enlargement of obstacles for the simple random walk. Ann. Probab. 23 1061-1101. · Zbl 0839.60064
[3] Ben Arous, G., Quastel, J. and Ramírez, A. F. (2000). Internal DLA in a random environment. Unpublished manuscript. · Zbl 1023.60089
[4] Dembo, A. and Zeitouni O. (1998). Large Deviations Techniques and Applications. Springer, New York. · Zbl 0896.60013
[5] Diaconis, P. and Fulton, W. (1991). A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Sem. Mat. Univ. Politec Torino 49 95-119. · Zbl 0776.60128
[6] Donsker, M. and Varadhan, S. R. S. (1975). Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28 525-565. · Zbl 0351.60070
[7] Donsker, M. and Varadhan, S. R. S. (1979). On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32 721-747. · Zbl 0418.60074
[8] Funaki, T. (1999). Free boundary problem from stochastic lattice gas model. Ann. Inst. H. Poincairé Probab. Statist. 35 573-603. · Zbl 0935.60094
[9] Gravner, J. and Quastel, J. (2000). Internal DLA and the Stefan problem. Ann. Probab. 28 1528-1562. · Zbl 1108.60318
[10] Grimmett, G. (1989). Percolation. Springer, New York. · Zbl 0691.60089
[11] Krug, J. and Spohn, H. (1991). Kinetic roughening of growing surfaces. In Solids Far From Equilibrium (C. Godr eche, ed.) 479-582. Cambridge Univ. Press.
[12] Lawler, G. (1991). Intersection of Random Walks. Birkhäuser, Ann Arbor. · Zbl 0735.60071
[13] Lawler, G., Bramson, M. and Griffeath, D. (1992). Internal diffusion limited aggregation. Ann. Probab. 20 2117-2140. · Zbl 0762.60096
[14] Stroock, D. and Zheng, W. (1997). Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré 33 619-649. · Zbl 0885.60065
[15] Sznitman, A. S. (1990). Lifschitz tail and Wiener sausage I. J. Funct. Anal. 94 223-246. · Zbl 0732.60088
[16] Sznitman, A. S. (1997). Capacity and principal eigenvalues: the method of enlargement of obstacles revisited. Ann. Probab. 25 1180-1209. · Zbl 0885.60063
[17] Sznitman, A. S. (1998). Brownian Motion Obstacles and Random Media. Springer, Berlin. · Zbl 0973.60003
[18] Sznitman, A. S. (1997). Fluctuations of principal eigenvalues and random scales. Comm. Math. Phys. 189 337-363. · Zbl 0888.60054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.