### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 1470-1527 |

Number of pages | 58 |

Journal | Annals of Probability |

Volume | 28 |

Issue number | 4 |

State | Published - Oct 2000 |

### Fingerprint

### Keywords

- Enlargement of obstacles
- Internal diffusion limited aggregation
- Principal eigenvalue
- Survival probability

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Probability*,

*28*(4), 1470-1527.

**Asymptotic survival probabilities in the random saturation process.** / Arous, Gerard Ben; Ramírez, Alejandro F.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 28, no. 4, pp. 1470-1527.

}

TY - JOUR

T1 - Asymptotic survival probabilities in the random saturation process

AU - Arous, Gerard Ben

AU - Ramírez, Alejandro F.

PY - 2000/10

Y1 - 2000/10

N2 - 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.

AB - 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.

KW - Enlargement of obstacles

KW - Internal diffusion limited aggregation

KW - Principal eigenvalue

KW - Survival probability

UR - http://www.scopus.com/inward/record.url?scp=0040364155&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040364155&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0040364155

VL - 28

SP - 1470

EP - 1527

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 4

ER -