# Invasion percolation with memory

## Abstract

Motivated by the problem of finding the minimum threshold path (MTP) in a lattice of elements with random thresholds {tau}{sub i}, we propose a new class of invasion processes, in which the front advances by minimizing or maximizing the measure S{sub n}={summation}{sub i}{tau}{sub i}{sup n} for real n. This rule assigns long-time memory to the invasion process. If the rule minimizes S{sub n} (case of minimum penalty), the fronts are stable and connected to invasion percolation in a gradient [J. P. Hulin, E. Clement, C. Baudet, J. F. Gouyet, and M. Rosso, Phys. Rev. Lett. {bold 61}, 333 (1988)] but in a correlated lattice, with invasion percolation [D. Wilkinson and J. F. Willemsen, J. Phys. A {bold 16}, 3365 (1983)] recovered in the limit {vert_bar}n{vert_bar}={infinity}. For small n, the MTP is shown to be related to the optimal path of the directed polymer in random media (DPRM) problem [T. Halpin-Healy and Y.-C. Zhang, Phys. Rep. {bold 254}, 215 (1995)]. In the large n limit, however, it reduces to the backbone of a mixed site-bond percolation cluster. The algorithm allows for various properties of the MTP and the DPRM to be studied. In the unstable case (case of maximum gain), themore »

- Authors:

- Department of Chemical Engineering, University of Southern California, Los Angeles, California 90089-1211 (United States)

- Publication Date:

- OSTI Identifier:
- 530154

- DOE Contract Number:
- FG22-93BC14899

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 6; Other Information: PBD: Jun 1997

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING NOT INCLUDED IN OTHER CATEGORIES; FLOW MODELS; ALGORITHMS; POROUS MATERIALS; CLUSTER MODEL

### Citation Formats

```
Kharabaf, H., and Yortsos, Y.C.
```*Invasion percolation with memory*. United States: N. p., 1997.
Web. doi:10.1103/PhysRevE.55.7177.

```
Kharabaf, H., & Yortsos, Y.C.
```*Invasion percolation with memory*. United States. doi:10.1103/PhysRevE.55.7177.

```
Kharabaf, H., and Yortsos, Y.C. Sun .
"Invasion percolation with memory". United States. doi:10.1103/PhysRevE.55.7177.
```

```
@article{osti_530154,
```

title = {Invasion percolation with memory},

author = {Kharabaf, H. and Yortsos, Y.C.},

abstractNote = {Motivated by the problem of finding the minimum threshold path (MTP) in a lattice of elements with random thresholds {tau}{sub i}, we propose a new class of invasion processes, in which the front advances by minimizing or maximizing the measure S{sub n}={summation}{sub i}{tau}{sub i}{sup n} for real n. This rule assigns long-time memory to the invasion process. If the rule minimizes S{sub n} (case of minimum penalty), the fronts are stable and connected to invasion percolation in a gradient [J. P. Hulin, E. Clement, C. Baudet, J. F. Gouyet, and M. Rosso, Phys. Rev. Lett. {bold 61}, 333 (1988)] but in a correlated lattice, with invasion percolation [D. Wilkinson and J. F. Willemsen, J. Phys. A {bold 16}, 3365 (1983)] recovered in the limit {vert_bar}n{vert_bar}={infinity}. For small n, the MTP is shown to be related to the optimal path of the directed polymer in random media (DPRM) problem [T. Halpin-Healy and Y.-C. Zhang, Phys. Rep. {bold 254}, 215 (1995)]. In the large n limit, however, it reduces to the backbone of a mixed site-bond percolation cluster. The algorithm allows for various properties of the MTP and the DPRM to be studied. In the unstable case (case of maximum gain), the front is a self-avoiding random walk. {copyright} {ital 1997} {ital The American Physical Society}},

doi = {10.1103/PhysRevE.55.7177},

journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},

number = 6,

volume = 55,

place = {United States},

year = {1997},

month = {6}

}