Invasion percolation with memory

Abstract

Motivated by the problem of finding the minimum threshold path (MTP) in a lattice of elements with random thresholds ${\mathrm{\ensuremath{\tau}}}_{\mathrm{i}}$, we propose a new class of invasion processes, in which the front advances by minimizing or maximizing the measure ${\mathrm{S}}_{\mathrm{n}}$=${\ensuremath{\sum}}_{\mathrm{i}}$${\mathrm{\ensuremath{\tau}}}_{\mathrm{i}}^{\mathrm{n}}$ for real n. This rule assigns long-time memory to the invasion process. If the rule minimizes ${\mathrm{S}}_{\mathrm{n}}$ (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. 61, 333 (1988)] but in a correlated lattice, with invasion percolation [D. Wilkinson and J. F. Willemsen, J. Phys. A 16, 3365 (1983)] recovered in the limit |n|=\ensuremath{\infty}. 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. 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.