Volume 77, Number 2, January 2007
Article Number 20003
Number of page(s) 4
Section General
Published online 11 January 2007
EPL, 77 (2007) 20003
DOI: 10.1209/0295-5075/77/20003

Stability of directed Min-Max optimal paths

E. Perlsman and S. Havlin

Department of Physics, Bar-Ilan University - Ramat-Gan 52900, Israel

received 18 July 2006; accepted in final form 17 November 2006; published January 2007
published online 11 January 2007

The stability of directed Min-Max optimal paths in cases of change in the random media is studied. Using analytical arguments it is shown that when small perturbations $\epsilon$ are applied to the weights of the bonds of the lattice, the probability that the new Min-Max optimal path is different from the original Min-Max optimal path is proportional to $t^{1/\nu _{\parallel }}\epsilon$, where t is the size of the lattice, and $\nu _{\parallel }$ is the longitudinal correlation exponent of the directed percolation model. It is also shown that in a lattice whose bonds are assigned with weights which are near the strong disorder limit, the probability that the directed polymer optimal path is different from the optimal Min-Max path is proportional to $t^{2/\nu _{\parallel }}/k^{2}$, where k is the strength of the disorder. These results are supported by numerical simulations.

05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion .
05.50.+q - Lattice theory and statistics (Ising, Potts, etc.) .

© Europhysics Letters Association 2007