Biased and greedy random walks on two-dimensional lattices with quenched randomness: the greedy ant within a disordered environment

The principle characteristics of biased greedy random walks (BGRWs) on two-dimensional lattices with real-valued quenched disorder on the lattice edges are studied. Here, the disorder allows for negative edge-weights. In previous studies, consideringthe negative-weight percolation (NWP) problem, this was shown to change the universality class of the existing, static percolation transition. In the presented study, four different types of BGRWs and an algorithm based on the ant colony optimization (ACO) heuristic were considered. Regarding the BGRWs, the precise configurations of the lattice walks constructed during the numerical simulations were influenced by two parameters: a disorder parameter rho that controls the amount of negative edge weights on the lattice and a bias strength B that governs the drift of the walkers along a certain lattice direction. Here, the pivotal observable is the probability that, after termination, a lattice walk exhibits a total negative weight, which is here considered as percolating. The behavior of this observable as function of rho for different bias strengths B is put under scrutiny. Upon tuning rho, the probability to find such a feasible lattice walk increases from zero to one. This is the key feature of the percolation transition in the NWP model. Here, we address the question how well the transition point rho_c, resulting from numerically exact and static simulations in terms of the NWP model can be resolved using simple dynamic algorithms that have only local information available, one of the basic questions in the physics of glassy systems.