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

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