Clearing out a maze: The hungry random walker and its anomalous diffusion

We study chemotaxis in a porous medium using as a model a biased ("hungry") random walk on a percolating cluster. In close resemblance to the 1980s arcade game Pac-Man, the hungry random walker consumes food, which is initially distributed in the maze, and biases its movement towards food-filled sites. We observe that, on the percolating cluster, the mean-squared displacement of the pacman process shows anomalous dynamics, which follow a power law with a dynamical exponent different from both that of a self avoiding random walk as well as that of an unbiased random walk. The change in dynamics with the propensity to move towards food is well described by a dynamical exponent that depends continuously on this propensity, and results in slower differential growth when compared to the unbiased random walk.