A recipe for an optimal power law tailed walk

Lévy-like movements, which are an asymptotic power law tailed distribution with an upper cutoff, are known to represent an optimal search strategy in an unknown environment. Organisms seem to show a Lévy walk when μ ≈ 2.0. In the present study, I investigate how such a walk can emerge as a result of the decision making process of a single walker. In my proposed algorithm, a walker avoids a certain direction; this may be related to the emergence of a Lévy walk. Instead of remembering all visited positions, the walker in my algorithm uses and remembers only the direction from which it has come. Moreover, the walker sometimes reconsiders and alters the directions it avoids if it experiences some directional inconsistencies in a series of recent directional moves, i.e., the walker moves in a different direction from the previous one. My results show that a walker can demonstrate power law tailed movements over a long period with an optimal μ.