Dissipative Lévy random searches: universal behavior at low target density

We investigate the problem of survival at the very low target-density limit and report on a second-order phase transition for one-dimensional random searches in which the energy cost of locomotion is a function of the distance traveled by the searcher. Surprisingly, from analytical calculations (also tested numerically) we find identical critical exponents for arbitrary energy cost functions. We conclude that there is a single universality class that describes this process.