Minimization of spatial cover times for impaired self-avoiding random walks: the mirage effect

Self-avoidance is a common mechanism to improve the efficiency of a random walker for covering a spatial domain. However, how this efficiency decreases when self-avoidance is impaired or limited by other processes has remained largely unexplored. Here we provide a numerical study in regular lattices for the case when the self-avoiding signal left by a walker both (i) saturates after successive revisits to a site, and (ii) evaporates, or disappears, after some characteristic time. We surprisingly reveal that the mean cover time becomes minimum for intermediate values of the evaporation time, leading to the existence of a nontrivial optimum management of the self-avoiding signal. We show that this is a consequence of a complex dynamics arising from the interplay between signal evaporation and signal saturation, in which evaporation has the capacity of creating some sort of mirages (sites or regions that the walker see as unvisited, though in fact they are not) that enhance the searcher mobility, so contributing to a more efficient exploration of the lattice that counteracts the effects of signal saturation. Remarkably, we argue both through scaling arguments and from numerical results, that this mirage effect will become more and more significant as long as the domain size increases.