Cover times of random searches

How long does it take a random searcher to visit all sites of a given domain? This time, known as the cover time, is a key observable to quantify the efficiency of exhaustive searches, which require a complete exploration of an area and not only the discovery of a single target; examples range from immune system cells chasing pathogens to animals harvesting resources, robotized exploration by e.g. automated cleaners or deminers, or algorithmics. Despite its broad relevance, the cover time has remained elusive and so far explicit results have been scarce and mostly limited to regular random walks. Here we determine the full distribution of the cover time for a broad range of random search processes, which includes the prominent examples of L\'evy strategies, intermittent strategies, persistent random walks and random walks on complex networks, and reveal its universal features. We show that for all these examples the mean cover time can be minimized, and that the corresponding optimal strategies also minimize the mean search time for a single target, unambiguously pointing towards their robustness.