From Maximum of Inter-Visit Times to Starving Random Walks

Very recently, a fundamental observable has been introduced and analyzed to quantify the exploration of random walks: the time τ_{k} required for a random walk to find a site that it never visited previously, when the walk has already visited k distinct sites. Here, we tackle the natural issue of the statistics of M_{n}, the longest duration out of τ_{0},…,τ_{n-1}. This problem belongs to the active field of extreme value statistics, with the difficulty that the random variables τ_{k} are both correlated and nonidentically distributed. Beyond this fundamental aspect, we show that the asymptotic determination of the statistics of M_{n} finds explicit applications in foraging theory and allows us to solve the open d-dimensional starving random walk problem, in which each site of a lattice initially contains one food unit, consumed upon visit by the random walker, which can travel S steps without food before starving. Processes of diverse nature, including regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, share common properties within the same universality classes.