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Very recently, a fundamental observable has been introduced and analyzed toquantify the exploration of random walks: the time $\tau_k$ required for arandom walk to find a site that it never visited previously, when the walk hasalready visited $k$ distinct sites. Here, we tackle the natural issue of thestatistics of $M_n$, the longest duration out of $\tau_0,\dots,\tau_{n-1}$.This problem belongs to the active field of extreme value statistics, with thedifficulty that the random variables $\tau_k$ are both correlated andnon-identically distributed. Beyond this fundamental aspect, we show that theasymptotic determination of the statistics of $M_n$ finds explicit applicationsin foraging theory and allows us to solve the open $d$-dimensional starvingrandom walk problem, in which each site of a lattice initially contains onefood unit, consumed upon visit by the random walker, which can travel$\mathcal{S}$ steps without food before starving. Processes of diverse nature,including regular diffusion, anomalous diffusion, and diffusion in disorderedmedia and fractals, share common properties within the same universalityclasses.

From Maximum of Intervisit Times to Starving Random Walks