Very recently, a fundamental observable has been introduced and analyzed to quantify the exploration of random walks: the time ${\tau }_k$ 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 ${\tau }_0,\dots ,{\tau }_{n-1}$. This problem belongs to the active field of extreme value statistics, with the difficulty that the random variables ${\tau }_k$ 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 $\mathcal{S}$ 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.

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• Random walks
• Stochastic processes