From Maximum of Intervisit Times to Starving Random Walks

Abstract

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 non-identically 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.

arXiv comment6 pages, 3 figures + 16 pages, 11 figures

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