From Maximum of Intervisit Times to Starving Random Walks
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909227464785920 |
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| author | Régnier, L. Dolgushev, M. Bénichou, O. |
| author_facet | Régnier, L. Dolgushev, M. Bénichou, O. |
| contents | 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 $τ_0,\dots,τ_{n-1}$. This problem belongs to the active field of extreme value statistics, with the difficulty that the random variables $τ_k$ are both correlated and 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09082 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | From Maximum of Intervisit Times to Starving Random Walks Régnier, L. Dolgushev, M. Bénichou, O. Statistical Mechanics Mathematical Physics Data Analysis, Statistics and Probability 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 $τ_0,\dots,τ_{n-1}$. This problem belongs to the active field of extreme value statistics, with the difficulty that the random variables $τ_k$ are both correlated and 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. |
| title | From Maximum of Intervisit Times to Starving Random Walks |
| topic | Statistical Mechanics Mathematical Physics Data Analysis, Statistics and Probability |
| url | https://arxiv.org/abs/2310.09082 |