Starving Random Walks
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917699740762112 |
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| author | Régnier, Léo Dolgushev, Maxim Bénichou, Olivier |
| author_facet | Régnier, Léo Dolgushev, Maxim Bénichou, Olivier |
| contents | In this chapter, we review recent results on the starving random walk (RW) problem, a minimal model for resource-limited exploration. Initially, each lattice site contains a single food unit, which is consumed upon visitation by the RW. The RW starves whenever it has not found any food unit within the previous $\mathcal{S}$ steps. To address this problem, the key observable corresponds to the inter-visit time $τ_k$ defined as the time elapsed between the finding of the $k^\text{th}$ and the $(k+1)^\text{th}$ food unit. By characterizing the maximum $M_n$ of the inter-visit times $τ_0,\dots,τ_{n-1}$, we will see how to obtain the number $N_\mathcal{S}$ of food units collected at starvation, as well as the lifetime $T_\mathcal{S}$ of the starving RW. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_14248 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Starving Random Walks Régnier, Léo Dolgushev, Maxim Bénichou, Olivier Statistical Mechanics Mathematical Physics In this chapter, we review recent results on the starving random walk (RW) problem, a minimal model for resource-limited exploration. Initially, each lattice site contains a single food unit, which is consumed upon visitation by the RW. The RW starves whenever it has not found any food unit within the previous $\mathcal{S}$ steps. To address this problem, the key observable corresponds to the inter-visit time $τ_k$ defined as the time elapsed between the finding of the $k^\text{th}$ and the $(k+1)^\text{th}$ food unit. By characterizing the maximum $M_n$ of the inter-visit times $τ_0,\dots,τ_{n-1}$, we will see how to obtain the number $N_\mathcal{S}$ of food units collected at starvation, as well as the lifetime $T_\mathcal{S}$ of the starving RW. |
| title | Starving Random Walks |
| topic | Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2406.14248 |