Cost of excursions until first crossing of the origin for random walks and Lévy flights: an exact general formula
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908801156775936 |
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| author | Mori, Francesco Majumdar, Satya N. Vivo, Pierpaolo |
| author_facet | Mori, Francesco Majumdar, Satya N. Vivo, Pierpaolo |
| contents | We consider a discrete-time random walk on a line starting at $x_0\geq 0$ where a cost is incurred at each jump. We obtain an exact analytical formula for the distribution of the total cost of a trajectory until the process crosses the origin for the first time. The formula is valid for arbitrary jump distribution and cost function (heavy- and light-tailed alike), provided they are symmetric and continuous. We analyze the formula in different scaling regimes, and find a high degree of universality with respect to the details of the jump distribution and the cost function. Applications are given to the motion of an active run-and-tumble particle in one dimension and extensions to multiple cost variables are considered. The analytical results are in perfect agreement with numerical simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16152 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cost of excursions until first crossing of the origin for random walks and Lévy flights: an exact general formula Mori, Francesco Majumdar, Satya N. Vivo, Pierpaolo Statistical Mechanics Probability We consider a discrete-time random walk on a line starting at $x_0\geq 0$ where a cost is incurred at each jump. We obtain an exact analytical formula for the distribution of the total cost of a trajectory until the process crosses the origin for the first time. The formula is valid for arbitrary jump distribution and cost function (heavy- and light-tailed alike), provided they are symmetric and continuous. We analyze the formula in different scaling regimes, and find a high degree of universality with respect to the details of the jump distribution and the cost function. Applications are given to the motion of an active run-and-tumble particle in one dimension and extensions to multiple cost variables are considered. The analytical results are in perfect agreement with numerical simulations. |
| title | Cost of excursions until first crossing of the origin for random walks and Lévy flights: an exact general formula |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2403.16152 |