First-passage statistics of random walks: a general approach via Riemann-Hilbert problems
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915390891753472 |
|---|---|
| author | Radice, Mattia Cristadoro, Giampaolo |
| author_facet | Radice, Mattia Cristadoro, Giampaolo |
| contents | We study first-passage statistics for one-dimensional random walks $S_n$ with independent and identically distributed jumps starting from the origin. We focus on the joint distribution of the first-passage time $τ_b$ and first-passage position $S_{τ_b}$ beyond a threshold $b\geq0$, as well as the distribution of $S_n$ for the walks that do not cross $b$ up to step $n$. By solving suitable Riemann-Hilbert problems, we are able to obtain exact and semi-explicit general formulae for the quantities of interest. Notably, such formulae are written solely in terms of the characteristic function of the jumps. In contrast with previous results, our approach is universally valid, applicable to both continuous and discrete, symmetric and asymmetric jump distributions. We complement our theoretical findings with explicit examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First-passage statistics of random walks: a general approach via Riemann-Hilbert problems Radice, Mattia Cristadoro, Giampaolo Statistical Mechanics We study first-passage statistics for one-dimensional random walks $S_n$ with independent and identically distributed jumps starting from the origin. We focus on the joint distribution of the first-passage time $τ_b$ and first-passage position $S_{τ_b}$ beyond a threshold $b\geq0$, as well as the distribution of $S_n$ for the walks that do not cross $b$ up to step $n$. By solving suitable Riemann-Hilbert problems, we are able to obtain exact and semi-explicit general formulae for the quantities of interest. Notably, such formulae are written solely in terms of the characteristic function of the jumps. In contrast with previous results, our approach is universally valid, applicable to both continuous and discrete, symmetric and asymmetric jump distributions. We complement our theoretical findings with explicit examples. |
| title | First-passage statistics of random walks: a general approach via Riemann-Hilbert problems |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2507.11196 |