Records, drift, and the longest increasing subsequence of biased Gaussian random walks
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913169544314880 |
|---|---|
| author | Mendonça, J. Ricardo G. |
| author_facet | Mendonça, J. Ricardo G. |
| contents | The longest increasing subsequence (LIS) of a random walk has so far been studied mainly for zero-mean, symmetric step increments. We numerically investigate the LIS of biased Gaussian random walks, with unit-variance increments and positive drift $μ_{p} = Φ^{-1}(p)$, where $p = \mathbb{P}(ξ>0)$. In contrast with the symmetric case, we find that for every fixed $p>1/2$ the mean LIS length grows linearly, $\langle L_{n}(p)\rangle \sim a(p)n$, with $a(p)$ increasing from $0$ at $p=1/2$ to $1$ as $p \to 1$. The record count is also linear, with coefficient $λ(p)$ given by Spitzer's formula for the mean ascending ladder epoch, and the LIS becomes increasingly aligned with this record skeleton as $p$ grows. At the symmetric point $p=1/2$, the record skeleton collapses to the Sparre Andersen $\sqrt{n}$ scale, while the LIS returns to the symmetric finite-variance $\sqrt{n}\log{n}$ regime. Near this limit, the excess $a(μ_{p})-λ(μ_{p})$ vanishes more slowly than linearly in the drift, although our data do not resolve a single power law. The empirical distribution of $L_{n}$ also changes across the singular point, from lognormal-like at $p=1/2$ to fluctuations consistent with Gaussian behavior for every sampled $p>1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29185 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Records, drift, and the longest increasing subsequence of biased Gaussian random walks Mendonça, J. Ricardo G. Statistical Mechanics Probability The longest increasing subsequence (LIS) of a random walk has so far been studied mainly for zero-mean, symmetric step increments. We numerically investigate the LIS of biased Gaussian random walks, with unit-variance increments and positive drift $μ_{p} = Φ^{-1}(p)$, where $p = \mathbb{P}(ξ>0)$. In contrast with the symmetric case, we find that for every fixed $p>1/2$ the mean LIS length grows linearly, $\langle L_{n}(p)\rangle \sim a(p)n$, with $a(p)$ increasing from $0$ at $p=1/2$ to $1$ as $p \to 1$. The record count is also linear, with coefficient $λ(p)$ given by Spitzer's formula for the mean ascending ladder epoch, and the LIS becomes increasingly aligned with this record skeleton as $p$ grows. At the symmetric point $p=1/2$, the record skeleton collapses to the Sparre Andersen $\sqrt{n}$ scale, while the LIS returns to the symmetric finite-variance $\sqrt{n}\log{n}$ regime. Near this limit, the excess $a(μ_{p})-λ(μ_{p})$ vanishes more slowly than linearly in the drift, although our data do not resolve a single power law. The empirical distribution of $L_{n}$ also changes across the singular point, from lognormal-like at $p=1/2$ to fluctuations consistent with Gaussian behavior for every sampled $p>1/2$. |
| title | Records, drift, and the longest increasing subsequence of biased Gaussian random walks |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2605.29185 |