One-dimensional lattice random walks in a Gaussian random potential
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908852763492352 |
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| author | Kalaj, Silvio Marinari, Enzo Oshanin, Gleb Peliti, Luca |
| author_facet | Kalaj, Silvio Marinari, Enzo Oshanin, Gleb Peliti, Luca |
| contents | We study random walks evolving in continuous time on a one-dimensional lattice where each site $x$ hosts a quenched random potential $U_x$. The potentials on different sites are independent, identically distributed Gaussian random variables. We analyze three distinct models that specify how the transition rates depend on $U_x$: the random-force-like model, random walks with randomized stepping times, and the Gaussian trap model. Our analysis focuses on five key disorder-dependent quantities defined for a finite chain with $N$ sites: the probability current, its reciprocal (the resistance), the splitting probability $E_-$, the mean first-passage time $T_N$, and the diffusion coefficient $D_N$ in a periodic chain. By determining the moments of these random variables, we demonstrate that the probability current and resistance are not self-averaging, which leads to pronounced differences between their average and typical behaviors. In contrast, $E_-$, $T_N$ and $D_N$ become self-averaging when $N \to \infty$, though they exhibit strong sample-to-sample fluctuations for finite $N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23985 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | One-dimensional lattice random walks in a Gaussian random potential Kalaj, Silvio Marinari, Enzo Oshanin, Gleb Peliti, Luca Statistical Mechanics We study random walks evolving in continuous time on a one-dimensional lattice where each site $x$ hosts a quenched random potential $U_x$. The potentials on different sites are independent, identically distributed Gaussian random variables. We analyze three distinct models that specify how the transition rates depend on $U_x$: the random-force-like model, random walks with randomized stepping times, and the Gaussian trap model. Our analysis focuses on five key disorder-dependent quantities defined for a finite chain with $N$ sites: the probability current, its reciprocal (the resistance), the splitting probability $E_-$, the mean first-passage time $T_N$, and the diffusion coefficient $D_N$ in a periodic chain. By determining the moments of these random variables, we demonstrate that the probability current and resistance are not self-averaging, which leads to pronounced differences between their average and typical behaviors. In contrast, $E_-$, $T_N$ and $D_N$ become self-averaging when $N \to \infty$, though they exhibit strong sample-to-sample fluctuations for finite $N$. |
| title | One-dimensional lattice random walks in a Gaussian random potential |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2509.23985 |