One-dimensional lattice random walks in a Gaussian random potential

2025/09/28 by Kalaj, Silvio, Marinari, Enzo, Oshanin, Gleb +1
#FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2509.23985

Abstract

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, 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, resistance, and splitting probability are not self-averaging, which leads to pronounced differences between their average and typical behaviors. In contrast, T_N and D_N become self-averaging when N → ∞, though they exhibit strong sample-to-sample fluctuations for finite N.

Citations

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