paper

Generalized arcsine laws for a sluggish random walker with subdiffusive growth

arXiv:2410.22097 · doi:10.1088/1742-5468/adb5fa

Abstract

We study a simple one dimensional sluggish random walk model with subdiffusive growth. In the continuum hydrodynamic limit, the model corresponds to a particle diffusing on a line with a space dependent diffusion constant D(x)= |x|^{-α} and a drift potential U(x)=|x|^{-α}, where α\geq 0 parametrizes the model. For α=0 it reduces to the standard diffusion, while for α>0 it leads to a slow subdiffusive dynamics with the distance scaling as x\sim t^μ at late times with μ= 1/(α+2)\leq 1/2. In this paper, we compute exactly, for all α\ge 0, the full probability distributions of three observables for a sluggish walker of duration T starting at the origin: (i) the occupation time t_+ denoting the time spent on the positive side of the origin, (ii) the last passage time t_{\rm l} through the origin before T, and (iii) the time t_M at which the walker is maximally displaced on the positive side of the origin. We show that while for α=0 all three distributions are identical and exhibit the celebrated arcsine laws of Lévy, they become different from each other for any α>0 and have nontrivial shapes dependent on α. This generalizes the Lévy's three arcsine laws for normal diffusion (α=0) to the subdiffusive sluggish walker model with a general α\geq 0. Numerical simulations are in excellent agreement with our analytical predictions.

27 pages, 8 Figures

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