Optimum exploration memory and anomalous diffusion in deterministic partially self-avoiding walks in one-dimensional random media

Abstract

Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $μ$, leaves from the leftmost point and moves, at each discrete time step, to the nearest point, which has not been visited in the preceding $μ$ steps. We have obtained analytically the probability $P_N(μ) = (1 - 2^{-μ})^{N - μ- 1}$ that all $N$ points are visited in this open system, with $N \gg μ\gg 1$. The expression for $P_N(μ)$ evaluated in the mentioned limit is valid even for small $N$ and leads to a transition region centered at $μ_1 = \ln N/\ln 2$ and with width $ε= e/\ln2$. For $μ< μ_1 - ε/2$, the walker gets trapped in cycles and does not fully explore the system. For $μ> μ_1 + ε/2$ the walker explores the whole system. In both cases the walker presents diffusive behavior. Nevertheless, in the intermediate regime $μ\sim μ_1 \pm ε/2$, the walker presents anoumalous diffusion behavior. Since the intermediate region increases as $\ln N$ and its width is constant, a sharp transition is obtained for one-dimensional large systems. The walker does not need to have full memory of its trajectory to explore the whole system, it suffices to have memory of order $μ_1$.