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 $\mu$, leaves from the leftmost point and moves, at each discrete time step, to the nearest point, which has not been visited in the preceding $\mu$ steps. We have obtained analytically the probability $P_N(\mu )=(1-2^{-\mu }{)}^{N-\mu -1}$ that all $N$ points are visited in this open system, with $N\gg \mu \gg 1$. The expression for $P_N(\mu )$ evaluated in the mentioned limit is valid even for small $N$ and leads to a transition region centered at ${\mu }_1=\ln N/\ln 2$ and with width $\epsilon =e/\ln 2$. For $\mu <{\mu }_1-\epsilon /2$, the walker gets trapped in cycles and does not fully explore the system. For $\mu >{\mu }_1+\epsilon /2$ the walker explores the whole system. In both cases the walker presents diffusive behavior. Nevertheless, in the intermediate regime $\mu \sim {\mu }_1\pm \epsilon /2$, 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 ${\mu }_1$.

12 pages and 3 figures