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

TL;DR

This paper analyzes a deterministic self-avoiding walk in a one-dimensional random medium, deriving the probability of full exploration and identifying a phase transition from trapped to fully exploring behavior based on the walker's memory size.

Contribution

The authors provide an analytical expression for the exploration probability and characterize the phase transition and anomalous diffusion in the system.

Findings

Probability of full exploration: P_N(μ) = (1 - 2^{-μ})^{N - μ - 1}

Transition centered at μ₁ = ln N / ln 2 with constant width

Anomalous diffusion occurs in the intermediate memory regime

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 $\epsilon = e/\ln2$. 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 $\mu_1$.