Exact analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media

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. Using open boundary conditions, we have calculated analytically the probability $P_N(\mu) = (1 - 2^{-\mu})^{N - \mu - 1}$ that all $N$ points are visited, with $N \gg \mu \gg 1$. This approximated expression for $P_N(\mu)$ is reasonable even for small $N$ and $\mu$ values, as validated by Monte Carlo simulations. We show the existence of a critical memory $\mu_1 = \ln N/\ln 2$. For $\mu < \mu_1 - e/(2\ln2)$, the walker gets trapped in cycles and does not fully explore the system. For $\mu > \mu_1 + e/(2\ln2)$ the walker explores the whole system. Since the intermediate region increases as $\ln N$ and its width is constant, a sharp transition is obtained for one-dimensional large systems. This means that the walker needs not to have full memory of its trajectory to explore the whole system. Instead, it suffices to have memory of order $\log_{2} N$.