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

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

2007-02-01

arxiv.org/abs/cond-mat/0702030v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

7 pages and 5 figures

Scientific paper

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 $\log_{2} N$.

Affiliated with

Also associated with

No associations

Rating

Exact analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Exact analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Exact analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-675634