Exact analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media
| dc.creator | Tercariol, Cesar Augusto Sangaletti | |
| dc.creator | Gonzalez, Rodrigo Silva | |
| dc.creator | Martinez, Alexandre Souto | |
| dc.date | 2007-02-01 | |
| dc.date.accessioned | 2026-07-07T07:44:25Z | |
| dc.date.available | 2026-07-07T07:44:25Z | |
| dc.description | 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. Using open boundary conditions, we have calculated analytically the probability $P_N(μ) = (1 - 2^{-μ})^{N - μ- 1}$ that all $N$ points are visited, with $N \gg μ\gg 1$. This approximated expression for $P_N(μ)$ is reasonable even for small $N$ and $μ$ values, as validated by Monte Carlo simulations. We show the existence of a critical memory $μ_1 = \ln N/\ln 2$. For $μ< μ_1 - e/(2\ln2)$, the walker gets trapped in cycles and does not fully explore the system. For $μ> μ_1 + e/(2\ln2)$ 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 $\log_{2} N$. | |
| dc.description | 7 pages and 5 figures | |
| dc.identifier | https://arxiv.org/abs/cond-mat/0702030 | |
| dc.identifier | http://arxiv.org/abs/cond-mat/0702030 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/123188 | |
| dc.subject | Disordered Systems and Neural Networks | |
| dc.subject | Statistical Mechanics | |
| dc.title | Exact analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media | |
| dc.type | text |