Abstract
Consider a random medium consisting of points randomly distributed so that there is no correlation among the distances separating them. This is the random link model, which is the high dimensionality limit (mean-field approximation) for the Euclidean random point structure. In the random link model, at discrete time steps, a walker moves to the nearest point, which has not been visited in the last steps (memory), producing a deterministic partially self-avoiding walk (the tourist walk). We have analytically obtained the distribution of the number of points explored by the walker with memory , as well as the transient and period joint distribution. This result enables us to explain the abrupt change in the exploratory behavior between the cases (memoryless walker, driven by extreme value statistics) and (walker with memory, driven by combinatorial statistics). In the case, the mean newly visited points in the thermodynamic limit is just while in the case, the mean number of visited points grows proportionally to . Also, this result allows us to establish an equivalence between the random link model with and random map (uncorrelated back and forth distances) with and the abrupt change between the probabilities for null transient time and subsequent ones.
- Received 13 June 2008
DOI:https://doi.org/10.1103/PhysRevE.78.031111
©2008 American Physical Society