Consider a random medium consisting of $N$ 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 $\mu$ steps (memory), producing a deterministic partially self-avoiding walk (the tourist walk). We have analytically obtained the distribution of the number $n$ of points explored by the walker with memory $\mu =2$, 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 $\mu =1$ (memoryless walker, driven by extreme value statistics) and $\mu =2$ (walker with memory, driven by combinatorial statistics). In the $\mu =1$ case, the mean newly visited points in the thermodynamic limit $(N⪢1)$ is just $⟨n⟩=e=2.72\dots$ while in the $\mu =2$ case, the mean number $⟨n⟩$ of visited points grows proportionally to $N^{1∕2}$. Also, this result allows us to establish an equivalence between the random link model with $\mu =2$ and random map (uncorrelated back and forth distances) with $\mu =0$ 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