  Consider a random medium consisting of points randomly distributed so that
there is no correlation among the distances. 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, the walker moves to the nearest site, which has not been visited in the
last $\mu$ steps (memory), producing a deterministic partially self avoiding
walk (the tourist walk). We have obtained analitically the distribution of the
number $n$ of points explored by a walker with memory $\mu = 2$, as well as the
transient and period joint distribution. This result enables to explain the
abrupt change in the exploratory behavior between the cases $\mu = 1$
(memoryless, driven by extremal statistics) and $\mu = 2$ (with memory, driven
by combinatorial statistics). In the $\mu = 1$ case, the mean newly visited
points in the thermodynamic limit $(N \gg 1)$ is just $<n > = e = 2.72 ...$
while in the $\mu = 2$ case, the mean number $<n>$ of visited points is
proportional to $N^{1/2}$. Also, this result allows us to stabilish an
equivalence between the random link model with $\mu=2$ and random map
(uncorrelated back and forth distances) with $\mu=0$ and the drastic change
between the cases where the transient time is null compared to non-null
transient times.
