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Consider a random medium consisting of points randomly distributed so thatthere is no correlation among the distances. This is the random link model,which is the high dimensionality limit (mean field approximation) for theeuclidean random point structure. In the random link model, at discrete timesteps, the walker moves to the nearest site, which has not been visited in thelast $\mu$ steps (memory), producing a deterministic partially self avoidingwalk (the tourist walk). We have obtained analitically the distribution of thenumber $n$ of points explored by a walker with memory $\mu = 2$, as well as thetransient and period joint distribution. This result enables to explain theabrupt change in the exploratory behavior between the cases $\mu = 1$(memoryless, driven by extremal statistics) and $\mu = 2$ (with memory, drivenby combinatorial statistics). In the $\mu = 1$ case, the mean newly visitedpoints 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 isproportional to $N^{1/2}$. Also, this result allows us to stabilish anequivalence between the random link model with $\mu=2$ and random map(uncorrelated back and forth distances) with $\mu=0$ and the drastic changebetween the cases where the transient time is null compared to non-nulltransient times.

The influence of memory in deterministic walks in random media: analytical calculation within a mean field approximation