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

TL;DR

This paper analytically studies how memory affects deterministic walks in a high-dimensional random medium, revealing a phase transition in exploration behavior between memoryless and memory-influenced walks.

Contribution

It provides the first analytical distribution of explored points for a walker with memory two in the random link model, explaining the transition in exploration dynamics.

Findings

For $=1$, the mean explored points is approximately 2.72.

For $=2$, the mean explored points scales as the square root of the total points.

The study establishes an equivalence between the random link model with memory 2 and random maps with no memory.

Abstract

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.