Multi-Dimensional Elephant Random Walk with Coupled Memory

The elephant random walk (ERW) is a microscopic, one-dimensional, discrete-time, non-Markovian random walk, which can lead to anomalous diffusion due to memory effects. In this study, I propose a multi-dimensional generalization in which the probability of taking a step in a certain direction depends on the previous steps in other directions. The original model is generalized in a straightforward manner by introducing coefficients that couple the probability of moving in one direction with the previous steps in all directions. I motivate the model by first introducing a two-elephant system and then elucidating it with a specific coupling. With the explicit calculation of the first moments, I show the existence of two newsworthy relative movement behaviours: one in which one elephant follows the other, and another in which they go in opposite directions. With the aid of a Fokker-Planck equation, the second moment is evaluated and two new super-diffusion regimes appear, not found in other ERWs. Then, I re-interpret the equations as a bidimensional elephant random walk model, and further generalize it to $N-$dimensions. I argue that the introduction of coupling coefficients is a way of extending any one-dimensional ERW to many dimensions.