The elephant random walk has memory: at each step, the walker chooses a past step uniformly at random and either repeats its direction (with probability p) or reverses it (with probability 1-p). The memory parameter p controls whether the walk is diffusive (p < 3/4), superdiffusive (p > 3/4), or at the critical point (p = 3/4). On the integer line, the asymptotic behavior is well-characterized.
The paper extends the elephant walk to the triangular lattice — a two-dimensional lattice where each vertex has six neighbors, arranged at 60° intervals. The extension is non-trivial because the memory mechanism must choose from a past step and then either repeat or reverse a direction in two dimensions, where “reverse” has multiple natural definitions.
On the triangular lattice, the walk retains its memory-dependent phase transition. Superdiffusive behavior persists above a critical memory parameter, and the walk is diffusive below it. The critical exponent is the same as in one dimension — the universality class of the elephant walk is insensitive to the lattice dimension and geometry.
The insensitivity is the content. The elephant walk's superdiffusion comes from its temporal correlations — the fact that steps are correlated with all past steps, not just recent ones. These temporal correlations dominate the spatial structure. Whether the walk happens on a line or a triangular lattice, the memory effect overwhelms the geometry. The lattice provides directions; the memory determines how far the walk goes. When the memory is strong enough to produce superdiffusion, it does so regardless of how many directions are available.