A two-dimensional non-Markovian random walk leading to anomalous diffusion

• We propose a non-Markovian random walk model in 2D.
• The model is analytically solved for the first two moments.
• Several anomalous regimes are exhibited (0<  Hurst exponent  <10<  Hurst exponent  <1).
• The full phase diagram with all anomalous regimes is drawn.
• The model can easily be upgraded to higher dimensions.

Exact solutions are rare for non-Markovian random walk models even in 1D, and much more so in 2D. Here we propose a 2D genuinely non-Markovian random walk model with a very rich phase diagram, such that the motion in each dimension can belong to one of 3 categories: (i) subdiffusive, (ii) superdiffusive, or (iii) normally diffusive. The main advance reported here is a different method, and the consequent physical insight, for analytically solving this model. Simpler non-Markovian models, such as Levy walks, have been solved in 2D, but it is not clear if the method of solution could be made to work for more complicated models such as the one studied here. We also report the exact solutions for the first two moments of the random walk propagator, along with the complete phase diagram. The latter is surprisingly rich and admits diverse diffusion regimes. Finally we discuss these results in the context of theoretical underpinnings as well as possible applications.