Phase Transitions for Elephant Random Walks with Two memory Channels

Elephant random walk, introduced to study the effect of memory on random walks, is a novel type of walk that incorporates the information of one randomly chosen past step to determine the future step. However, memory of a process can be multifaceted and can arise due to interactions of more than one underlying phenomena. To model this, random walks with multiple memory channels were introduced in the statistical physics literature by Saha (2022) - here the information on a bunch of independently chosen past steps is needed to decide the future step. With the help of variance heuristics, this work analyzed the two-channel case and predicted a double phase transition: from diffusive to superdiffusive and from superdiffusive to ballistic regimes. We prove these conjectures rigorously (with some corrections), discover a mildly superdiffusive regime at one of the conjectured transition boundaries, and observe a new second-order phase transition. We also carry out a detailed investigation of the asymptotic behavior of the walk at different regimes.