Phase Transition for Elephant Random Walks with Two Memory Channels

Sunday, October 27, 2024, 4:45 -- 17:15

Random processes with strong memory and/or self-excitation arise naturally in various disciplines including physics, economics, biology, geology, etc. Memory can be multifaceted and can arise due to interactions of more than one underlying phenomena. Many of these processes exhibit superdiffusive growth due to the effect of self-excitation. A class of one-dimensional, discrete-time such models called “random walk with m memory channels” was introduced and discussed in a recent paper on statistical physics by Saha (2022). In these models, the information of m independently chosen steps from the walker’s entire history is needed to decide the future step. The aforementioned work carried out heuristic calculations of variance, and conjectured phase transitions from diffusive to superdiffusive and from superdiffusive to ballistic regimes in the m=2 case. We have proved these conjectures rigorously (with mild corrections), and discovered a new regime beyond the predicted ones. These results will be presented along with several open problems.

This talk is based on a joint work with Krishanu Maulik and Tamojit Sadhukhan.