Random processes with strong memory and/or self-excitation properties arise naturally in various disciplines including physics, economics, biology, engineering, geology, etc. Many of these processes exhibit superdiffusive growth due to the effect of self-excitation. In order to model such situations, two statistical physicists Schütz and Trimper (2004) introduced a class of processes called Elephant Random Walks, in which the random walker (an elephant 🐘 with strong memory!) remembers the past steps and repeats them with some probability. In the first half of this talk, we shall discuss the basics of random walks and the elephant random walk in a lucid language. The second half will focus on a couple of models investigated by the speaker and his collaborators leading to phase transition results that verify a bunch of conjectures of Saha (2022).

Special care will be taken so that students can follow both halves of this talk. Most of the notions will be introduced and nothing more than basic probability will be prerequisite for this talk. (Based on collaborations with A. Haldar, K. Maulik, S. S. Manna and T. Sadhukhan.)