Isabella Goncalves de Alvarenga (Univ. Orléans) : Forgetful elephant random walk

The elephant random walk is a non-Markovian nearest-neighbour random walk which, before taking each step, recalls an increment chosen uniformly from its entire history and then either repeats or reverses it. But what happens when the elephant forgets? In this talk, I will introduce a family of forgetful elephant random walks in which only a random subset of the past is available at each step. We encode this random memory through a directed percolation structure on the time indices: an edge from i to j means that the increment taken at time i may be recalled at time j. If no past increment is available, the walker instead takes a fresh biased step. I will discuss four mechanisms for generating the memory environment: Bernoulli percolation, geometric lifetimes, heavy-tailed lifetimes and Catalan percolation. These examples produce rather different memory geometries and require a variety of techniques, including martingale methods, renewal structures and random genealogical representations. I will explain how these structures determine the long-term behaviour of the walk and present laws of large numbers and fluctuation results in several regimes. This is a joint work with Deniz Luiz Araujo.