A self-interacting random walk is a random process evolving in an environment which depends on its history. In this talk, we will discuss a few examples of these walks including the Lorentz gas, the mirror walk and the cyclic walk in the interchange process. I will present a method to analyze these walks in high dimensions and prove that they behave diffusively. If time permits, I will also mention a related result about the Poisson–Dirichlet distribution for the interchange process.

The talk is based on joint works with Allan Sly, Felipe Hernandez, Antoine Gloria and Gady Kozma.