Simple random covering, disconnection, late and favorite points

Abstract

We review recent advances in the study of the fractal nature of certain random sets, the key to which is a multi-scale truncated second moment method. We focus on some of the fine properties of the sample path of the most basic stochastic processes such as the simple random walk and the Brownian motion. As we shall see, probability on trees inspires many of our proofs, with trees used to model the relevant correlation structure. Along the way we also mention a few open problems.