The Renormalization Group as a Stochastic Exploration Process

Abstract

The Renormalization Group (RG) is a powerful and versatile framework for analyzing complex physical systems. Here, we reinterpret it as a stochastic process that explores physical phase spaces, scale by scale, progressively revealing finer details of small-scale structures. This perspective establishes a natural connection to other random exploration processes, such as the Schramm-Loewner evolution, and is more suited for a probabilist audience. It links RG concepts such as RG transformations, effective actions, etc, to usual probabilistic tools such as conditional expectation values, martingales, etc, but also makes contact with stochastic quantization.