Statistical Physics Out of Equilibrium: Quantitative Results and Universality

Abstract

Over the last 30 years, there has been spectacular progress in deriving the well-known hydrodynamic limits from stochastic interacting particle systems, as well as characterizing the fluctuations of locally conserved quantities around this limit. Many interesting results on the aforementioned topic have been derived from stochastic integrability, an approach relying on very specific combinatorial and algebraic properties of the underlying dynamics which allow deriving several scaling limits. However, a microscopic change on the dynamics can dramatically impact the macroscopic level, in the sense that scaling limits are no longer tractable by this methodology. Moreover, microscopic perturbations can lead to evolution equations with a variety of behaviours and at the critical parameter of the underlying dynamics, several universal anomalous laws can emerge, both in hydrodynamics and in fluctuations. More generally, understanding critical points where physical systems undergo phase transitions, and establishing that the phenomenology is described by universal mathematical objects that do not depend on the specific properties of the underlying microscopic dynamics, is a cornerstone of modern probability and mathematical physics, both from a pure and an applied point of view.