Large-scale behavior of out-of-equilibrium systems at and above the critical dimension

Abstract

The mathematical understanding of out-of-equilibrium, statistical physics systems at and above their critical space dimension has witnessed spectacular progress in the last few years. These systems include, for instance, driven diffusive systems (such as stochastic interacting particle systems with asymmetric jump rates) in dimension $d\ge 2$, stochastic $(d+1)$-dimensional growth models with $d\ge 2$, non-linear singular (super-)critical stochastic partial differential equations (SPDEs) and (self-)interacting diffusions. We give here a review of recent results on their large-scale (i.e., infrared) behavior, focusing on singular SPDEs at and above the so-called critical dimension (in particular, the stochastic Burgers equation (SBE) in space dimension $d\ge 2$ and the anisotropic KPZ (AKPZ) equation in $d=2$) as well as on critical diffusions in random divergence-free vector field and self-repelling diffusions. A highlight of the main results includes sharp bounds on the large-time logarithmic divergence of the diffusion coefficient at the critical dimension, and Gaussian infrared scaling limits, at both the critical and super-critical dimensions.

Most of the results presented here have been obtained in collaboration with Dirk Erhard, Harry Giles, Massimiliano Gubinelli, Levi Haunschmid-Sibitz and Quentin Moulard, in different combinations.