Statistical Mechanics and Singular SPDEs

Overview

Statistical Mechanics and Stochastic Partial Differential Equations (SPDEs) are two major areas of research in probability theory, both of which have witnessed fundamental breakthroughs in recent years. Statistical Mechanics models are typically discrete while SPDEs live in the continuum. One connection between the two is through scaling limits: SPDEs often arise as the continuum limits of stochastic fluctuations of discrete models in Statistical Mechanics. A deeper connection has emerged in recent years between the study of critical phenomena in statistical physics and the study of singular SPDEs. Both are connected to quantum field theories and are concerned with the phenomena of universality, namely that in suitable space-time scaling limits, different models may converge to the same universal limit.

What universal limit one obtains often depends crucially on the spatial dimension. For example, the Ising model and the closely related Phi^4 model have critical dimension 4, above which the model at the phase transition point has a universal Gaussian scaling limit, below which the model at the phase transition point should have universal but non-Gaussian scaling limits. For many singular SPDEs, there is also a critical dimension, below which we now have satisfactory solution theories including the theories of regularity structures, paracontrolled calculus, etc, while for singular SPDEs at and above the critical dimension, progress has only started to emerge in a few special cases.

For both statistical mechanics models and singular SPDEs, the emergence of universal scaling limits and the existence of a critical dimension can be explained heuristically via renormalization group arguments and are closely connected to quantum field theories. However, mathematically, much remains to be understood. This program seeks to explore these connections further and to advance our understanding of both critical phenomena and singular SPDEs.