Renormalization Group and Statistical Field Theory

This proposed project aims to develop rigorous renormalization group arguments, based on ideas from analysis and probability, and to apply them to problems in statistical physics, probability, and fluid mechanics. The emphasis is on physical phenomena which exhibit behavior across a large range of length scales, often with erratic and irregular behavior on small scales and possibly diverging correlation lengths. The idea is to gradually coarse-grain the system using ideas from the quantitative theory of stochastic homogenization, to obtain a macroscopic description which is simpler and thus more tractable, but also accurately describes the behavior of the system. Models that display anomalous diffusion, intermittent and/or multifractal behavior are of particular interest, examples of which can be found in simplified models of turbulent fluids and Euclidean field theory.