Universality Classes for Strongly Correlated Random Fields

The objective of this project is to develop the mathematical theory of critical phenomena.

In spite of recent breakthroughs, second-order phase transitions have remained largely elusive in dimensions 3 and higher and represent a formidable challenge.

The project will make headway on this and related fundamental problems, especially in the hard intermediate dimensions.The expected outcomes of this proposal will provide prime rigorous results that determine the scaling behaviour of certain non-planar percolation models near their critical point. Its findings are meant to put a heuristic scaling theory originating from physics on firm mathematical foundations.

This theory is at the heart of modern computational methods and goes back to foundational (Nobel-prize winning) work on renormalisation.

Universality is the guiding principle by which the behaviour of many physical systems ought to be largely independent of the specifics of the model considered and instead only depend on a few key parameters.

The project is designed as a full-fledged study of several judiciously chosen models with long-range (LR) correlations, which my recent works have identified as ideal candidates for a rigorous investigation.

These benefit from a rich mathematical structure stemming from the interplay between percolation and random walks, which can be forcefully exploited.Building on my recent results, I plan to tackle a series of questions, in order to: (1) identify and describe their off-critical phases;(2) prove the existence of critical exponents exhibiting (hyper-)scaling;(3) investigate scaling limits of critical clusters and study their fractal geometry.The outcomes of steps (1)-(3) will give rise to a rigorous (near-)critical scaling theory for the continuous phase transition associated to these LR-models, and identify their universality classes.

In so doing the project will also tackle several related problems in disordered systems and random media.