The grant will support his project:

Statistical Physics through Random Paths and Random Functions

Abstract:

Statistical physics seeks to explain the macroscopic behavior of complex systems by analyzing the interactions between their microscopic components. For over a century, lattice models – that is, random systems defined on discrete lattices – have been introduced as simplified yet powerful descriptions of phase transitions in a wide range of phenomena, from ferroelectricity to lattice gases.

Over the past decades, significant advances have been made in the rigorous study of models such as percolation, self-avoiding-walks, and Ising. Many of the techniques and insights developed in these contexts have unexpectedly found applications far beyond their original scope, illuminating areas as diverse as quantum spin chains and models of electron localization and delocalization. The objective of this project is to achieve decisive progress in understanding phase transitions across a broad range of statistical physics models by pioneering the combined use of recently developed techniques – both those introduced by the PI and others emerging from the community – alongside new ideas from probability, combinatorics, analysis and integrable systems yet to be discovered. Specifically, we aim to pursue three main objectives: Objective A Deepen the understanding of spin O(n) models by leveraging recent advances in the graphical representation of spin systems.

Objective B Exploit more efficiently random height models to understand critical phenomena, with the goal of elucidating a broad class of lattice models.

Objective C Make significant advances in the study of statistical physics in high dimensions by applying novel path expansion techniques to a wider range of lattice models. Most of the questions we propose to address are notoriously difficult open problems. We believe that breakthroughs on these fronts would profoundly reshape our mathematical understanding of phase transitions.

Image: @Fabien Scotti