Conformally invariant statistical mechanics models

Columbia University, New York NY

Investigators

Abstract

Recently, spectacular progress in the field of conformally invariant processes has been achieved, in particular with the introduction of the SLE processes. Thanks to these advances, it appears now possible to understand at an unprecedented level of resolution a wide class of models, and to construct their universal limits. The Ising model, which is probably the most studied model for phase transitions, has seen recent breakthroughs, and its further investigation seems very promising: it is one of the models that offer the richest behaviors and is among the best understood from a mathematical point of view. A unified picture, describing the limit of this model as random curves and fields, capturing subtle geometry and boundary conditions effects, seems to be rigorously obtainable. This picture will also deepen the comprehension of other models and offer many applications. Phase transitions are abrupt changes in the nature of systems: vapor that suddently condensates into water, metals that gain supraconductivity, social networks whose activity explode once critical mass has been reached. A major question is to understand how such global-scale phase transitions occur in large systems. We plan to develop tools and techniques to study models where random macroscopic geometries arise. Such models have found applications for instance in chemistry, image processing, ecology, economics or machine learning. Thanks to new ideas and methods introduced in the recent years, it now appears possible to give a more complete description of the phase transition of such models, which will hopefully become useful tools for both theoretical and applied researchers.