Universality, Phase Transitions and Disorder Effects in Statistical Physics

Mathematical statistical physics has seen spectacular progress in recent years.

Existing problems which were previously unattainable were solved, opening a way to approach some of the classical open questions in the field.

The proposed research focuses on phenomena of universality, phase transitions and the effect of disorder in physical systems of large size, identifying several fundamental questions at the interface of Statistical Physics and Probability Theory.One circle of questions concerns the fluctuation behavior of random surfaces, where the PI recently resolved the 1975 delocalization conjecture of Brascamp-Lieb-Lebowitz.

The PI proposes to establish some of the long-standing universality conjectures for random surfaces, including their scaling limit, localization properties and behavior of integer-valued surfaces.

A second circle of questions regards specific two-dimensional models on which there are exact predictions in the physics literature concerning their critical properties which remain elusive from the mathematical standpoint. The PI proposes several ways to advance the state of the art.

The PI further proposes to investigate the dependence of two-dimensional phenomena on the underlying planar graph structure, in the spirit of conjectures of Benjamini to which the PI recently supplied significant support.A third circle of questions revolves around random-field models.

Imry-Ma predicted in 1975, and Aizenman-Wehr proved in 1989, that an arbitrarily weak random field can eliminate the magnetization phase transition of systems in low dimensions including the spin O(n) models. Quantitative aspects of this phenomenon remain unclear, in the mathematical and physical literature.

Following recent substantial progress of the PI in the Ising model case, a quantitative analysis of the phenomenon for the classical models is proposed.Further emphasis is placed on the problem of finding new methods for proving the breaking of continuous symmetries.