Description du projet
Un examen (mathématique) plus approfondi des systèmes physiques de grande taille
En ce qui concerne le monde physique et notre univers de particules, la plupart des propriétés et des comportements que nous voyons au niveau global sont le résultat des interactions de nombreuses «unités», qu’il s’agisse de particules comme les électrons ou de structures multiparticulaires plus complexes comme les protons ou les neutrons. La physique statistique s’appuie sur la théorie des probabilités et les statistiques nous aident à résoudre des problèmes physiques résultant des interactions de ces grands nombres d’unités. Le domaine de la physique statistique mathématique a énormément progressé ces dernières années, mais des questions restent ouvertes. Le projet Transitions, financé par l’UE, aborde désormais plusieurs thèmes importants à l’intersection de la physique statistique et de la théorie des probabilités.
Objectif
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.
Champ scientifique
Mots‑clés
Programme(s)
Régime de financement
ERC-COG - Consolidator GrantInstitution d’accueil
69978 Tel Aviv
Israël