Thermodynamical and geometrical approach to multi dimensional aperiodic structures

• Grant number: 24/04685-2
• Support Opportunities: Regular Research Grants
• Start date: December 01, 2024
• End date: November 30, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Agreement: ANR
• Principal Investigator: Eduardo Garibaldi
• Grantee: Eduardo Garibaldi
• Principal researcher abroad: Philippe Thieullen
• Institution abroad: Université de Bordeaux, Carreire/Victoire, France
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated researchers: Ali Messaoudi ; João Tiago Assunção Gomes ; Manuel Stadlbauer ; Paulo César Rodrigues Pinto Varandas
• Associated scholarship(s): 25/05034-8 - Frozen States for Short-Range Interaction Models with Four Spin States, BP.IC

Abstract

Ordered structure is understood in the sense that the structure possesses a low complexity as a quasicrystal, or an aperiodic geometric structure as a tiling or a Delone set, or a computable combinatorial structure as a sub-shift of finite type, a substitution, a recursively enumerable language. A disordered system is understood in the sense that, above a critical value of the temperature, each unit of the system behaves randomly independently from the others, and below that value, the system undergoes a sharp transition to a well structured pattern. The emergence of an order is understood in the sense that the set of Gibbs measures at positive temperature tends to structure itself as the system is cooling. Three axes of research will be developed: a first axe about the study of minimizing configurations of the Frenkel-Kontorova model, a second axe on the study of gradient Gibbs measures on lattices, a third axe on the thermodynamical formalism of the Delone sets or tilings. The three axes are tightly inter-winded in the sense that a Delone set is deformed version of a lattice, a Frenkel-Kontorova model is also a gradient type model coupled with en external periodic potential. (AU)