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Self-Interacting Random Processes, Stochastic Approximations with Applications. & Two exactly solvable models of statistical mechanics: Ising and Dimer models
Responsable du projet Michel Benaim
Béatrice De Tilière
   
Collaborateur Grégory Roth
Erwan Hillion
Faure Mathieu
   
Résumé This research is divided in two parts. Part I consist of two projects - A: Stochastic Approximation and Applications. This is a direct continuation of the research conducted under the preceding SNF grants. This new proposal is strongly motivated by applications of our results to several questions arising in the theory of learning in games. Four directions have been precisely identified: - [A1] In collaboration with M. Faure (post-doc) we will work on stochastic approximation with decreasing step size associated to a cooperative vector field. This will be applied to the rigorous analysis of learning processes for supermodular games. - [A2] G. Roth (PhD student) will continue his investigation of stochastic approximations that are associated to differential inclusions. - [A3] In collaboration with O. Raimond (Paris 10) and M.~Faure we will conduct a precise analysis of Markovian fictitious play in term of consistency and efficiency. - [A4] In collaboration with J. Hofbauer and S. Sorin, we will consider stochastic approximation with constant step sizes that are associated to a differential inclusion. - B: Permanence in ecological systems. This second part is a continuation of the work initiated in collaboration with J. Hofbauer, W. Sandholm (Madison) and S. Schreiber (UC Davis) under the preceding grants. Our aim is to understand and derive conditions ensuring permanence for models of interacting species subjected to random fluctuations. Part II: Two exactly solvable models of statistical mechanics: Ising and Dimer models. We are specifically interested in two models of statistical mechanics: the 2-dimensional Ising model, a model of ferromagnet; and the dimer model, representing the adsorption of di-atomic molecules on the surface of a crystal. Both have the rare feature of being exactly solvable, thus yielding hope of obtaining deep and exact results. -A: The critical 2-dimensional Ising model. Our goal is to prove two predictions of Conformal Field Theory: convergence of the Ising contours to a Gaussian Free Field, and asymptotics of the Ising spin-spin correlation function. -B: The dimer model. Relying on the expertise acquired during our previous contributions to the field, our research project aims at understanding more on the dimer model defined on isoradial graphs. We would like to get insight into the cases were: - weights are Z-invariant, but not critical, - the underlying graph is isoradial but not bipartite, - the underlying graph is a `general' decorated graph obtained from an isoradial one. These questions are all related to the quest, dear to statistical mechanics today, of understanding universality, and critical behaviors.
   
Mots-clés Self interacting random processes, Stochastic approximation, Game theory, Differential equations, Differential inclusions, Ising model, Dimer model, Isoradial graphs, Critical phenomena, Exact local formulae
   
Type de projet Recherche fondamentale
Domaine de recherche Mathématiques
Source de financement FNS - Encouragement de projets (Div. I-III)
Etat Terminé
Début de projet 1-4-2010
Fin du projet 30-9-2011
Budget alloué 272'482.00
Autre information http://p3.snf.ch/projects-130574#
Contact Michel Benaim