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 |