Ising models with periodical external fields: phase diagrams and stochastic evolution

Abstract

In this project we propose the study of a ferromagnetic Ising model with a periodical external field. In particular, we consider the external field forming a chess-board, and alternately in each cell we put the value $h_1$ or $-h_2$, where $h_1, h_2> 0$.We address two particular mathematical questions, usually studied in the areas of probability and statistical mechanics: (1) characterization of the low-temperature phase diagram, and (2) the dynamical behaviors under a stochastic evolution (Glauber dynamics). These questions represent a complete characterization of an Ising model with periodical external field.In the first line of research, based on the applicant's doctoral thesis, we conjecture the presence of a phase transition for any low temperature, when $h_1 - h_2 = \varepsilon$ positive, but small. In the case of uniqueness regions of Gibbs measure, our focus is the study of sophisticated methods, such as the Pirogov-Sinai theory, cluster expansion and disagreement percolation.In order to attack the problem of the dynamical behaviors, we propose the study of the following two topics. On the one hand, we investigate mestastability, our aim is to characterize the metastable behaviors for $T> 0$, around the coexistence lines ($T = 0$). On the other hand, the phase transitions associated to the studied model, one of them showed in the doctoral thesys and the other conjectured in this project, seem to imply that the model presents loss and recovery of Gibbsianness by stochastic evolution. Our plan is to answer this last question. (AU)