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

Grant number: 15/02801-6
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): June 01, 2015
Effective date (End): October 04, 2017
Field of knowledge:Physical Sciences and Mathematics - Probability and Statistics - Probability
Cooperation agreement: Coordination of Improvement of Higher Education Personnel (CAPES)
Principal Investigator:Luiz Renato Gonçalves Fontes
Grantee:Manuel Alejandro González Navarrete
Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
Associated scholarship(s):16/07183-1 - Metastability for Ising models with periodical-alternating external field, BE.EP.PD


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)

Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
GONZALEZ-NAVARRETE, MANUEL; LAMBERT, RODRIGO. Non-Markovian random walks with memory lapses. Journal of Mathematical Physics, v. 59, n. 11 NOV 2018. Web of Science Citations: 0.
GONZALEZ-NAVARRETE, MANUEL. TYPE-DEPENDENT STOCHASTIC ISING MODEL DESCRIBING THE DYNAMICS OF A NON-SYMMETRIC FEEDBACK MODULE. Mathematical Biosciences and Engineering, v. 13, n. 5, p. 981-998, OCT 2016. Web of Science Citations: 0.

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