Variational Description of Gibbs-Non-Gibbs Dynamical Transitions for Spin-Flip Systems with a Kac-Type Interaction

• Autores: Fernández, Roberto; Hollander, Frank den; Martínez Linares, Julián Facundo
• Año de publicación: 2014
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Gibbs Non Gibbs; Kac Potential; Large Deviations; Spin-Flip Dynamics
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

We continue our study of Gibbs-non-Gibbs dynamical transitions. In the present paper we consider a system of Ising spins on a large discrete torus with a Kac-type interaction subject to an independent spin-flip dynamics (infinite-temperature Glauber dynamics). We show that, in accordance with the program outlined in van Enter et al. (Moscow Math. J. 10:687–711, 2010), in the thermodynamic limit Gibbs-non-Gibbs dynamical transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the empirical density conditional on their endpoint. More precisely, the time-evolved measure is non-Gibbs if and only if this set is not a singleton for some value of the endpoint. A partial description of the possible scenarios of bifurcation is given, leading to a characterization of passages from Gibbs to non-Gibbs and vice versa, with sharp transition times. Our analysis provides a conceptual step-up from our earlier work on Gibbs-non-Gibbs dynamical transitions for the Curie–Weiss model, where the mean-field interaction allowed us to focus on trajectories of the empirical magnetization rather than the empirical density.
Fil: Fernández, Roberto. Utrecht Univeristy; Países Bajos
Fil: Hollander, Frank den. Leiden University; Países Bajos
Fil: Martínez Linares, Julián Facundo. Leiden University; Países Bajos. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina