Perfect simulation of Gibbs measures of infinite range on compact state space for particle system and its finite approximation

Abstract

The subject of this work is the construction and perfect simulation of Gibbs measures of infinite order on compact spaces for interacting particle systems with infinite components and its finite approximation. To construct and perfect simulate a Gibbs measure of infinite order we propose to couple stationary versions of Markov Processes, called Gibbs sampler, having as reversible measures the original Gibbs measure and its finite approximation. The processes can be constructed using the so called Harris graphical construction together with a particular version of the coupling from the past algorithm of Propp and Wilson. In order to construct the stationary process in infinite volume, we will follow the ideas developed by Fernández, Ferrari and Garcia for the study of the equilibrium measure of systems with exclusions in the high-temperature regime. Then, the results of this research will be in this high-temperature regime.The approach which will be followed in this research is based on recent works of Galves et al. (2009) and Grynberg (2009, PH-D Thesis). The result we expect to get generalizes the results of Galves et al to continuous spins and the results of Grynberg to Gibb measure compatible with the local specification induced by an infinite range potential. We do believe that this kind of result will offer a renewal alternative to the classical techniques in statistical mechanics making the analysis of Gibbs measures simpler. We expect to publish the main result of this research in a specialized journal. Also, it is expected that the visit of Professor Sebastian Pablo Grynberg will contribute to the consolidation of Probability as a genuine area of research at Centro de Matemática, Computação e Cognição (CMCC), UFABC. (AU)