New classes of Gibbs model and their applications

Abstract

The goal of the project is the investigation of a new class of statistical physics models. Besides the study of the properties of the model we will investigate its possible applications in physics and biology. The main feature of the first class of models is the fact that the interaction at each site is a function of the configuration as seen by the site. Our first goal is to develop the thermodynamical formalism for such interacting systems. This is a generalization of the notion of stochastic chain with memory of variable length introduced by J. Rissane, 1983, and extensively studied by the probability research group of IME-USP and their collaborators: A. Galves, F. Leonardi, P. Collet, E. Loecherbach, V. Maume-Deschamps, B. Schmitt e outros. In particular we are interested in the investigation of the possibility of extending the classical uniqueness result of R. Dobrushin and E. Pechersky(1983) "A criterion of the uniqueness of Gibbsian fields in the non-compact case" and results recently suggested by A.Galves, E. Loecherbach, E. Orlandi (2009) "Coupled perfect simulation of infinite range Gibbs measures and their finite range approximations". In the second class of the models which we call discrete gas models we study percolation problems. The existence of a threshold between the percolation almost sure and the non-percolation almost sure can be connected to the phase transition insulator-conductor. In E. Pechersky, A. Yambartsev (2009) "Percolation properties of non-ideal gas" it was obtained a similar result for the continuous model of the gas. We hope that the discrete case can provide a more clear picture of the physical phenomena and sharpening of the results already known. (AU)