Renormalization of Gibbs States

Equilibrium statistical mechanics is a branch of probability theory that has born between the end of 19th century and the beginning of the 20th century due to the work of Boltzmann and Gibbs. The latter, in particular, has been the fist one to introduced a statistical approach to thermodynamics to deduce collective macroscopic behaviours from individual microscopic information. Gibbs measures are the central object of this field, and the study of their existence and uniqueness is of great importance to understand the behaviour of a large number of physical infinite-volume models. In particular, the existence of more than one Gibbs measures is associated with statistical phenomena such as symmetry breaking and phase coexistence. In order to further investigate the behaviour of these systems, theoretical physicists developed a powerful tool: renormalization transformations. Simply speaking, they allows systematic investigation of the changes of a physical system as viewed at different distance scales. However, during the second half of 20th century, it has been noticed that these kind of transformations should be applied carefully: indeed, they may be ill-defined and present some pathologies. The aim of this work is double: first of all we want to present in details Gibbs formalism and the renormalization transformation. Second of all, we want to analyse one of the most famous examples in literature: the two-dimension Ising model.