Critical behavior of two-dimensional disordered Potts model

Phase transitions describe abrupt changes in the physical properties of systems and represent fundamental complex phenomena. While in the past, ideal models were used in order to obtain analytical results, in the last decades much attention has been directed towards more realistic representations. In fact, real materials are naturally characterized by defects and inhomogeneities which, even in small amounts, can have a strong influence on the features of second order phase transitions. Understanding the impact of disorder, or randomness, on critical statistical models is the main goal of this thesis. It represents the first step to describe more peculiar phenomena such as random lines of impurities. In particular this work focuses on the disordered Potts model on a square lattice, which, as a generalization of the Ising model, is able to describe different and broader classes of phenomena and phase transitions. The disorder is treated using the replica method, while the study of criticality is carried out using renormalization group techniques in real space and conformal field theories. In addition, these approaches are used for two general disorder distributions which are able to capture possible short-and long-range interactions. The two disorder critical points found confirm the existence of new universality classes, and their stability study is considered to represent the renormalization flow in the parametric space. The theoretical development is supported by numerical results using Monte Carlo methods with non-local updates, with a particular interest in the magnetization critical exponents. The study of the relevancy of the random fixed points is carried out numerically in several cases and provides additional evidence for the importance of disordered models.