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.