Critical Properties of the Potts Glass with many states

The main goal of this thesis is the extensive analysis of the phase structure of the Potts Glass (the disordered version of the Potts Model) in three dimensions for several values of the parameter p, the number of allowed state for the spin. In particular we concentrated on the values p = 4; 5; 6. The study of this model is important both from a theoretical standpoint than from the point of view of the applications: it is considered one of the best candidates to describe orientational glasses: In this way there is an opportunity to apply the information obtained from the study of theoretical models to the description of real glasses and, in general, to those of systems in which frustration and disorder play a leading role. There is no way to study the model in three dimensions using only analytical methods: in this thesis we used Monte Carlo simulations, which we were run on the Janus dedicated computer, to locate the para-spin glass transition and evaluated the critical exponents. We simulated systems of size L = 4; 6; 8; 12; 16 with p = 4; 5; 6. While the results may not appear incredibly precise, due the difficulties in the simulations of the model, they enrich the phenomenology of the model and will serve as a reference for future analysis. Each size has been analyzed independently but, also, we studied the system as a function of p: we obtained an empirical law the critical inverse temperature as a function of p (β_{c} ≈ p) and the critical exponents, which are compatible, for all p considered, with a continuous transition, in contrast with what is expected from Mean Field Theory, which describes a change in the nature of the transition from continuous to discontinuous for p > 4. From the analysis of Parallel Tempering for p = 5, L = 16 we infer that the model, as L increases, has apparently a different behaviour: we could be looking at a change in the nature of the transition as a function of L, opening different scenarios: as a function of p the transition of a three dimensional system could be continuous up to some value of p > 6, or could be continuous for all p. In the case in which the transition becomes discontinuous, this could have happened in the range of p we study, but the discontinuous transition could be rounded due to finite size effects. Understanding if this crossover exists increasing L will be doable once we have machines and techniques able to study system of bigger sizes.