Phase transition in the 2D Ising model: the theory behind and simulations via the Metropolis-Hastings algorithm

This thesis discusses the 2D Ising model and the phenomenon of phase transition, both from a theoretical and a simulation perspective, using the language of probability. We prove the existence of a phase transition first by using Peierls’ argument, an elegant and effective geometric-combinatorial approach to the problem, then simulating the Ising model as a Markov chain through the Metropolis-Hasting algorithm, via the Matlab platform. A self-contained part is presented with all the theoretical results necessary for a good understanding of the topics, ranging from Markov chains to Markov and Gibbs fields.