Ising Model I

Abstract: We will introduce the Ising model on a finite lattice and discuss the infinite volume thermodynamic limit. We will prove that the pressure (a.k.a. free energy) converges, and describe the magnetization in the limit in terms of differentiability of the pressure. This will lead us to a definition of a first-order phase transition in the system. Lastly, we'll analyze the 1D Ising model, for which we can compute the pressure exactly and prove that there is no phase transition: the magnetization always concentrates at 0, unlike what we saw with the Curie-Weiss model.