Phase Transitions in the Two Dimensional Ising Model and Integer Gaussian Free Field

Abstract

Various physical systems can be modeled effectively using the approach of statistical mechanics. The most famous perhaps in the Ising model, which serves as a mathematical model of spontaneous magnetization in two dimensions. We will begin by developing the necessary mathematics for understanding statistical mechanics on integer lattices. We then adapt various proofs to show the two dimensional Ising model demonstrates spontaneous magnetization for sufficiently low temperatures. Our study will generalize to the Gaussian Free Field, where we will study height correlation between points on the lattice. We finally restrict the Gaussian Free Field to the integers and show computationally there exists a delocalization phase transition as temperature increases.