Phase Transitions

Contents 1 Introduction and motivation: physical phase transitions 2 2 Mathematical abstraction: macroscopic order from microscopic disorder 4 3 The Thermodynamic Formalism and lattice gases 7 3. 5 Bond percolation 22 6 The Ising ferromagnet and bond percolation in two dimensions, and Peierls' argument 23 7 Other important ideas 28 7. Prerequisites This essay covers a number of different mathematical topics, none of them in very great depth. I will, however, have to assume basic notions of graph theory and some more sophisticated probability , measure theory and functional analysis. In particular, I will assume a basic working knowledge of topological measure theory (including the Riesz Representation Theorem, Radon measures and the vague topology on the space of Radon probabilities on a Polish space) and also ergodic theory 1 and Choquet's theory of barycentric decompositions. Much of the necessary technology is developed in the more comprehensive standard texts on the mathematical foundations of statistical mechanics (Chapter I of Simon [23] is a particularly useful introduction); alternatively, the more general background ideas can be found in any advanced probability text such as Kallenberg [15], or in comprehensive detail and generality in Volume 4 of Fremlin [9].