Universality, Phase Transitions and Statistical Mechanics

Abstract

AbstractThis survey will describe some mathematical results and conjectures related to phase transitions and statistical mechanics. In particular, we will focus on the principle of universality which asserts that singularities associated with second order phase transitions are universal. This means that they do not depend on the details of the model. Hence, although the temperature at which a phase transition occurs is typically model dependent, the macroscopic or long distance behavior at the transition is believed to depend on only a few general features such as the dimension of space and the symmetry of the model. Universality explains why relatively simple mathematical models can give quantitatively accurate information about transitions for a wide class of physical and mathematical systems. After a brief digression about random walks and self-avoiding walks we shall focus on the Ising model for interacting spins. The phase transition of the Ising model is believed to describe liquid-gas transitions, coupled chains of quantum anharmonic oscillators, certain quantum field theories and models of probabilistic cellular automata. The last section is devoted to a review of some problems in random Schrödinger operators and GOE and GUE matrix ensembles. The universality of eigenvalue correlations is explored.