The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in @math whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation @math has weight @math , where @math is a positive real parameter, and @math is the total length of the edges in @math . Empirically, this model exhibits a "phase transition" at @math (corresponding to the uniform distribution): for @math very large regions of aligned edges appear. We substantiate this picture as follows. For @math we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.
A state-of-the-art survey of both classical and quantum lattice gas models, this two-volume work will cover the rigorous mathematical studies of such models as the Ising and Heisenberg, an area in which scientists have made enormous strides during the past twenty-five years. This first volume addresses, among many topics, the mathematical background on convexity and Choquet theory, and presents an exhaustive study of the pressure including the Onsager solution of the two-dimensional Ising model, a study of the general theory of states in classical and quantum spin systems, and a study of high and low temperature expansions. The second volume will deal with the Peierls construction, infrared bounds, Lee-Yang theorems, and correlation inequality.This comprehensive work will be a useful reference not only to scientists working in mathematical statistical mechanics but also to those in related disciplines such as probability theory, chemical physics, and quantum field theory. It can also serve as a textbook for advanced graduate students.Originally published in 1993.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. These notes have been the subject of a course I gave in the summer 1997 for the school in probability theory in Saint-Flour. I review in a self-contained way the state of the art, sometimes providing new and simpler proofs of the most relevant results, of the theory of Glauber dynamics for classical lattice spin models of statistical mechanics. The material covers the dynamics in the one phase region, in the presence of boundary phase transitions, in the phase coexistence region for the two dimensional Ising model and in the so-called Griffiths phase for random Systems.