Conformal invariance and $2D$ statistical physics

Abstract

A number of two-dimensional models in statistical physics are conjectured to have scaling limits at criticality that are in some sense conformally invariant. In the last ten years, the rigorous understanding of such limits has increased significantly. I give an introduction to the models and one of the major new mathematical structures, the Schramm-Loewner Evolution (SLE). 1. Critical Phenomena Critical phenomena in statistical physics refers to the study of systems at or near the point at which a phase transition occurs. There are many models of such phenomena. We will discuss some discrete equilibrium models that are defined on a lattice. These are measures on paths or configurations where configurations are weighted by their energy E with a preference for paths of smaller energy. These measures depend on at least one parameter. A standard parameter in physics β = c/T where c is a fixed constant, T stands for temperature and the measure given to a configuration is e . Phase transitions occur at critical values of the temperature corresponding to, e.g., the transition from a gaseous to liquid state. We use β for the parameter although for understanding the literature it is useful to know that large values of β correspond to “low temperature” and small values of β correspond to “high temperature”. Small β (high temperature) systems have weaker correlations than large β (low temperature) systems. In a number of models, there is a critical βc such that qualitatively the system has three regimes β βc (low temperature). A standard procedure is to define a model on a finite subset of a lattice, and then let the lattice size grow. Equivalently, one can consider a bounded region and consider finer and finer lattices inside the region. In either case, one would like to know the scaling or continuum limit of the system. For many models, it can be difficult to even describe what kind of object one has in the limit, and then it can be much more difficult to prove such a limit exists. The behavior of the models I will describe varies significantly in different dimensions. In two dimensions, Belavin, Polyakov, and Zamolodchikov predicted [3, 4] that many systems at criticality (at βc) had scaling limits that were in some sense conformally invariant. This assumption and nonrigorous techniques of conformal field theory allowed for exact calculation of a number of “critical exponents” and other quantities in the limit. For mathematicians, many of the arguments were 2000 Mathematics Subject Classification. Primary 82B27 ; Secondary 30C35, 60J65, 82B27. Research supported by National Science Foundation grant DMS-0734151. 1I use the word “predicted” to mean that the result was mathematically nonrigorous, but had significant theoretical argument behind it. There is much nontrivial, deep mathematics in the conformal field theory arguments and other theories in mathematical physics. I use the word “predict” rather then “conjecture” to acknowledge this.