Two-dimensional lattice models at criticality and the Schramm-Loewner evolution

点击翻译按钮获取中文摘要

One of the goals of statistical mechanics is to understand the behaviour of a physical system at criticality; that is, at the temperature where a phase transition occurs. For example, a phase transition occurs when water freezes at 0 degrees Celsius (i.e., it changes state from liquid to solid) or when it boils at 100 degrees Celsius (i.e., it changes state from liquid to gas). In more elaborate situations, the continuous physical system is often described by a discrete, or lattice, model. These lattice models are often more tractable mathematically, and often physical or chemical predictions about the continuous system can be proved rigorously using results established for the lattice model. An important class of systems are two-dimensional; by first understanding two-dimensional models, we may better hope to understand our three-dimensional world. An exciting develop- ment occurred in 1999 when the late O. Schramm introduced the stochastic Loewner evolution (SLE) while studying loop-erased random walk. SLE is a new class of conformally invariant stochastic processes which has revolutionized the study of critical phenomenon in statistical mechanics. In fact, its importance was punctuated by the awarding of the Fields Medals in 2006 to W. Werner "for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory" and in 2010 to S. Smirnov "for the proof of conformal invariance in percolation and the planar Ising model." Although this program of study has already produced several deeply significant results, there is still much more work to be done. For instance, the self-avoiding random walk, a model of polymer chains introduced by the Nobel prize-winning chemist P. Flory in the 1940's, is a model where minimal mathematical progress has been made and is one of the motivations for much of this entire program of study. Another important area of study that has not yet received much attention is the case of multiple interfaces that occur naturally in, for example, ferromagnetism and the corresponding Ising model. Thus, the primary goal of this research program is to help facilitate a better understanding of the interaction between SLE and conformal field theory.