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

统计力学的目标之一是理解物理系统在临界状态下的行为；也就是说，在发生相变的温度下。例如，当水在0摄氏度结冰时（即从液态变为固态）或当水在100摄氏度沸腾时（即从液态变为气态），就会发生相变。在更复杂的情况下，连续的物理系统通常用离散的或点阵的模型来描述。这些点阵模型通常在数学上更容易处理，并且通常可以使用为点阵模型建立的结果严格证明关于连续系统的物理或化学预测。二维系统是一类重要的系统；通过首先了解二维模型，我们可能更有希望了解我们的三维世界。一个令人兴奋的进展发生在1999年，已故的O. Schramm在研究环擦除随机游走时引入了随机洛厄纳进化（SLE）。SLE是一类新的共形不变随机过程，它使统计力学中临界现象的研究发生了革命性的变化。事实上，2006年授予W. Werner“对随机洛伊纳演化、二维布朗运动几何和共形场论的发展做出的贡献”以及2010年授予S. Smirnov“对渗流和平面Ising模型的共形不变性的证明”的菲尔兹奖，都强调了它的重要性。尽管这一研究项目已经产生了一些意义深远的成果，但仍有许多工作要做。例如，由诺贝尔奖得主化学家P. Flory在20世纪40年代提出的聚合物链的自我避免随机游走模型，是一个数学上取得最小进展的模型，也是整个研究计划的主要动机之一。另一个尚未受到重视的重要研究领域是自然出现的多界面的情况，例如铁磁性和相应的Ising模型。因此，本研究计划的主要目标是帮助更好地理解SLE与适形场理论之间的相互作用。

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.