The Schramm-Loewner evolution and scaling limits of discrete planar processes

统计力学的广泛目标之一是理解物理系统在临界点的行为；也就是在(或接近)相变发生的温度下。相变的一个众所周知的例子是当水结冰(即从液体变为固体)或沸腾(即从固体变为气体)时发生。在更精细的模型中，连续的物理系统由离散的或格子模型很好地描述。这些晶格模型通常在数学上更容易处理，并且通常可以使用为晶格模型建立的结果严格地证明关于连续系统的物理或化学预测。一类重要的系统是二维的；通过首先理解二维模型，我们可能更有希望理解我们的三维世界。一个令人兴奋的发展发生在2000年，Oded Schramm在研究环擦除随机游动的标度极限时引入了随机Loewner演化(SLE)。SLE是一类新的共形不变随机过程，它彻底改变了统计力学中临界现象的研究。事实上，它的重要性被授予温德林·沃纳的菲尔兹奖，以表彰他对随机洛夫纳演化、二维布朗运动的几何和共形场论的发展做出的贡献。虽然这项研究已经取得了几项意义深远的成果，但仍有很多工作要做。例如，诺贝尔奖获得者、化学家保罗·弗洛里在1940年的《S》中提出的高分子链模型--自我回避随机游走--就是一个在数学上取得了最小进展的模型，也是整个研究计划的大部分动机之一。另一个尚未得到太多关注的重要研究领域是自然出现的多个界面的情况，例如，铁磁性和相应的伊辛模型。因此，本研究计划的主要目标是研究多条SLE曲线及其相互作用，这是考虑多个界面时自然连续研究的对象。

One of the broad goals of statistical mechanics is to understand the behaviour of a physical system at criticality; that is, at (or near) the temperature at which a phase transition occurs. A well-known example of a phase transition occurs when water freezes (i.e., it changes state from liquid to solid) or when it boils (i.e., it changes state from solid to gas). In more elaborate models, the continuous physical system is well-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 development occurred in 2000 when Oded Schramm introduced the stochastic Loewner evolution (SLE) while studying scaling limits of 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 Medal to Wendelin Werner "for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.'' 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 Paul 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. Therefore, the primary goal of this research program is to study multiple SLE curves and their interactions which is the natural continuous object to study when considering multiple interfaces.