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

统计力学的主要目标之一是理解物理系统在临界状态下的行为，即在（或接近）相变发生的温度下。 当水结冰时发生相变的一个众所周知的例子（即，它从液体变为固体）或当它沸腾时（即，它从固态变为气态）。在更复杂的模型中，连续的物理系统可以用离散的或格点的模型来描述。这些格子模型在数学上通常更容易处理，并且通常可以使用为格子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。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，当Oded Schramm介绍了随机Loewner演化（SLE），同时研究循环擦除随机行走的标度极限。 SLE是一类新的共形不变随机过程，它的出现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，它的重要性是由颁发菲尔兹奖给温德林沃纳打断“他的贡献发展的随机Loewner演变，几何的二维布朗运动，共形场理论。虽然这项研究计划已经产生了一些非常重要的结果，但还有很多工作要做。例如，自避免随机行走，一种由诺贝尔奖获得者化学家保罗·弗洛里在20世纪40年代引入的聚合物链模型，是一种数学进步最小的模型，也是整个研究计划的动机之一。另一个重要的研究领域，尚未得到太多的关注是自然发生的多个界面的情况下，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个SLE曲线及其相互作用，这是考虑多个界面时自然连续的研究对象。

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.