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

离散平面过程的 Schramm-Loewner 演化和缩放限制

• 批准号: 312354-2008
• 负责人: Kozdron, MichaelJohn
• 金额: $ 1.02万
• 依托单位: University of Regina
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2009
• 资助国家: 加拿大
• 起止时间: 2009-01-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=428158
• 关键词: Schramm; Loewner; evolution; scaling; limits

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.

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；也就是说，在（或接近）发生相变的温度下。相变的一个众所周知的例子发生在水冻结时（即，它从液体状态变为固体）或沸腾时（即，它从固体状态变为气体）。在更精细的模型中，连续的物理系统可以用离散的或点阵的模型很好地描述。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。2000年，Oded Schramm在研究环擦除随机游走的尺度限制时引入了随机下限进化（SLE），这是一个令人兴奋的发展。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，由于温德林·维尔纳在发展随机洛厄纳演化、二维布朗运动几何和共形场论方面的贡献，菲尔兹奖进一步强调了它的重要性。尽管这个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，自我避免随机游走，这是一个聚合物链模型，由诺贝尔奖得主化学家保罗·弗洛里在20世纪40年代提出，是一个数学上进展最小的模型，也是整个研究项目的动力之一。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，本研究项目的首要目标是研究多个SLE曲线及其相互作用，这是考虑多个界面时自然的连续研究对象。