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

临界点的二维晶格模型和 Schramm-Loewner 演化

• 批准号: 312354-2013
• 负责人: Kozdron, MichaelJohn
• 金额: $ 1.09万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571727
• 关键词: Two; dimensional; lattice; models; criticality

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.

统计力学的目标之一是了解物理系统在临界状态下的行为;也就是说，在相变发生的温度下。例如，当水在0摄氏度结冰时发生相变（即，它从液体变为固体）或当它在100摄氏度沸腾时（即，它将状态从液体变为气体）。在更复杂的情况下，连续的物理系统通常由离散或格模型描述。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。1999年出现了一个令人兴奋的发展，已故的O. Schramm在研究环擦除随机游动时引入了随机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，2006年菲尔兹奖授予W。维尔纳“他的贡献，发展随机Loewner演化，二维布朗运动的几何，共形场理论”，并在2010年到S。Smirnov“证明了渗流和平面伊辛模型中的共形不变性。“虽然这项研究计划已经产生了一些非常重要的成果，但还有更多的工作要做。例如，自避免随机行走，一种由诺贝尔奖获得者化学家P. Flory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，这项研究计划的主要目标是帮助促进SLE和共形场理论之间的相互作用更好地理解。