Conformal invariance and the renormalization group in some critical systems

物理学中的许多模型在非常小的尺度(通常是一个原子的大小)上都是固有的离散的，在这种离散的水平上存在固有的随机性。在宏观层面上，随机性通常是看不到的。但在一定条件下，这种随机性在宏观层面表现出来。这被称为一种临界现象。例如，磁性材料中的自旋在其方向上具有随机性。在一定的温度下，这些自旋可以相互对准，从而产生宏观磁区，这在硬盘驱动器等技术应用中起着至关重要的作用。在宏观尺度上看到的随机性通常不取决于微观随机性的细节。在物理学中，这被称为普遍性，并已发展成为理解临界现象的一个关键概念。重整化群是一套来自物理学的方法，已经成为现代理解临界现象和量子场论--基本粒子理论--的基础。尽管重整化群在物理学上取得了巨大的成功，但我们对这一套思想在数学上并没有深入的理解。本研究将进一步加深对这套思想和方法的数学发展和理解，并利用它们来理解模型中临界现象的数学，如自回避随机行走和磁自旋的伊辛模型。大部分研究将致力于随机行走模型和伊辛型模型。智能动态行走(也称为拉普拉斯-b随机行走的一种极限情况)是一种用于自我回避行走的动态模型。在六方晶格上，它与渗流密切相关。渗流方法被用来证明它的标度极限是参数值为6的Schramm-Loewner演化。本研究将在其他晶格上研究这种标度极限，并推广跃迁几率。我们的目标是了解这一比例限制的普遍性。对于普通的随机游动，区域的出口分布的标度极限是调和测度。研究的另一个目标是了解普通随机行走和智能运动行走对这种收敛的一阶修正。本研究还将研究伊辛型模型的实空间重正化群，特别是一维和二维的精确重正化群变换。这些变换有一个潜在的优势，即映射将作用于有限维空间。其目的是证明映射可以被严格定义，然后利用有限维的性质来证明不动点的存在性以及由此存在的所有令人兴奋的结果。最后，将Schramm-Loewner演化作为重整化群不动点进行研究。众所周知，这种随机过程是许多关键模型的比例极限。这里的目标是定义一个将这个过程作为固定点的重整化群映射，然后使用这个映射来理解接近临界点的模型。

Many models in physics are inherently discrete at the very smallest scale (typically the size of an atom) and there is inherent randomness at this discrete level. At the macroscopic level the randomness is typically not seen. But under certain conditions this randomness manifests itself at the macroscopic scale. This is known as a critical phenomena. For example, the spins in a magnetic material have randomness in their orientation. At a certain temperature these spins can align with each other to produce macroscopic magnetic domains that play a crucial role in technological applications such as hard drives. The randomness seen at the macroscopic scale often does not depend on the details of the microscopic randomness. In physics this is called universality and has developed into a key idea in the understanding of critical phenomena. The renormalization group is set of methods from physics that has become the basis for the modern understanding of both critical phenomena and of quantum field theory - the theory of elementary particles. Despite the tremendous success of the renormalization group in physics, we do not have a deep mathematical understanding of this set of ideas. This research will further the mathematical development and understanding of this set of ideas and methods and use them to understand the mathematics of critical phenomena in models such as self-avoiding random walks and Ising models of magnetic spins.Most of the research will be devoted to random walk models and Ising-type models. The smart kinetic walk (also known as a limiting case of the Laplacian-b random walk) is a dynamic model for self-avoiding walks. On the hexagonal lattice it is closely related to percolation. Percolation methods have been used to prove its scaling limit is the Schramm-Loewner evolution with parameter value 6. The research will study this scaling limit on other lattices and for generalizations of the transition probabilities. The goal is to understand the universality of this scaling limit. For the ordinary random walk the scaling limit of the exit distribution for a domain is harmonic measure. Another goal of the research is to understand the first order correction for this convergence for both the ordinary random walk and the smart kinetic walk. The research will also study real space renormalization groups for Ising type models, in particular exact renormalization group transformations in one and two dimensions. These transformation have the potential advantage that the map would act on a finite dimensional space. The goal is to prove that the map can be rigorously defined and then take advantage of the finite dimensional nature to prove existence of a fixed point and all the exciting consequences that follow from this existence. Finally the research will study Schramm-Loewner evolution as a renormalization group fixed point. This stochastic process is known to be the scaling limit of many critical models. The goal here is to define a renormalization group map that has this process as a fixed point and then use this map to understand models that are near criticality.