Critical planar systems and conformal invariance

在过去的十年里，共形不变系统的研究取得了一系列重大成就，特别是在Schramm-Loewner演化的引入之后。这一新方法在描述渗流和伊辛模型等二维统计物理临界模型的界面标度极限时，与共形场理论(CFT)的早期预测一致，被证明是非常有效的。本文提出的研究方案主要集中在路径和回路上的系统，特别是与CFT形式有关的系统；自由场不变性原理和柯西-黎曼算符的分析；以及连续标度极限的几何表示和泛函表示之间的联系。相变描述了物理系统在外部参数变化下的急剧质变，如水的冻结(随着温度的降低从液体转变为固体)。在相变时，可能会出现一个随机的宏观几何形状，类似于分离不混合流体(如油和水)的摆动界面，或磁性材料中的正负电荷区域。对这些波动界面的研究是以概率论和理论物理为基础的。将特别强调这两种方法之间的相互作用，以及它们与数学其他领域的关系。

The last decade has seen a series of major achievements in the study of conformally invariant systems, in particular following the introduction of Schramm-Loewner Evolutions. This new approach has proved extremely effective in describing the scaling limit of interfaces of critical models of two-dimensional Statistical Physics, such as percolation and the Ising model, in accordance with the earlier predictions of Conformal Field Theory (CFT). The research program proposed here focuses on systems on paths and loops, in particular in relation with the CFT formalism; on free field invariance principles and the analysis of Cauchy-Riemann operators; and on the connections between geometric and functional representations of continuous scaling limits.The goal of this proposal is to analyze mathematical models of the physical phenomenon of phase transition. A phase transition describes a sharp qualitative change in a physical system under variation of an external parameter, such as freezing of water (transition from liquid to solid phase as temperature decreases). At the phase transition, a random macroscopic geometry may emerge, akin to the wiggly interfaces separating non mixing fluids like oil and water, or positively and negatively charged regions in a magnetic material. The study of those fluctuating interfaces is grounded in both Probability Theory and Theoretical Physics. Special emphasis will be put on the interplay between these two approaches, and their relation to other areas of mathematics.