Critical planar systems and conformal invariance

在过去的十年中，在共形不变系统的研究中取得了一系列重大成就，特别是在引入Schramm-Loewner进化之后。根据早期共形场论（CFT）的预测，这种新方法在描述二维统计物理的关键模型（如渗透和Ising模型）的界面缩放极限方面被证明是非常有效的。这里提出的研究计划侧重于路径和循环系统，特别是与CFT形式主义的关系；自由场不变性原理及Cauchy-Riemann算子的分析以及连续尺度极限的几何表示和函数表示之间的联系。本文的目标是分析相变物理现象的数学模型。相变描述了物理系统在外部参数变化下的急剧质的变化，例如水的冻结（随着温度的降低从液态转变为固相）。在相变过程中，可能会出现随机的宏观几何形状，类似于分离油和水等非混合流体的弯曲界面，或者磁性材料中带正电荷和负电荷的区域。这些波动界面的研究是基于概率论和理论物理。特别强调的是这两种方法之间的相互作用，以及它们与其他数学领域的关系。

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.