Discrete models and conformally invariant limits

在统计力学和概率论中，由大量微观元素（如分子）组成的系统，它们各自受到随机性或噪声的影响，并相互作用，发挥着突出的作用。理解简单的微观相互作用规则如何在宏观尺度上产生复杂的随机特征是一个基本问题，这些特征可以通过连续随机结构来建模。这个问题的一般领域与许多重要的物理现象有关，如孔隙度和铁磁性。该项目的重点是二维系统的情况下，在过去的15年里已经看到了壮观的和持续的进展，涉及各种各样的数学概念和技术，沿着数学和理论物理之间的和解。PI将继续参与学生和初级研究人员的数学和跨学科培训活动，特别是与物理和数学接口的研究项目有关。该项目中描述的研究计划主要集中在统计物理学中的二维临界模型，特别强调共形不变性以及场公式与其缩放限制的几何方面之间的相互作用。Schramm-Loewner演化的引入对共形不变随机系统的研究产生了深远的影响。这类新的随机连续平面曲线在描述统计物理临界模型（如渗流和伊辛模型）界面的标度极限方面被证明是非常有效的。PI的研究项目有三个主要重点。首先是研究系统的SLE型路径和回路，特别是在与共形场论形式主义。第二个问题的二聚体模型和自由场描述其标度极限方面，建立在分析的柯西-黎曼算子的家庭。第三个焦点涉及SLE路径和环境高斯场之间的精确关系。PI的期望是，这些研究将有助于更深入地了解关键模型以及组合学，概率，分析，复杂几何和表示理论中相关概念和工具的相互作用。

In statistical mechanics and probability theory, systems consisting of a large number of microscopic elements - such as molecules - individually subject to randomness or noise, and interacting with each other, play a prominent role. It is a fundamental problem to understand how simple, microscopic interaction rules produce intricate random features on a macroscopic scale, features that can be modeled by continuous stochastic structures. This general area of problems is relevant to many important physical phenomena, such as porosity and ferromagnetism. The project's emphasis is on the case of two-dimensional systems, where the last fifteen years have seen spectacular and ongoing progress, involving a wide variety of mathematical concepts and techniques, along with a rapprochement between mathematical and theoretical physics. The PI will continue to engage in mathematical and interdisciplinary training activities of students and junior researchers, in particular in relation with his research projects at the interface of Physics and Mathematics. The program of research described in this project is mainly focused on two-dimensional critical models in statistical physics, with special emphasis on conformal invariance and the interplay between the field formulation and the geometric aspects of their scaling limits. The introduction of Schramm-Loewner evolutions (SLE) has deeply influenced the study of conformally invariant random systems. This new class of stochastic continuous planar curves has proved extremely effective in describing the scaling limit of interfaces of critical models of statistical physics, such as percolation and the Ising model. The research projects of the PI have three main focuses. The first is the study of systems of SLE-type paths and loops, in particular in relation with the Conformal Field Theory formalism. The second concerns the dimer model and aspects of the free field description of its scaling limit, building on the analysis of families of Cauchy-Riemann operators. The third focus involves the exact relations between SLE paths and an ambient Gaussian field. The expectation of the PI is that these studies will help provide a deeper understanding of critical models and the interplay of the relevant concepts and tools in combinatorics, probability, analysis, complex geometry and representation theory.