Algebraic methods for lattice models of statistical physics

物理学通过原子或分子结构来理解物质。这种描述在预测物理性质和发现新材料方面取得了压倒性的成功。它特别适合于原子和分子数量有限（虽然不是太多）的系统。只有当大量原子或分子相互作用时才会出现的现象需要其他技术。相变就是这样一种现象：众所周知，除非相互作用的物体的数量趋于无穷大，否则不会发生相变。该计划旨在理解从有限系统（例如，具有有限数量原子的系统）到它们的连续体极限（从一开始就假设有无限数量的相互作用物体的理论）的过渡。特别是，它想要识别有限系统的性质，这些性质揭示了通过增加物体数量而获得的无限系统的性质。

Physics understands matter through its atomic or molecular structure. This description has been overwhelmingly successful to predict physical properties and discover new materials. It is particularly well-suited to systems with a finite, though not too large, number of atoms and molecules. Phenomena occurring only when a large number of atoms or molecules interact call for other techniques. Phase transition is one such phenomenon: it is known that no phase transitions may occur unless the number of interacting objects tends to infinity. The proposed project aims at understanding the passage from finite systems (say, those with a finite number of atoms) to their continuum limit (the theories that assume from the start an infinite number of interacting objects). In particular it wants to identify the properties of finite systems that reveal those of the infinite ones obtained by increasing the number of objects. The main mathematical tool of the project will be algebra. Several chapters of mathematics have played a role in the description of phase transitions. Analysis and algebra are probably the central ones. Algebra provides the tools to identify the symmetries of physical systems. Symmetries are operations that transforms a system into another one without changing its overall physical properties. These symmetries offer fundamental ways to approach physical systems and define them mathematically. The proposed project puts an emphasis on studying the algebraic structures arising in both the finite lattice models and their infinite continuum limits. The research will focus on two-dimensional lattice models of microscopic interactions. These models are known as percolation, the Ising model, the XXZ spin chain, dense and dilute loop models, etc. They offer a natural laboratory to probe physical properties and prove them rigorously. They have a finite number of “particles”, they can be probed on the computer and they are believed to go to (logarithmic) conformal field theories (a distinguished set of well-studied continuum models). Most importantly they rest upon an algebraic description that lends itself naturally to the study of the limit to large number of particles. Previous works in these directions, others' and mine, have contributed to both physics and mathematics. For example it is useful in physics to recognize emerging properties of a finite system, even though there are only partially realized. In mathematics, the study of algebraic structures of physical systems has suggested many new avenues of development or new ways of looking at existing results.