RUI: Critical Models of Two Dimensional Statistical Mechanics. A Mathematical Approach

数学物理界长期以来对二维统计力学临界模型的兴趣最近有了强烈的复兴，这是由于来自概率的两个主要思想：根据Schramm-Lowner-Evolution （SLE）描述随机界面，以及通过离散复分析研究大尺度性质的保形不变性。本研究项目涉及统计力学理论的这一方面，即二维中的普遍性，但有一个独立的起源和不同的具体目标。它使用一种自然而有效的数学方法来研究相关性的关键指数：物理学家在严格的数学重新表述中的重整化群（RG）。本项目主要关注两种类型的系统：a)库仑系统；B)自旋、二聚体和顶点模型。经过多年的研究，除了极少数可以精确解决的例外，似乎不可能对这些模型的关键指数进行严格的计算。PI最近在独立和合作的情况下，在RG技术方面取得了重大进展，这表明在这两种情况下，实际上都可以获得临界指数。本项目旨在完成和扩展这些成果；为田野的重新开放开辟了道路。物理学家已经付出了相当大的努力在实际材料上检验统计力学的预测。最近在太空中进行的实验（航天飞机、和平号、国际空间站）表明，收集到的数据与二阶相变临界指数的理论计算之间存在惊人的一致性。然而，这个理论几乎完全缺乏数学上的严谨性。PI的研究活动，已经完成或计划在这个项目中，代表了一个具体的刺激，重新开始填补这一空白的宏伟努力。更广泛的影响：调和实验物理与数学是迷人的和智力回报。PI将通过指导硕士研究生进行相关的教育或研究工作，重点是对二聚体模型和Ising模型的不同精确解方法进行比较研究。在PI的经验中，这个主题总是让学生们感到震惊，因为它是那些幸运的案例之一，其中线性代数和基本组合学的知识，伴随着足够的耐心，足以到达研究的前沿！PI在一所西班牙裔入学率很高的大型本科院校任教，很多学生都是家里第一个上大学的人。学生培训的目标是培养有才华的学生继续攻读博士学位的自信心：在最佳情况下，将产生与主要研究项目相关的原创成果；在不那么幸运的情况下，学生们可能仍然会在数学和物理科学方面培养出热情和有限但重要的专业知识。

The long-standing interest of the Mathematical Physics community in critical models of two-dimensional Statistical Mechanics has recently had an intense revival due to two major ideas coming from Probability: the description of random interfaces in terms of the Schramm-Lowner-Evolution (SLE), and the study of the conformal invariance of the large scale properties by means of the discrete complex analysis. This research project deals with that aspect of the theory of Statistical Mechanics, the universality in dimension two, but has an independent origin and a different specific objective. It uses a mathematical approach that is natural and effective for studying the critical exponents of correlations: the physicists' Renormalization Group (RG) in a rigorous mathematical reformulation. There are two types of systems of major interest in this project: a) Coulomb systems; b) spin, dimer and vertex models. After many years of studies, it seemed that, with very few exactly solvable exceptions, it was not possible to arrive at a rigorous computation of critical exponents of these models. The PI has recently achieved, independently and in collaborations, significant progress in the RG technique, which has shown that critical exponents in both cases are in fact accessible. This project aims for completions and extensions of such results; and opens the way to a re-blossoming of the field. Physicists have devoted considerable efforts to test the predictions of Statistical Mechanics on real materials. Recent experiments, performed in space (Shuttle space missions, MIR, ISS), showed a spectacular agreement between collected data and theoretical computations of critical exponents of second-order phase transitions. Yet the theory lacks, almost completely, mathematical rigor. The PI's research activity, already accomplished or planned in this project, represents a concrete stimulation for re-starting the grand endeavor of filling this gap. Broader Impact: Reconciling experimental physics with mathematics is fascinating and intellectually rewarding. The PI will engage masters degree students in such an activity by mentoring them in related educational or research works that are focused on comparative studies of different methods of exact solution of dimer and Ising models. In the experience of the PI this subject always strikes the students as being one of those fortunate cases in which the knowledge of linear algebra and rudimentary combinatorics, accompanied by enough patience, suffices to arrive at the forefront of the research! The PI teaches in a large undergraduate institution with high Hispanic enrollment and a large number of students who are first in their families to go to college. The student training has the ambition of building in talented students the self-confidence for continuing their studies in a PhD program: in the optimal case, original results in connection with the main research project will be produced; in a less fortunate case, students may still develop enthusiasm and a limited, but significant, expertise in mathematical and physical sciences.