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

数学物理界对二维统计力学临界模型的长期兴趣最近由于来自概率的两个主要思想而得到了强烈的复兴：用Schramm-Lowner-演化(SLE)描述随机界面，以及用离散复分析来研究大尺度性质的共形不变性。这项研究项目涉及统计力学理论的这一方面，即二维共性，但具有独立的起源和不同的具体目标。它使用了一种自然而有效的数学方法来研究关联的临界指数：在严格的数学重新表述中，物理学家的重整化群(RG)。在这个项目中有两种主要感兴趣的系统：a)库仑系统；b)自旋、二聚体和顶点模型。经过多年的研究，似乎除了极少数精确可解的例外情况外，不可能对这些模型的临界指数进行严格的计算。PI最近独立地和合作地在RG技术方面取得了重大进展，这表明在这两种情况下的临界指数实际上都是可以获得的。该项目旨在完成和扩大这些成果；并为该领域的重新开花开辟了道路。物理学家们花了相当大的精力来检验统计力学对真实材料的预测。最近在太空(航天飞机太空任务、和平号、国际空间站)进行的实验表明，收集的数据与二级相变临界指数的理论计算之间存在惊人的一致性。然而，这一理论几乎完全缺乏数学上的严谨性。国际和平研究所的研究活动，已经完成或计划在这个项目中，代表了一个具体的刺激，重新开始宏伟的努力，以填补这一空白。更广泛的影响：将实验物理与数学相结合是令人着迷的，在智力上也是有益的。PI将通过指导硕士研究生从事相关的教育或研究工作，重点是比较研究二聚体和伊辛模型的准确解的不同方法，从而使他们参与到这样的活动中来。在PI的经历中，这门学科总是给学生们留下深刻的印象，因为在这些案例中，线性代数和基本组合数学的知识，加上足够的耐心，足以走到研究的前沿！PI在一所大型本科院校任教，拉美裔入学率高，有大量家庭中第一个上大学的学生。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：在最理想的情况下，将产生与主要研究项目相关的原创结果；在不太幸运的情况下，学生可能仍然会培养热情，在数学和物理科学方面拥有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.