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

RUI：二维统计力学的关键模型。

• 批准号: 1306571
• 负责人: Pierluigi Falco
• 金额: $ 10.17万
• 依托单位: The University Corporation, Northridge
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1306571&HistoricalAwards=false
• 关键词: RUI; Critical; Models; Two; Dimensional

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.

数学物理界对二维统计力学临界模型的长期兴趣最近得到了强烈的复兴，这是由于来自概率论的两个主要思想：用Schramm-Lowner-Evolution（SLE）描述随机界面，以及用离散复分析研究大尺度性质的共形不变性。本研究项目涉及统计力学理论的这一方面，即第二维的普遍性，但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相关性的临界指数：物理学家的重整化群（RG）在严格的数学重新表述。在这个项目中主要有两种类型的系统：a）库仑系统; B）自旋，二聚体和顶点模型。经过多年的研究，似乎除了极少数可解的例外，不可能对这些模型的临界指数进行严格的计算。PI最近在RG技术方面取得了独立和合作的重大进展，这表明这两种情况下的关键指数实际上都是可访问的。该项目旨在完成和扩展这些成果;并为该领域的重新繁荣开辟道路。物理学家们投入了相当大的努力来检验统计力学在真实的材料上的预测。最近在空间进行的实验（航天飞机航天飞行任务、和平号、国际空间站）表明，收集的数据与二级相变临界指数的理论计算之间惊人的一致性。然而，这个理论几乎完全缺乏数学上的严密性。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奋进。更广泛的影响：将实验物理学与数学结合起来是迷人的，在智力上是有益的。PI将通过指导他们从事相关的教育或研究工作，专注于二聚体和伊辛模型的精确解的不同方法的比较研究，让硕士学位的学生参与这样的活动。在经验的PI这一主题总是罢工的学生作为一个幸运的情况下，知识的线性代数和基本的组合，伴随着足够的耐心，足以到达前沿的研究！PI在一个大型本科院校任教，西班牙裔入学率很高，有大量的学生是家里第一个上大学的。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有关的原创成果;在不太幸运的情况下，学生仍然可以培养热情和有限的，但重要的数学和物理科学专业知识。