Large-Scale Phenomena in Models of Statistical Mechanics

本提案的目的是研究一类介于概率论和数学物理之间的问题。总共提出了十个具体项目，可分为四类。第一类处理一阶相变问题。这里的具体研究对象包括通过与平均场理论的比较证明相变，稀释对反铁磁体对称性破缺的影响，以及连续自旋系统中伴随相变的各种效应。第二类关注的是液滴和界面现象。相应的项目包括研究溶剂-溶质系统的松弛平衡和相共存系统的介观压力概念。第三类项目涉及二维临界模型的保形不变性问题。这里将研究所谓的SLE边界的分形性质，并分析随机分形树模型。最后一类涉及小世界现象的问题。这里感兴趣的具体问题是基于欧几里得的随机图的临界模型中图距离的增长。在更广泛的层面上，该提案希望解决一系列问题，这些问题都源于物理、化学、工程和/或社会科学。将特别注意相变现象，在相变现象中，系统的总体特征经历了突然的，通常是相当剧烈的变化，而外部条件通过特定的过渡值平滑地变化。这种转变的例子在物理科学（例如，物理化学中的冻结或融化）和工程（例如，互联网流量的阻塞）中比比皆是；但它们在社会科学领域也有天然的对应物（例如，社会网络中的观点传播）。这个建议的大部分都花在研究这些类型的非常具体的数学问题上。例如，其中一个提议的项目涉及过渡金属化合物的磁性外观，另一个提供了对海冰中盐袋形成的理论方面的一些启发，还有一个项目致力于在连接（通信）网络中的两个个体（或计算机服务器）的“分离程度”。这些问题的共同点是需要适当的数学工具来成功地解决它们。本提案的目标之一是开发这样的工具，并向整个科学界传播它们的主要思想。

The aim of this proposal is to investigate a class of problems on the borderline between probability and mathematical physics. Altogether ten specific projects are proposed which can be grouped into four categories. The first category deals with problems of first-order phase transitions. Here the specific objects of study include proofs of phase transitions by comparison to mean-field theory, influence of dilution on symmetry breaking in antiferromagnets, and various effects accompanying phase transitions in systems with continuous spins. The second category is focused on droplet and interface phenomena. The corresponding projects include a study of relaxation to equilibrium in solvent-solute systems and the mesoscopic concept of pressure in systems at phase coexistence. The third category of projects deals with problems related to conformal invariance in 2D critical models. Here the fractal properties of the so-called SLE boundaries will be investigated and a model of random fractal trees will be analyzed. The final category involves problems of small-world phenomena. The specific problem of interest here is the growth of the graph distance in a critical model of Euclidean-based random graphs.On a broader level, the proposal hopes to address a series of questions that all have their origin in physics, chemistry, engineering and/or social sciences. A particular attention will be paid to the phenomena of phase transitions in which the overall character of the system undergoes a sudden, and often rather drastic, change while the external conditions vary smoothly through a particular, transitional, value. Examples of such transitions are ample in physical sciences (e.g., freezing or melting in physical chemistry) and engineering (e.g., jams in internet traffic); but they also have their natural counterparts in social sciences (e.g., opinion spreading in social networks). Much of this proposal is spent on studying very specific mathematical problems of these types. For instance, one of the proposed projects deals with the appearance of magnetism in transition-metal compounds, another offers to shed some light on the theoretical aspects of brine-pocket formation in the sea ice, yet another project is devoted to the "degree of separation" of two individuals (or computer servers) in a thinly connected (communication) network. The common ground of several of these problems is the need of proper mathematical tools for their successful resolution. One of the goals of the present proposal is to develop such tools and disseminate their main ideas to the scientific community at large.