Ising models, random currents, percolation and random walks. Part I

湘潭大学数学与计算科学学院      向开南   

贴于2021年元旦下午3: 50

      附件是我花了大量精力写的一个131页科研笔记的压缩版（60页），它的姊妹篇会后续推出。写此综述的目的是展示相变和临界现象中产生的迷人概率论以及鼓励和吸引中国的有志概率青年投身于此类型概率论的研究。实际上，SLE理论的中心问题（临界2维离散统计物理模型的共形不变尺度极限问题）属于临界现象的概率论。由于自己的能力不足和无知，此综述有其自身局限性。它的一个价值是对大量文献的收集和整理以及散布在各节的各种Level的问题。

       The science of phase transitions is a source of fascinating mathematical problems and important mathematical discoveries of real physical significance, and the most difficult and interesting part of the phase transitions is the critical phenomena (behaviors). Borrowing ideas/techniques from probability theory and other mathematical fields, statistical physics and quantum field theory, the mathematical theory of phase transitions and critical phenomena has been undergoing intense developments. This theory is inherently probabilistic since the systems are described by the ensemble of random configurations.

       To a large extent, the theory of critical phenomena is now guided to a highly profound insight, called universality, from physics. At present, universality is not so much a mathematical theorem as a philosophy. The universality means that many essential features of the transition at a critical point depend on relatively few attributes of the considered system; and as predicted by theoretical physics, large-scale features near the critical point are universal. Note simple mathematical models (even if they greatly simplify the local interactions of the real systems) can capture some qualitative and quantitative features of critical behavior in real physical systems, which has helped to focus attention on particular mathematical models among both physicists and mathematicians.

       Bernoulli percolation is a key mathematical model for the study of phase transitions and critical phenomena. The Ising model, a fundamental model of the ferromagnetic phase transition, is unrelated to percolation at first sight, but actually they are closely connected within the framework of random-cluster models; and the random-cluster model is a unification of percolation, Ising and Potts models and an extrapolation of electrical networks. For physics, the Ising model is more important than Bernoulli percolation. Note the Ising model can be viewed naturally as a site percolation model with dependence between the sites. Additionally, studies of these two models are closely similar, and techniques arising from studying one model can usually (but not always) be effective to study another.

      The aim of this review is to show fascinating and important mathematics arising from phase transitions and critical phenomena through percolation and Ising models and related topics. The review is divided into two separate parts: Part I and Part II. Part I focuses on the Ising models and related topics. Random current is a powerful tool to study the ferromagnetic Ising models; and when using it to study the nearest-neighbour ferromagnetic Ising models onone often compares some of its behaviors with those of the d-dimensional simple random walk. Part II focuses on percolation (Bernoulli percolation, long-range percolation and first-passage percolation), random walks and related topics. The two parts are independent but intertwined. There are many big or famous problems scattering in the two parts. And the two parts have their own shortcomings due to the author's ignorance and academic limitations.

     具体的内容见附件：Ising-Current.pdf

一个麻烦的问题是只有科学网的博客用户在登录博客系统后才能下载此附件。因此，若你对它感兴趣，请写邮件给我：kainan.xiang@xtu.edu.cn

     用赖内·马利亚·里尔克的诗《秋日》结束此博文：

       主啊！是时候了。夏日曾经很盛大。

       把你的阴影落在日规上，

       让秋风刮过田野。

       让最后的果实长得丰满，

       再给它们两天南方的气候，

       迫使它们成熟，

       把最后的甘甜酿入浓酒。

       谁这时没有房屋，就不必建筑，

       谁这时孤独，就永远孤独，

       就醒着，读着，写着长信，

       在林荫道上来回

       不安地游荡，当着落叶纷飞。