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已有 12365 次阅读 2021-1-1 15:50 |个人分类:专题综述|系统分类:科研笔记

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



       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.

















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