The Ising model: highlights and perspectives

IF 1.1 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2025-08-02 DOI:10.1007/s11040-025-09515-1

Christof Külske

引用次数: 0

Abstract

We give a short non-technical introduction to the Ising model, and review some successes as well as challenges which have emerged from its study in probability and mathematical physics. This includes the infinite-volume theory of phase transitions, and ideas like scaling, renormalization group, universality, SLE, and random symmetry breaking in disordered systems and networks. This note is based on a talk given on 15 August 2024, as part of the Ising lecture during the 11th Bernoulli-IMS world congress, Bochum.

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伊辛模型：亮点和视角

我们对伊辛模型进行了简短的非技术介绍，并回顾了在概率和数学物理研究中出现的一些成功和挑战。这包括相变的无限体积理论，以及无序系统和网络中的缩放、重整化群、普适性、SLE和随机对称性破缺等思想。这篇笔记是基于2024年8月15日在波鸿举行的第11届伯努利- ims世界大会上的一次演讲。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.