The Ising model: Highlights and perspectives

TL;DR

This survey synthesizes how the Ising framework explains ferromagnetic ordering and phase transitions across finite and infinite volumes, highlighting the roles of boundary conditions, contour methods, and exact solutions. It surveys critical phenomena, universality, and renormalization group perspectives, with rigorous 2D results linked to SLE and lattice-scale interfaces, alongside RG-driven insights and challenges. The discussion extends to disordered systems, detailing random-field and spin-glass scenarios, metastates, and mean-field theories with Parisi-type formulas, illustrating a broad landscape of methods and open questions. Collectively, the notes emphasize the interplay between rigorous probability, mathematical physics, and statistical mechanics in understanding critical behavior, scaling limits, and disorder effects in the Ising setting.

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.