Random currents expansion of the Ising model

TL;DR

This paper reviews the random currents expansion approach for analyzing the Ising model, a key method that has deepened understanding of its critical behavior through interdisciplinary mathematical techniques.

Contribution

It summarizes the development and application of the random currents method, highlighting its role in advancing the mathematical understanding of the Ising model.

Findings

Deepened understanding of the Ising model's critical behavior.

Unified various mathematical techniques in analyzing phase transitions.

Provided a comprehensive review of the random currents approach.

Abstract

Critical behavior at an order/disorder phase transition has been a central object of interest in statistical physics. In the past century, techniques borrowed from many different fields of mathematics (Algebra, Combinatorics, Probability, Complex Analysis, Spectral Theory, etc) have contributed to a more and more elaborate description of the possible critical behaviors for a large variety of models. The Ising model is maybe one of the most striking success of this cross-fertilization, for this model of ferromagnetism is now very well understood both physically and mathematically. In this article, we review an approach, initiated in \cite{GriHurShe70,Aiz82} and based on the notion of random currents, enabling a deep study of the model.