Percolation and Magnetization in the Continuous Spin Ising Model

TL;DR

This paper explores the connection between percolation and magnetization in the continuous spin Ising model, deriving a cluster representation and demonstrating the equivalence of magnetic and percolation transitions through theoretical analysis and simulations.

Contribution

It introduces a Fortuin-Kasteleyn cluster representation for the continuous spin Ising model and establishes the equivalence of magnetic and percolation transitions.

Findings

Percolation transition coincides with magnetic transition in the model.

Cluster distribution properties are characterized through the Fortuin-Kasteleyn transformation.

Numerical simulations support the theoretical results.

Abstract

In the strong coupling limit the partition function of SU(2) gauge theory can be reduced to that of the continuous spin Ising model with nearest neighbour pair-interactions. The random cluster representation of the continuous spin Ising model in two dimensions is derived through a Fortuin-Kasteleyn transformation, and the properties of the corresponding cluster distribution are analyzed. It is shown that for this model, the magnetic transition is equivalent to the percolation transition of Fortuin-Kasteleyn clusters, using local bond weights. These results are also illustrated by means of numerical simulations.