Percolation and Magnetization for Generalized Continuous Spin Models

TL;DR

This paper extends the percolation framework to a broad class of generalized continuous spin models, showing their thermal behavior aligns with the Ising universality class and deriving corresponding percolation formulations.

Contribution

It introduces generalized continuous spin models with varied interactions and derives their percolation representations, broadening the understanding of critical phenomena beyond traditional Ising models.

Findings

Models exhibit Ising universality class behavior.

Percolation formulations extend Fortuin-Kasteleyn clusters.

Results illustrate recent rigorous theoretical findings.

Abstract

For the Ising model, the spin magnetization transition is equivalent to the percolation transition of Fortuin-Kasteleyn clusters; this result remains valid also for the conventional continuous spin Ising model. The investigation of more general continuous spin models may help to obtain a percolation formulation for the critical behaviour in SU(2) gauge theory. We therefore study a broad class of theories, introducing spin distribution functions, longer range interactions and self-interaction terms. The thermal behaviour of each model turns out to be in the Ising universality class. The corresponding percolation formulations are then obtained by extending the Fortuin-Kasteleyn cluster definition; in several cases they illustrate recent rigorous results.