# Geometric properties of the complete-graph Ising model in the loop representation

## TL;DR

This paper investigates the geometric properties of the complete-graph Ising model in the loop representation, providing exact results and insights into complex phenomena through numerical and theoretical analysis.

## Contribution

It introduces a detailed study of the loop representation of the CG-Ising model, revealing new geometric insights and connecting them with the random-cluster representation.

## Key findings

- Exact volume fractal dimensions derived
- Identification of two configuration sectors and length scales
- Demonstration of the loop representation's intuitive power

## Abstract

The exact solution of the Ising model on the complete graph (CG) provides an important, though mean-field, insight for the theory of continuous phase transitions. Besides the original spin, the Ising model can be formulated in the Fortuin-Kasteleyn random-cluster and the loop representation, in which many geometric quantities have no correspondence in the spin representations. Using a lifted-worm irreversible algorithm, we study the CG-Ising model in the loop representation, and, based on theoretical and numerical analyses, obtain a number of exact results including volume fractal dimensions and scaling forms. Moreover, by combining with the Loop-Cluster algorithm, we demonstrate how the loop representation can provide an intuitive understanding to the recently observed rich geometric phenomena in the random-cluster representation, including the emergence of two configuration sectors, two length scales and two scaling windows.