Cluster Distribution in Mean-Field Percolation: Scaling and Universality

TL;DR

This paper derives a closed-form cluster distribution near the percolation transition in the Potts model, revealing scaling and universality, and compares predictions with simulations showing generally good agreement.

Contribution

It provides a new analytical expression for cluster distribution in the Potts model near criticality, highlighting scaling and universality features.

Findings

Good agreement between theory and simulations

Scaling behavior and emergence of spanning clusters

Slow convergence in large systems

Abstract

The partition function of the finite $1+\epsilon$ state Potts model is shown to yield a closed form for the distribution of clusters in the immediate vicinity of the percolation transition. Various important properties of the transition are manifest, including scaling behavior and the emergence of the spanning cluster. The predictions are compared with simulations. Agreement is found to be good, although convergence between theory and numerical results as the system size is increased is, in some cases, unaccountably slow.