Clusters' size-degree distribution for bond percolation

TL;DR

This paper introduces a method using a modified q-state Potts model in the q→1 limit to analytically derive the size and degree distributions of finite clusters in classical bond percolation on Bethe lattices and complete graphs.

Contribution

It presents a novel analytical approach linking a modified Potts model to cluster degree and size distributions in bond percolation, expanding potential applications.

Findings

Derived analytical distributions for Bethe lattice and complete graph.

Established connection between Potts model and cluster characteristics.

Discussed extensions to other percolation models.

Abstract

To address some physical properties of percolating systems it can be useful to know the degree distributions in finite clusters along with their size distribution. Here we show that to achieve this aim for classical bond percolation one can use the $q \to 1$ limit of suitably modified q-state Potts model. We consider a version of such model with the additional complex variables and show that its partition function gives generating function for the size and degree distribution in this limit. We derive this distribution analytically for bond percolation on Bethe lattice and complete graph. The possibility to expand the applications of present method to other clusters' characteristics and to models of correlated percolation is discussed.