Percolation on the average and spontaneous magnetization for q-states Potts model on graph

TL;DR

This paper establishes a precise equivalence between spontaneous magnetization in the q-states Potts model on a graph and the occurrence of percolation on the average, linking thermodynamic behavior to a combinatorial percolation property.

Contribution

It proves that spontaneous magnetization occurs if and only if the graph exhibits percolation on the average, connecting statistical physics with graph percolation theory.

Findings

Magnetization is linked to percolation on the average.

An inequality relates average probability of percolation to magnetization.

Spontaneous magnetization occurs precisely when percolation on the average is present.

Abstract

We prove that the q-states Potts model on graph is spontaneously magnetized at finite temperature if and only if the graph presents percolation on the average. Percolation on the average is a combinatorial problem defined by averaging over all the sites of the graph the probability of belonging to a cluster of a given size. In the paper we obtain an inequality between this average probability and the average magnetization, which is a typical extensive function describing the thermodynamic behaviour of the model.