Percolation Phenomena in Low and High Density Systems

TL;DR

This paper investigates percolation behavior in 2D disordered Potts models, revealing unexpected non-percolation of most species at self-dual points and persistent percolation of a single species above threshold, with implications for related coloring models.

Contribution

It uncovers novel percolation phenomena in disordered Potts systems, including non-percolation of multiple species at self-dual points and single-species percolation above threshold, supported by a unified model and coloring analogy.

Findings

Amalgamation of q-1 species fails to percolate at self-dual points.

Single species percolates in the dilute bond model just above threshold.

Phenomena demonstrated in a perturbed self-dual model and a coloring model.

Abstract

We consider the 2D quenched--disordered $q$--state Potts ferromagnets and show that at self--dual points any amalgamation of $q-1$ species will fail to percolate despite an overall (high) density of $1-q^{-1}$. Further, in the dilute bond version of these systems, if the system is just above threshold, then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density. Finally, we demonstrate both phenomena in a single model by considering a ``perturbation'' of the dilute model that has a self--dual point. We also demonstrate that these phenomena occur, by a similar mechanism, in a simple coloring model invented by O. H\"aggstr\"om.