Polydispersity in Percolation

TL;DR

This paper explains why the percolation threshold remains largely unaffected by polydispersity in systems, showing it depends mainly on the first moments of the size distribution, using branching process analysis.

Contribution

It provides a unified explanation for the insensitivity of percolation thresholds to polydispersity across different systems, via branching process modeling.

Findings

Percolation threshold depends only on the first few moments of the distribution.

Polydispersity does not significantly alter the network structure at the threshold.

Critical parameters of monodisperse systems can be extended to polydisperse cases.

Abstract

Every realistic instance of a percolation problem is faced with some degree of polydispersity, e.g., the pore-size distribution of an inhomogeneous medium, the size distribution of filler particles in composite materials, or the vertex degree of agents in a social network. Studies on different classes of systems have independently found very similar conceptual results for the percolation problem, i.e., that the percolation threshold is insensitive to the particular distribution controlling the polydispersity. Rather, the percolation threshold depends only on the first few moments of the distribution. In this article, we explain this frequently observed pattern using branching processes. The key observation is that a reasonable degree of polydispersity effectively does not alter the structure of the network that forms at the percolation threshold. As a consequence, the critical parameters of the monodisperse system can be analytically continued to account for polydispersity.