Continuum percolation of polydisperse hyperspheres in infinite dimensions

TL;DR

This paper investigates the percolation behavior of polydisperse hyperspheres in high dimensions, revealing that the critical connectivity depends on the largest spheres and that the percolation threshold is non-universal for unbounded size distributions.

Contribution

It provides a detailed analysis of continuum percolation in high dimensions, showing the influence of size distribution and the dominance of largest spheres on the percolation threshold.

Findings

Clusters are tree-like in high dimensions with no loops.

Percolation threshold approaches 2^{-d} for bounded sizes.

Unbounded size distributions can have thresholds smaller than 2^{-d}.

Abstract

We analyze the critical connectivity of systems of penetrable $d$-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers and edges are formed between any two overlapping spheres. Edge weights naturally arise from the different radii of two overlapping spheres. For the case in which the spheres have bounded size distributions, we show that clusters of connected spheres are tree-like for $d\rightarrow \infty$ and they contain no closed loops. In this case, we find that the mean cluster size diverges at the percolation threshold density $\eta_c\rightarrow 2^{-d}$, independently of the particular size distribution. We also show that the mean number of overlaps for a particle at criticality $z_c$ is smaller than unity, while $z_c\rightarrow 1$ only for spheres with fixed radii. We explain these features by showing that in the large dimensionality limit the critical connectivity is dominated by the spheres with the largest size. Assuming that closed loops can be neglected also for unbounded radii distributions, we find that the asymptotic critical threshold for systems of spheres with radii following a lognormal distribution is no longer universal, and that it can be smaller than $2^{-d}$ for $d\rightarrow\infty$.