Continuum percolation of polydisperse hyperspheres in infinite dimensions
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 $η_c\rightarrow 2^{-d}$, 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 $d\rightarrow\infty$.