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 treelike for $d\to \infty$ and they contain no closed loops. In this case, we find that the mean cluster size diverges at the percolation threshold density ${\eta }_c\to 2^{-d}$, 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 $z_c\to 1$ 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 log-normal distribution is no longer universal, and that it can be smaller than $2^{-d}$ for $d\to \infty$.

Figure 1

Connectedness criterion for spheres with different radii. (a) The spheres of radii $R_1$ and $R_2$ overlap the sphere of radius $R_3$, forming links between $R_3$ and $R_1$ and between $R_3$ and $R_2$. (b) Corresponding cluster formed by nodes (sphere centers) labeled by the sphere radii and weighted links (solid lines) connecting the nodes.

Figure 2

Schematic representation of a finite treelike cluster formed by connected spheres of radii $R_1,R_2$, and $R_3$. Each node label corresponds to the value of the radius of the sphere attached to the node.

Figure 3

(a) Percolation threshold ${\eta }_c$ in units of the asymptotic value $2^{-d}$ as a function of dimensionality for a discrete distribution of radii with $M=2$ and $R_1/R_2=1/2$. In this case, Eq. (32) is a system of two linear equations so that ${\eta }_c$ can be calculated exactly for any $d$. $x_2=0.8$, $0.6,0.4$, and $0.2$ from the uppermost to the lowermost curves. (b) Corresponding values of the critical coordination number $z_c$. As $d\to \infty ,z_c$ tends asymptotically to $x_2$.

Figure 4

(a) Dimensional dependence of the percolation threshold ${\eta }_c$ in units of the asymptotic value $2^{-d}$ obtained from a numerical solution of Eq. (43) for rectangular (upper curve), semicircular (middle curve), and triangular (lower curve) distributions of the sphere radii. (b) Corresponding critical average connectivity per particle $z_c$ (solid curves). Dashed lines are the asymptotic results for $d\gg 1$: $z_c=4/d$ (rectangular distribution), $z_c=32/\left(\sqrt{\pi }{d}^{3/2}\right)$ (semicircular distribution), and $z_c=32/d^2$ (triangular distribution).

Figure 5

(a) Dimensional dependence of the percolation threshold ${\eta }_c$ in units of the asymptotic value $exp(-{\sigma }^2{d}^2/2)$ for a log-normal distribution of the radii obtained from a numerical solution of Eq. (54); $\sigma =0.25,0.3,0.4$, and $0.5$ from the lowermost to the uppermost curves. (b) Critical average connectivity per particle $z_c$ for $\sigma =0.25,0.3,0.4$, and $0.5$ from the uppermost to the lowermost curves. All curves tend to $z_c/{\eta }_c\to 2$ as $d\to \infty$.