We study the classical and quantum percolation of spheres in a three-dimensional continuum. Each sphere has an impenetrable hard core of diameter σ, and two spheres are considered to be directly connected if the distance between their centers is less than d. We calculate the critical percolation density as a function of σ/d. In the classical problem this is the density ρc at which an infinite cluster of connected spheres first forms. In the quantum problem, we study a tight-binding model where the hopping matrix element between two spheres is nonzero only if they are directly connected. In this case the critical density ρq is the density at which the eigenstates of the Hamiltonian first become extended. Our method uses Monte Carlo simulation and finite-size scaling techniques, and for the quantum problem, the concept of quantum connectivity. We find that both ρc and ρq exhibit nonmonotonic behavior as a function of σ/d. We also find that for all values of σ/d, ρq>ρc, although the ratio of the thresholds decreases with increasing σ/d. We argue that a better understanding of this ratio is obtained by considering the average coordination number. We speculate about the nature of both classical and quantum percolation as σ/d approaches 1.

Saven, J. G., Skinner, J. L., & Wright, J. R. (1991). Classical and quantum continuum percolation with hard core interactions. The Journal of Chemical Physics, 94(9), 6153–6159.