Percolation phenomena are ubiquitous in many aspects of natural, technological, and social sciences, and they arise when system-spanning clusters or components of, in some sense, connected objects form. In most systems of practical interest, percolation is established among particles that occupy random positions in space and that are not constrained by an underlying regular lattice. In such continuum percolating systems, the critical threshold depends on the shape, the orientation, the size distributions and the quality of the dispersion of the percolating objects, as well as on the connectedness criterion.
Continuum percolation is particulary relevant for the understanding of the electrical transport properties in conductor-insulator nanocomposites such as carbon nanotubes or graphene platelets dispersed in polymeric matrices. In these systems, or more generally in dispersions of conducting nano-fillers, the electron transfer between the conducting particles is mediated by quantum tunneling or hopping processes. In this case, the percolation formalism amounts to find the minimum distance between the particles such that a macroscopic cluster of tunneling-connected particles exists. This method, also known as the critical distance (or critical path) approximation, is able to reproduce quantitatively and to a high accuracy the results of fully numerical calculations of the conductivity.