  In random systems consisting of grains with size distributions the transport
properties are difficult to explore by network models. However, the
concentration dependence of effective conductivity and its critical properties
can be considered within coarsened lattice model proposed that takes into
account information from experimentally known size histograms. For certain
classes of size distributions the specific local arrangements of grains can
induce either symmetrical or unsymmetrical critical behaviour at two threshold
concentrations. Using histogram related parameters the non-monotonic behaviour
of the conductor-insulator and conductor-superconductor threshold is
demonstrated.
