  Recently, the effective medium approach using 2x2 basic cluster of model
lattice sites to predict the conductivity of interacting droplets has been
presented by Hattori et al. To make a step aside from pure applications, we
have studied earlier a multi-scale percolation, employing any kxk basic cluster
for non-interacting particles. Here, with interactions included, we examine in
what way they alter the percolation threshold for any cluster case. We found
that at a fixed length scale k the interaction reduces the range of shifts of
the percolation threshold. To determine the critical concentrations, the
simplified model is used. It diminishes the number of local conductivities into
two main ones. In the presence of a dominance of the repulsive interaction over
the thermal energy, the exact percolation thresholds at scales k=2 and 3 can be
obtained from analytical formulas. Furthermore, by a simple reasoning, we
obtain the limiting threshold formula for odd k. When k>>1, the odd-even
difference becomes negligible. Hence, the 0.75 is the highest possible value of
the threshold.
