Percolation in networks with voids and bottlenecks

A general method is proposed for predicting the asymptotic percolation threshold of networks with bottlenecks, in the limit that the subnet mesh size goes to zero. The validity of this method is tested for bond percolation on filled checkerboard and "stack-of-triangle" lattices. Thresholds for the checkerboard lattices of different mesh sizes are estimated using the gradient percolation method, while for the triangular system they are found exactly using the triangle-triangle transformation. The values of the thresholds approach the asymptotic values of 0.64222 and 0.53993, respectively, as the mesh is made finer, consistent with a direct determination based upon the predicted critical corner-connection probability.