Scaling of the spanning threshold in gradient percolation

A simple and fast way to apply correlations in percolation simulations is to apply a uniform gradient to the occupancy probabilities. For small networks, exact results are presented here for the spanning thresholds in site percolation with a gradient for networks up to 4 x 4 in two dimensions and 2 x 2 x 2 in three dimensions. Numerical results are provided for larger networks that extrapolate to a linear modification of the threshold proportional to the gradient for moderate values of the gradient.