Structural properties of invasion percolation with and without trapping: Shortest path and distributions

We study several structural properties including the shortest path l between two sites separated by a Euclidean distance r of invasion percolation with trapping (TIP) and without trapping (NIP). For the trapping case we find that the mass M scales with l as M~l^dₗ with dₗ=1.510±0.005 and l scales with r as l~r^dₘᵢₙ with dₘᵢₙ=1.213±0.005, whereas in the nontrapping case dₗ=1.671±0.006 and dₘᵢₙ=1.133±0.005. These values further support previous results that NIP and TIP are in distinct universality classes. We also study numerically using scaling approaches the distribution $N(l,r)$ of the lengths of the shortest paths connecting two sites at distance r in NIP and TIP. We find that it obeys a scaling form N(l,r)~r^df-1-dₘᵢₙf(l/r^dₘᵢₙ). The scaling function has a power-law tail for large x values, f(x)~x^-h, with a universal value of h≈2 for both models within our numerical accuracy.