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 [Formula Presented] with [Formula Presented] and l scales with r as [Formula Presented] with [Formula Presented] whereas in the nontrapping case [Formula Presented] and [Formula Presented] These values further support previous results that NIP and TIP are in distinct universality classes. We also study numerically using scaling approaches the distribution [Formula Presented] 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 [Formula Presented] The scaling function has a power-law tail for large x values, [Formula Presented] with a universal value of [Formula Presented] for both models within our numerical accuracy.