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

We study several structural properties including the shortest path I betwee n two sites separated by a Euclidean distance r of invasion percolation wit h trapping (TIP) and without trapping (NIP). For the trapping case we find that the mass M scales with l as M similar to I-dt with d(l) = 1.510+/-0.00 5 and l scales with r as l similar to r(dmin) with d(min) = 1.213+/-0.005, whereas in the nontrapping case d(l) = 1.671+/-0.006 and d(min) = 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 app roaches the distribution N(l,r) of the lengths of the shortest paths connec ting two sites at distance r in NIP and TIP. We find that it obeys a scalin g form N(l,r)similar to r(df-1-dmin)f(l/r(dmin)). The scaling function has a power-law tail for large x values, f(x)similar to x(-h), with a universal value of h approximate to 2 for both models within our numerical accuracy. [S1063-651X(99)12603-5].