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 $M\ensuremath{\sim}{l}^{{d}_{l}}$ with ${d}_{l}=1.510\ifmmode\pm\else\textpm\fi{}0.005$ and l scales with r as $l\ensuremath{\sim}{r}^{{d}_{\mathrm{min}}}$ with ${d}_{\mathrm{min}}=1.213\ifmmode\pm\else\textpm\fi{}0.005,$ whereas in the nontrapping case ${d}_{l}=1.671\ifmmode\pm\else\textpm\fi{}0.006$ and ${d}_{\mathrm{min}}=1.133\ifmmode\pm\else\textpm\fi{}0.005.$ 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 $N(l,r)\ensuremath{\sim}{r}^{{d}_{f}\ensuremath{-}1\ensuremath{-}{d}_{\mathrm{}\mathrm{min}}}{f(l/r}^{{d}_{\mathrm{min}}}).$ The scaling function has a power-law tail for large x values, $f(x)\ensuremath{\sim}{x}^{\ensuremath{-}h},$ with a universal value of $h\ensuremath{\approx}2$ for both models within our numerical accuracy.