Dynamics of Invasion Percolation

Abstract

Simulations of the growth of planar invasion percolation clusters exhibit novel dynamic scaling. The probability to invade a site at a distance $r$ from a reference site at a time $t$ after that site was invaded behaves as $N(r,t)={r}^{\ensuremath{-}1}f(\frac{{r}^{D}}{t})$, where $D$ is the fractal dimension of the invaded region. The scaling function has the unusual timing behavior $f(u)\ensuremath{\sim}{u}^{a}(u\ensuremath{\ll}1)$ and $\ensuremath{\sim}{u}^{\ensuremath{-}b}(u\ensuremath{\gg}1)$, with $a\ensuremath{\simeq}1.4$, $b\ensuremath{\simeq}0.6$, and growth occurring mainly around $r\ensuremath{\sim}{t}^{\frac{1}{D}}$.