Large clusters in a correlated percolation model

We consider a correlated site percolation problem on a cubic lattice of size $L^3$, with $16\le L\le 512$. The sites of an initially full lattice are removed by a random walk of $\mathcal{N}=u{L}^3$ steps. When the parameter $u$ crosses a threshold $u_c=3.15$, a large system transitions between percolating and non-percolating states. We study the $L$-dependence of the mean mass (number of sites) $M_r$ of the $r$th largest cluster, as well as $r$-dependence of $M_r$ for various system sizes $L$ at $u_c$. We demonstrate that $M_r\sim L^{5\mathrm{/}2}\mathrm{/}{r}^{5\mathrm{/}6}$ for moderate or large $L$ and $r\gg 1$, and also conclude that for {\em any} $r$ the fractal dimensions of the clusters are $5\mathrm{/}2$.