The Shape and Size distribution of HII Regions near the percolation transition

Using Shapefinders, which are ratios of Minkowski functionals, we study the morphology of neutral hydrogen (HI) density fields, simulated using semi-numerical technique (inside-out), at various stages of reionization. Accompanying the Shapefinders, we also employ the 'largest cluster statistic' (LCS), originally proposed in Klypin and Shandarin (1993), to study the percolation in both neutral and ionized hydrogen. We find that the largest ionized region is percolating below the neutral fraction $x_{HI} \lesssim 0.728$ (or equivalently $z \lesssim 9$). The study of Shapefinders reveals that the largest ionized region starts to become highly filamentary with non-trivial topology near the percolation transition. During the percolation transition, the first two Shapefinders - 'thickness' ($T$) and 'breadth' ($B$) - of the largest ionized region do not vary much, while the third Shapefinder - 'length' ($L$) - abruptly increases. Consequently, the largest ionized region tends to be highly filamentary and topologically quite complex. The product of the first two Shapefinders, $T\times B$, provides a measure of the 'cross-section' of a filament-like ionized region. We find that, near percolation, the value of $T\times B$ for the largest ionized region remains stable at $\sim 7$ Mpc$^2$ (in comoving scale) while its length increases with time. Interestingly all large ionized regions have similar cross-sections. However their length shows a power-law dependence on their volume, $L\propto V^{0.72}$, at the onset of percolation.