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.
