  Ly$\alpha$ absorption spectra of QSOs at redshifts $z\simeq6$ show complete
Gunn-Peterson absorption troughs (dark gaps) separated by tiny leaks. The dark
gaps are from the intergalactic medium (IGM) where the density of neutral
hydrogen are high enough to produce almost saturated absorptions, however,
where the transmitted leaks come from is still unclear so far. We demonstrate
that leaking can originate from the lowest density voids in the IGM as well as
the ionized patches around ionizing sources using semi-analytical simulations.
If leaks were produced in lowest density voids, the IGM might already be highly
ionized, and the ionizing background should be almost uniform; in contrast, if
leaks come from ionized patches, the neutral fraction of IGM would be still
high, and the ionizing background is significantly inhomogeneous. Therefore,
the origin of leaking is crucial to determining the epoch of
inhomogeneous-to-uniform transition of the the ionizing photon background. We
show that the origin could be studied with the statistical features of leaks.
Actually, Ly$\alpha$ leaks can be well defined and described by the equivalent
width $W$ and the full width of half area $W_{\rm H}$, both of which are less
contaminated by instrumental resolution and noise. It is found that the
distribution of $W$ and $W_{\rm H}$ of Ly$\alpha$ leaks are sensitive to the
modeling of the ionizing background. We consider four representative
reionization models. It is concluded that the leak statistics provides an
effective tool to probe the evolutionary history of reionization at
$z\simeq5-6.5$. Similar statistics would also be applicable to the reionization
of He II at $z \simeq 3$(Abridged)

Astrophysics