The reionization of intergalactic hydrogen has received intense theoretical scrutiny over the past two decades .
Here , we approach the process formally as a percolation process and phase transition .
Using semi-numeric simulations , we demonstrate that an infinitely-large ionized region abruptly appears at an ionized fraction of x _ { i } \approx 0.1 and quickly grows to encompass most of the ionized gas : by x _ { i } \sim 0.3 , nearly ninety percent of the ionized material is part of this region .
Throughout most of reionization , nearly all of the intergalactic medium is divided into just two regions , one ionized and one neutral , and both infinite in extent .
We also show that the discrete ionized regions that exist before and near this transition point follow a near-power law distribution in volume , with equal contributions to the total filling factor per logarithmic interval in size up to a sharp cutoff in volume .
These qualities are generic to percolation processes , with the detailed behavior a result of long-range correlations in the underlying density field .
These insights will be crucial to understanding the distribution of ionized and neutral gas during reionization and provide precise meaning to the intuitive description of reionization as an ‘ ‘ overlap '' process .