  We model the escape of ionizing radiation from high-redshift galaxies using
high-resolution Adaptive Mesh Refinement N-body + hydrodynamics simulations.
Our simulations include time-dependent and spatially-resolved transfer of
ionizing radiation in three dimensions, including effects of dust absorption.
For galaxies of total mass M > 10^11 Msun and star formation rates SFR ~ 1-5
Msun/yr, we find angular averaged escape fractions of 0.01-0.03 over the entire
redshift interval studied (3<z<9). In addition, we find that the escape
fraction varies by more than an order of magnitude along different
lines-of-sight within individual galaxies, from the largest values near
galactic poles to the smallest along the galactic disk. The escape fraction
declines steeply at lower masses and SFR. We show that the low values of escape
fractions are due to a small fraction of young stars located just outside the
edge of HI disk. We compare our predicted escape fraction of ionizing photons
with previous results, and find a general agreement with both other simulation
results and available direct detection measurements at z ~ 3. We also compare
our simulations with a novel method to estimate the escape fraction in galaxies
from the observed distribution of neutral hydrogen column densities along the
lines of sights to long duration gamma-ray bursts. Using this method we find
escape fractions of the GRB host galaxies of 2-3%, consistent with our
theoretical predictions. [abridged]
