  The cross-correlation between high redshift galaxies and 21 cm emission from
the high redshift intergalactic medium (IGM) promises to be an excellent probe
of the Epoch of Reionization (EoR). On large scales, the 21 cm and galaxy
fields are anti-correlated during most of the reionization epoch. However, on
scales smaller than the size of the H II regions around detectable galaxies,
the two fields become roughly uncorrelated. Consequently, the 21 cm-galaxy
cross power spectrum provides a tracer of bubble growth during reionization,
with the signal turning over on progressively larger scales as reionization
proceeds. The precise turnover scale depends on the minimum host mass of the
detectable galaxies, and the galaxy selection technique. Measuring the turnover
scale as a function of galaxy luminosity constrains the characteristic bubble
size around galaxies of different luminosities. The cross spectrum becomes
positive on small scales if ionizing photons fail to escape from low mass
galaxies, and these galaxies are detectable longward of the hydrogen ionization
edge, because in this case some identifiable galaxies lie outside of ionized
regions. LOFAR can potentially measure the 21 cm-galaxy cross spectrum in
conjunction with mild extensions to the existing Subaru survey for $z=6.6$
Lyman-alpha emitters, while the MWA is slightly less sensitive for detecting
the cross spectrum. A futuristic galaxy survey covering a sizable fraction of
the MWA field of view ($\sim 800$ deg$^2$) can probe the scale dependence of
the cross spectrum, constraining the filling factor of H II regions at
different redshifts during reionization, and providing other valuable
constraints on reionization models.
