  The earliest generation of stars and black holes must have established an
early 'Lyman-Werner' background (LWB) at high redshift, prior to the epoch of
reionization. Because of the long mean free path of photons with energies
E<13.6 eV, the LWB was nearly uniform. However, some variation in the LWB is
expected due to the discrete nature of the sources, and their highly clustered
spatial distribution. In this paper, we compute the probability distribution
function (PDF) of the LW flux that irradiates dark matter (DM) halos collapsing
at high-redshift (z~10). Our model accounts for (i) the clustering of DM halos,
(ii) Poisson fluctuations in the number of corresponding star forming galaxies,
and (iii) scatter in the LW luminosity produced by halos of a given mass
(calibrated using local observations). We find that > 99% of the DM halos are
illuminated by a LW flux within a factor of 2 of the global mean value.
However, a small fraction, ~1e-8 to 1e-6, of DM halos with virial temperatures
above 1e4 K have a close luminous neighbor within < 10 kpc, and are exposed to
a LW flux exceeding the global mean by a factor of > 20, or to J_(21,LW)> 1e3
(in units of 1e-21 erg/s/Hz/sr/cm^2). This large LW flux can photo--dissociate
H_2 molecules in the gas collapsing due to atomic cooling in these halos, and
prevent its further cooling and fragmentation. Such close halo pairs therefore
provide possible sites in which primordial gas clouds collapse directly into
massive black holes (M_BH~ 1e4 - 1e6 M_sun), and subsequently grow into
supermassive (M_BH > 1e9 M_sun) black holes by z~6.
