  We examine both analytically and numerically the validity of factorization
for the double dipole scattering amplitude T^{(2)} which appears on the right
hand side of the BK--JIMWLK equation. We demonstrate that, if one uses a dilute
object (e.g., a proton in DIS) as the initial condition, the correlation in the
transverse plane induced by the leading order BFKL evolution is generally
strong, resulting in a violation of the mean field approximation T^{(2)}
\approx TT even at zero impact parameter by a factor ranging from 1.5 to O(10)
depending on the relative size of the scatterers and rapidity. This suggests
that, within the experimentally accessible energy interval, the transverse
correlation can significantly affect the nonlinear evolution of the dipole
scattering amplitude. It also suggests that the nonlinear effects may set in
earlier, already in the weak scattering regime. In the case of the simulation
with a running coupling, the violation of factorization is somewhat milder, but
still noticeable.
