  We continue the study of the maximum of the scale-inhomogeneous discrete
Gaussian free field in dimension two. In this paper, we consider the regime of
weak correlations and prove the convergence in law of the centred maximum to a
randomly shifted Gumbel distribution. In particular, we obtain limiting
expressions for the random shift. As in the case of variable speed branching
Brownian motion, the shift is of the form CY, where C is a constant that
depends only on the variance at the shortest scales, and Y is a random variable
that depends only on the variance at the largest scales. Moreover, we
investigate the geometry of highest local maxima. We show that they occur in
clusters of finite size that are separated by macroscopic distances. The poofs
are based on Gaussian comparison with branching random walks and second moment
estimates.
