  Many low temperature disordered systems are expected to exhibit
Poisson-Dirichlet (PD) statistics. In this paper, we focus on the case when the
underlying disorder is a logarithmically correlated Gaussian process $\phi_N$
on the box $[-N,N]^d\subset\mathbb{Z}^d$. Canonical examples include branching
random walk, $*$-scale invariant fields, with the central example being the two
dimensional Gaussian free field (GFF), a universal scaling limit of a wide
range of statistical mechanics models. The corresponding Gibbs measure obtained
by exponentiating $\beta$ (inverse temperature) times $\phi_N$ is a discrete
version of the Gaussian multiplicative chaos (GMC) famously constructed by
Kahane. In the low temperature or supercritical regime, the GMC is expected to
exhibit atomic behavior on suitable renormalization, dictated by the extremal
statistics of $\phi_N$. Moreover, it is predicted, going back to a conjecture
made in 2001 by Carpentier and Le Doussal, that the weights of this atomic GMC
has a PD distribution. In a series of works, Biskup and Louidor carried out a
comprehensive study of the near maxima of the 2D GFF, and established the
conjectured PD behavior throughout the super-critical regime. In another
direction, Ding, Roy and Zeitouni established universal behavior of the maximum
for a general class of log-correlated Gaussian fields.
  In this paper we continue this program simply under the assumption of
log-correlation and nothing further. We prove that the GMC concentrates on an
$O(1)$ neighborhood of the local extrema and the PD prediction holds, in any
dimension $d$, throughout the supercritical regime, significantly generalizing
past results. Unlike for the 2D GFF, in absence of any Markovian structure for
general Gaussian fields, we develop and use as our key input a sharp estimate
of the size of level sets, which could have other applications.
