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Many low temperature disordered systems are expected to exhibitPoisson-Dirichlet (PD) statistics. In this paper, we focus on the case when theunderlying disorder is a logarithmically correlated Gaussian process $\phi_N$on the box $[-N,N]^d\subset\mathbb{Z}^d$. Canonical examples include branchingrandom walk, $*$-scale invariant fields, with the central example being the twodimensional Gaussian free field (GFF), a universal scaling limit of a widerange of statistical mechanics models. The corresponding Gibbs measure obtainedby exponentiating $\beta$ (inverse temperature) times $\phi_N$ is a discreteversion of the Gaussian multiplicative chaos (GMC) famously constructed byKahane. In the low temperature or supercritical regime, the GMC is expected toexhibit atomic behavior on suitable renormalization, dictated by the extremalstatistics of $\phi_N$. Moreover, it is predicted, going back to a conjecturemade in 2001 by Carpentier and Le Doussal, that the weights of this atomic GMChas a PD distribution. In a series of works, Biskup and Louidor carried out acomprehensive study of the near maxima of the 2D GFF, and established theconjectured PD behavior throughout the super-critical regime. In anotherdirection, Ding, Roy and Zeitouni established universal behavior of the maximumfor a general class of log-correlated Gaussian fields.  In this paper we continue this program simply under the assumption oflog-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 anydimension $d$, throughout the supercritical regime, significantly generalizingpast results. Unlike for the 2D GFF, in absence of any Markovian structure forgeneral Gaussian fields, we develop and use as our key input a sharp estimateof the size of level sets, which could have other applications.

Universality of Poisson-Dirichlet law for log-correlated Gaussian fields via level set statistics