Universality of Poisson–Dirichlet Law for Log-Correlated Gaussian Fields via Level Set Statistics

Abstract

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 \([0,N]^d\subset \mathbb {Z}^d\). 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 (Ann Sci Math Québec 9(2): 105–150, 1985). In the low temperature or supercritical regime, i.e., \(\beta \) larger than a critical \(\beta _c,\) the GMC is expected to exhibit atomic behavior on suitable renormalization, dictated by the extremal statistics or near maximum values of \(\phi _N\). Moreover, it is predicted going back to a conjecture made in 2001 in Carpentier and Le Doussal (Phys Rev E 63(2): 026110, 2001), that the weights of this atomic GMC has a PD distribution. In a series of works culminating in Biskup and Louidor (Adv Math 330, 589–687, 2018), 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 (\(\beta > 2\)). In another direction Ding et al. (Ann Probab 5(6A), 3886–3928, 2017), 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 made in Carpentier and Le Doussal (Phys Rev E 63(2): 026110, 2001) holds, in any dimension d, throughout the supercritical regime \(\beta > \sqrt{2d}\), significantly generalizing past results. While many of the arguments for the GFF make use of the powerful Gibbs–Markov property, 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, a result we believe could have other applications.