We consider the discrete Gaussian Free Field in a square box in ( Z ^2 ) of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as ( N ). Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an r N -neighborhood thereof, we prove that the associated process tends, whenever ( r_N ) and ( r_N N 0 ), to a Poisson point process with intensity measure ( Z ( dx) e ^ - h d h ), where ( := 2 g ) with g: = 2 π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field.
A version of the Gibbs measure appears also in the studies of Gaussian multiplicative chaos by Robert and Vargas @cite_22 , Allez, Rhodes and Vargas @cite_12 and Duplantier, Rhodes, Sheffield and Vargas @cite_30 @cite_18 ; see Rhodes and Vargas @cite_48 for a recent review of this subject going back to Kahane @cite_0 . In particular, the papers @cite_30 @cite_18 are concerned with the construction of a which corresponds to the @math limit of (unnormalized) measure This is an object closely related to the @math -derivative of @math at @math . On the basis of various conjectural statements, the references @cite_30 @cite_18 predict the derivative martingale to appear in the place of our @math . In Theorem we thus prove a version of this prediction for the DGFF. (The apparent discrepancy in the factor @math in front of the variance in the exponents of and comes from the fact that the measure @math in has been scaled by a factor of @math .)
@cite_30 @cite_18 @cite_22 @cite_48 @cite_0 @cite_12
2102318662 1573609569 2022385268 2266756552
We study the maximum of a Gaussian field on @math ( @math ) whose correlations decay logarithmically with the distance. Kahane Kah85 introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas DRSV12a DRSV12b extended Kahane's construction to the critical case and established the KPZ formula at criticality. Moreover, they made in DRSV12a several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in DRSV12a : we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale. We introduce a Gibbs measure on nearest-neighbour paths of length @math in the Euclidean @math -dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature @math . We prove that, for all @math , the random walk condensates to a set of diameter @math in dimension @math , up to a multiplicative constant. In all dimensions @math , we also prove that the volume is bounded above by @math and the diameter is bounded below by @math . Similar results hold for a random walk conditioned to have local time greater than @math everywhere in its range when @math is larger than some explicit constant, which in dimension two is the logarithm of the connective constant. In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of logcorrelated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables. Abstract Building upon the theory of graph limits and the Aldous–Hoover representation and inspired by Panchenko’s work on asymptotic Gibbs measures [Annals of Probability 2013], we construct continuous embeddings of discrete probability distributions. We show that the theory of graph limits induces a meaningful notion of convergence and derive a corresponding version of the Szemeredi regularity lemma. Moreover, complementing recent work (2015), we apply these results to Gibbs measures induced by sparse random factor graphs and verify the “replica symmetric solution” predicted in the physics literature under the assumption of non-reconstruction.