Log-correlated gaussian fields: An overview

Abstract

We survey the properties of the log-correlated Gaussian field (LGF), which is a centered Gaussian random distribution (generalized function) h on Rd, defined up to a global additive constant. Its law is determined by the covariance formula (Formula Found) which holds for mean-zero test functions ϕ1, ϕ2. The LGF belongs to the larger family of fractional Gaussian fields obtained by applying fractional powers of the Laplacian to a white noise W on Rd. It takes the form h = (−Δ)−d/4W. By comparison, the Gaussian free field (GFF) takes the form (−Δ)−1/2W in any dimension. The LGFs with d ∈ {2, 1} coincide with the 2D GFF and its restriction to a line. These objects arise in the study of conformal field theory and SLE, random surfaces, random matrices, Liouville quantum gravity, and (when d = 1) finance. Higher-dimensional LGFs appear in models of turbulence and early-universe cosmology. LGFs are closely related to cascade models and Gaussian branching random walks. We review LGF approximation schemes, restriction properties, Markov properties, conformal symmetries, and multiplicative chaos applications.