# 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 $\mathbb R^d$, defined up to a global additive constant. Its law is determined by the covariance formula $\mathrm{Cov}\bigl[ (h, \phi_1), (h, \phi_2) \bigr] = \int_{\mathbb R^d \times \mathbb R^d} -\log|y-z| \phi_1(y) \phi_2(z)dydz$ which holds for mean-zero test functions $\phi_1, \phi_2$. 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 $\mathbb R^d$. It takes the form $h = (-\Delta)^{-d/4} W$. By comparison, the Gaussian free field (GFF) takes the form $(-\Delta)^{-1/2} W$ in any dimension. The LGFs with $d \in \{2,1\}$ 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.
Keywords :
Document type :
Book sections
Domain :

Cited literature [34 references]

https://hal-upec-upem.archives-ouvertes.fr/hal-01071012
Contributor : Rémi Rhodes Connect in order to contact the contributor
Submitted on : Thursday, October 2, 2014 - 9:15:51 PM
Last modification on : Saturday, January 15, 2022 - 4:11:37 AM
Long-term archiving on: : Saturday, January 3, 2015 - 11:26:51 AM

### File

lcgf.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-01071012, version 1

### Citation

Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, Vincent Vargas. Log-correlated Gaussian fields: an overview. Geometry, Analysis and Probability, Progress in Mathematics, (In Honor of Jean-Michel Bismut)., 310, 2014, 978-3-319-49636-8 (Print) 978-3-319-49638-2 (Online). ⟨hal-01071012⟩

### Metrics

Les métriques sont temporairement indisponibles