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## Log-correlated Gaussian fields: an overview

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Bertrand Duplantier
• Function : Author
• PersonId : 974394
Rémi Rhodes
• Function : Author
Scott Sheffield
• Function : Author
Vincent Vargas

#### 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.

#### Domains

Physics [physics]

### Dates and versions

cea-01252004 , version 1 (07-01-2016)

### Identifiers

• HAL Id : cea-01252004 , version 1
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### Cite

Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, Vincent Vargas. Log-correlated Gaussian fields: an overview. 2016. ⟨cea-01252004⟩

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