excursion set theory, Hausdorff dimension, Hölder regularity, etc.) can be found in [@adler2007random; @adler2010geometry].

The log-correlated Gaussian field (LGF) is a random element $h$ of the space of tempered distributions modulo constants and has covariance given by $$\operatorname{Cov}[(h,\phi_1),(h,\phi_2)] = -\int_{{\mathbb{R}}^d}\int_{{\mathbb{R}}^d}\log|x-y|\phi_1(x)\phi_2(y)\,dx\,dy,$$ In two dimensions, the LGF coincides with the GFF (up to a constant factor). We will see in Section \[sec:whole-space\] that the $d$-dimensional LGF is a multiple of $\operatorname{FGF}_{d/2}({\mathbb{R}}^d)$. In recent years the log-correlated Gaussian field has enjoyed renewed interest because of its relationship to Gaussian multiplicative chaos. For a survey article of Gaussian multiplicative chaos see [@rhodes2013gaussian]. Furthermore, the LGF in ${\mathbb{R}}^3$ plays an important role in early universe cosmology, where it approximately describes the gravitational potential function of the universe at a fixed time shortly after the big bang; see [@duplantier2013log] for more discussion and references.

Another noteworthy subclass of the fractional Gaussian fields is $\operatorname{FGF}_2({\mathbb{R}}^d)$, which is known as the bi-Laplacian Gaussian field. The discrete counterpart of the bi-Laplacian Gaussian field is called the membrane model in physics literature; for a mathematical point of view see [@sakagawa2003entropic], [@kurt2007entropic], [@kurt2009maximum], and [@sakagawa2012free]. In dimension at least five, there is a natural discrete field associated with the uniform spanning 