= Liouville quantum gravity
{c}
<Liouville quantum gravity> gives a random geometry in two dimensions whose area is formally weighted by $e^{\gamma h}$, where $h$ is a <Gaussian free field>. Since the field is a random distribution rather than a pointwise <function>, this exponential needs regularization. For $0<\gamma<2$, a standard circle-average construction on a planar <domain> is
$$
\mu_\gamma(dz)=\lim_{\varepsilon\downarrow0}\varepsilon^{\gamma^2/2}e^{\gamma h_\varepsilon(z)}\,d^2z,
$$
with the normalization of the field fixed as in the source. Here $h_\varepsilon$ is the average around a circle of radius $\varepsilon$; the limit is a random <measure>, not a pointwise density.<Liouville Quantum Gravity and KPZ by Bertrand Duplantier and Scott Sheffield>[Abstract and construction of the quantum area measure]{ref}
For example, adding a constant $c$ to the field multiplies the area measure by $e^{\gamma c}$ without changing its coordinate domain. This separates random area from ordinary Euclidean area. The construction relates scaling exponents of random sets measured in these two geometries; it does not constitute a model of four-dimensional spacetime. Constructing an associated random distance requires more than defining the area measure alone.
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