Liouville quantum gravity

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 iswith the normalization of the field fixed as in the source. Here $h_{\epsilon }$ is the average around a circle of radius $\epsilon$; the limit is a random measure, not a pointwise density.^[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.