Liouville quantum gravity is a model of random planar geometry specified by a logarithmically correlated field 
and a parameter 
. For 
, its area measure
is obtained by Gaussian multiplicative
chaos. With the Gaussian free field normalized
to have unit logarithmic singularity, the circle-average construction locally has
the form
 |
where the limit is a weak limit of measures and 
is the circle average of 
. The coordinate-change rule is 
, where 
and 
is a conformal mapping.
It makes the random area unchanged by the coordinate used
to represent the quantum surface (Duplantier and Sheffield 2008).
The geometry also carries an intrinsic random metric. A Liouville quantum sphere is the spherical finite-area version, whereas Liouville Brownian motion is the diffusion with quantum area as its reference measure.
