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.
