Gaussian multiplicative chaos is a random measure constructed by renormalizing exponentials of a Gaussian field. For a regularized centered Gaussian
field 
and a reference measure 
, the approximating measures are
![mu_(gamma,epsilon)(dz)=exp(gammaX_epsilon(z)-(gamma^2)/2E[X_epsilon(z)^2])m(dz).](/images/equations/GaussianMultiplicativeChaos/NumberedEquation1.svg) |
For logarithmically correlated planar fields with planar Lebesgue measure as 
and unit logarithmic singularity, the subcritical range is
.
Under the usual regularization hypotheses, the weak
limit gives the subcritical chaos measure. A critical
measure at 
requires a different normalization, rather than simply
substituting 2 into the subcritical limit. Liouville
quantum gravity uses such random area measures
(Duplantier and Sheffield 2008).
