Introduction to the Liouville quantum gravity metric (Q6118120)

In this paper the authors review and describe the construction and the properties of Liouville quantum gravity, a family of random surfaces which were introduced by Knizhnik, Polyakov and Zamolodzhikov in physics. Formally they are obtained by exponentiating a parametrised Gaussian free field (GFF) in front of the metric tensor. They also occur as scaling limits of random planar maps. As the GFF is quite rough -- not a function, but a distribution -- the construction is technically challenging. Many detailed properties have been derived, for the subcritical, critical and supercritical regimes. The limiting measure, and the more complicated limiting metric, are the objects of interest, and many results have been obtained over the last years, which are presented and discussed. The parameter dependence, as well as a number of still open questions are explained as well as some of the various physical and mathematical connections the models have. For the entire collection see [Zbl 07816360].