  2D Liouville quantum gravity (LQG) is used as a toy model for 4D quantum
gravity and is the theory of world-sheet in string theory. Recently there has
been growing interest in studying LQG in the realm of probability theory:
David, Kupiainen, Rhodes, Vargas (2014) and Duplantier, Miller, Sheffield
(2014) both provide a probabilistic perspective of the LQG on the 2D sphere. In
particular, in each of them one may find a definition of the so-called unit
area quantum sphere. We examine these two perspectives and prove their
equivalence by showing that the respective unit area quantum spheres are the
same. This is done by considering a unified limiting procedure for defining
both objects.
