  A rigorous probabilistic construction of Liouville conformal field theory
(LCFT) on the Riemann sphere was recently given by David-Kupiainen and the last
two authors. In this paper, we focus on the connection between LCFT and the
classical Liouville field theory via the semiclassical approach. LCFT depends
on a parameter $\gamma \in (0,2)$ and the limit $\gamma \to 0$ corresponds to
the semiclassical limit of the theory. Within this asymptotic and under a
negative curvature condition (on the limiting metric of the theory), we
determine the limit of the correlation functions and of the associated
Liouville field. We also establish a large deviation result for the Liouville
field: as expected, the large deviation functional is the classical Liouville
action. As a corollary, we give a new (probabilistic) proof of the
Takhtajan-Zograf theorem which relates the classical Liouville action (taken at
its minimum) to Poincar\'e's accessory parameters. Finally, we gather
conjectures in the positive curvature case (including the study of the
so-called quantum spheres introduced by Duplantier-Miller-Sheffield).
