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                <term xml:lang="en">singular Liouville equation.</term>
                <term xml:lang="en">Liouville field theory</term>
                <term xml:lang="en">large deviation principle</term>
                <term xml:lang="en">semiclassical limit</term>
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              <p>Liouville Field Theory (LFT for short) is a two dimensional model of random surfaces, which is for instance involved in $2d$ string theory or in the description of the fluctuations of metrics in $2d$ Liouville quantum gravity. This is a probabilistic model that consists in weighting the classical   Free Field action with an interaction term given by the exponential of a Gaussian multiplicative chaos. The main input of our work is the study of the semiclassical limit of the theory, which is a prescribed asymptotic regime of LFT of interest in physics literature (see \cite{witten} and references therein). We derive exact formulas for the Laplace transform of the Liouville field in the case of flat metric on the unit disk with Dirichlet boundary conditions. As a consequence, we prove that the Liouville field  concentrates on the solution of the classical Liouville equation   with explicit negative  scalar curvature. We also characterize the leading fluctuations, which are Gaussian and massive, and establish a large deviation principle. Though considered as an ansatz in the whole physics literature, it seems that it is the first rigorous probabilistic derivation of the semiclassical limit of LFT.  On the other hand, we carry out the same analysis  when we further weight the Liouville action with heavy matter operators. This procedure appears when computing the $n$-points correlation functions of LFT. </p>
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