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Around the semi-classical limit of boundary Liouville theory
Abstract
Liouville theory provides a notion of random surface that "fluctuates" around a deterministic (=classical) one. This classical geometry corresponds to the unique solution of the problem of finding, within a given conformal class, a Riemannian metric with prescribed scalar and geodesic curvatures as well as conical singularities and corners. The level of randomness in Liouville theory is measured by the coupling constant $\gamma \in (0,2)$, the semi-classical limit corresponding to taking $\gamma \to 0$.
In this talk we will first discuss this classical geometry and the analytic tools used to study it. In a second part we will explain, thanks to its probabilistic formulation based on Gaussian Free Fields and Gaussian Multiplicative Chaos, that the semi-classical limit of boundary Liouville CFT indeed describes this classical geometry. If time permits we will discuss some implications of this semi-classical limit in relation with uniformisation of open Riemann surfaces.
