Three-point functions in c <= 1 Liouville theory and conformal loop ensembles

The possibility of extending the Liouville Conformal Field Theory from values of the central charge $c geq 25$ to $c leq 1$ has been debated for many years in condensed matter physics as well as in string theory. It was only recently proven that such an extension -- involving a real spectrum of critical exponents as well as an analytic continuation of the DOZZ formula for three-point couplings -- does give rise to a consistent theory. We show in this Letter that this theory can be interpreted in terms of microscopic loop models. We introduce in particular a family of geometrical operators, and, using an efficient algorithm to compute three-point functions from the lattice, we show that their operator algebra corresponds exactly to that of vertex operators $V_{hat{alpha}}$ in $c leq 1$ Liouville. We interpret geometrically the limit $hat{alpha} to 0$ of $V_{hat{alpha}}$ and explain why it is not the identity operator (despite having conformal weight $Delta=0$).