  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 $\Delta=0$).
