metric induces an effective action known as the Liouville one. Quantization of this action (in the conformal gauge) leads to the Liouville CFT.
After a brief survey of the of the bootstrap construction of the latter field theory, we use it to formulate the perturbed Lioville gravity, i.e., a perturbed matter CFT coupled to the corresponding induced 2D gravity.
Although the formalism is perturbative, in certain cases it allows, under some natural assumptions, to obtain rather detailed information at all scales.
Commonly assumed relation to the critical behavior of the statistical systems on random lattices is verified in some cases when the latter allow exact solution through the matrix model techniques.
Otherwise, the results can be compared against numerical simulations of the dynamical lattices.