Perturbed Conformal Field Theory on Fluctuating Sphere

General properties of perturbed conformal field theory interacting with quantized Liouville gravity are considered in the simplest case of spherical topology. We discuss both short distance and large distance asymptotic of the partition function. The crossover region is studied numerically for a simple example of the perturbed Yang-Lee model, complemented in general with arbitrary conformal ``spectator'' matter. The latter is not perturbed and remains conformal along the flow, thus giving a control over the Liouville central charge. The partition function is evaluated numerically from combined analytic and perturbative information. In this paper we use the perturbative information up to third order. At special points the four-point integral can be evaluated and compared with our data. At the solvable point of minimal Liouville gravity we are in remarkably good agreement with the matrix model predictions. Possibilities to compare the result with random lattice simulations is discussed.