  We study the renormalization group for nearly marginal perturbations of a
minimal conformal field theory M_p with p >> 1. To leading order in
perturbation theory, we find a unique one-parameter family of ``hopping
trajectories'' that is characterized by a staircase-like renormalization group
flow of the C-function and the anomalous dimensions and that is related to a
recently solved factorizable scattering theory. We argue that this system is
described by interactions of the form t phi_{(1,3)} - t' \phi_{(3,1)} . As a
function of the relevant parameter t, it undergoes a phase transition with new
critical exponents simultaneously governed by all fixed points M_p, M_{p-1},
..., M_3. Integrable lattice models represent different phases of the same
integrable system that are distinguished by the sign of the irrelevant
parameter t'.
