Boundary flows in minimal models

We discuss in this paper the behaviour of minimal models of conformal theory perturbed by the operator Φ 13 at the boundary. Using the RSOS restriction of the sine-Gordon model, adapted to the boundary problem, a series of boundary flows between different set of conformally invariant boundary conditions are described. Generalizing the " staircase " phenomenon discovered by Al. Zamolodchikov, we find that an analytic continuation of the boundary sinh-Gordon model provides a flow interpolating not only between all minimal models in the bulk, but also between their possible conformal boundary conditions. In the particular case where the bulk sinh-Gordon coupling is turned to zero, we obtain a boundary roaming trajectory in the c = 1 theory, that interpolates between all the possible spin s Kondo models.