  The massless flow between successive minimal models of conformal field theory
is related to a flow within the sine-Gordon model when the coefficient of the
cosine potential is imaginary. This flow is studied, partly numerically, from
three different points of view. First we work out the expansion close to the
Kosterlitz-Thouless point, and obtain roaming behavior, with the central charge
going up and down in between the UV and IR values of $c=1$. Next we
analytically continue the Casimir energy of the massive flow (i.e. with real
cosine term). Finally we consider the lattice regularization provided by the
O(n) model in which massive and massless flows correspond to high- and
low-temperature phases. A detailed discussion of the case $n=0$ is then given
using the underlying N=2 supersymmetry, which is spontaneously broken in the
low-temperature phase. The ``index'' $\tr F(-1)^F$ follows from the Painleve
III differential equation, and is shown to have simple poles in this phase.
These poles are interpreted as occuring from level crossing (one-dimensional
phase transitions for polymers). As an application, new exact results for the
connectivity constants of polymer graphs on cylinders are obtained.
