  We consider two classes of backgrounds of Type IIB supergravity obtained by
wrapping D5-branes on a two-cycle inside the conifold. The field theory dual
exhibits confinement and, in addition, a region in which the dynamics is
walking, at least in the weak sense that the running of the coupling is
anomalously slow. We introduce quenched matter in the fundamental, modelled by
probe D7-branes which wrap an internal three-dimensional manifold and lie at
the equator of the transverse two-sphere. In the space spanned by the remaining
internal angle and the radial coordinate the branes admit two embeddings. The
first one is U-shaped: the branes merge at some finite value of the radius. The
second one is disconnected and extends along the entire radial direction at
fixed angular separation. We interpret these two configurations as
corresponding to chiral-symmetry breaking and preserving phases, respectively.
We present a simple diagnostic tool to examine the classical stability of the
embedding, based on the concavity/convexity conditions for the relevant
thermodynamic potentials. We use this criterion to show that U-shaped probes
that explore the walking region are unstable, hence providing a dynamical
origin for the tachyonic mode found in the literature. Whenever this occurs,
the disconnected solution becomes favored energetically. We find that in one of
the two classes of backgrounds the U-shaped embedding is always unstable, and
thus never realised dynamically. Consequently, these models cannot be used to
describe chiral-symmetry breaking. In the second category of solutions, our
analysis reveals the presence of a first-order phase transition between
chiral-symmetry broken and restored phases. Interestingly, this is in the same
class that contains a parametrically light scalar in the spectrum of glueballs
of the dual field theory.
