  Several cool-core clusters are known to host a radio mini-halo, a diffuse,
steep-spectrum radio source located in their cores, thus probing the presence
of non-thermal components as magnetic field and relativistic particles on
scales not directly influenced by the central AGN. The nature of the mechanism
that produces a population of radio-emitting relativistic particles on the
scale of hundreds of kiloparsecs is still unclear. At the same time, it is
still debated if the central AGN may play a role in the formation of mini-halos
by providing the seed of the relativistic particles.
  We aim to investigate these open issues by studying the connection between
thermal and non-thermal components of the intra-cluster medium. We performed a
point-to-point analysis of the radio and the X-ray surface brightness of a
compilation of mini-halos. We find that mini-halos have super-linear scalings
between radio and X-rays, with radio brightness declining more steeply than the
X-ray brightness. This trend is opposite to that generally observed in giant
radio halos, thus marking a possible difference in the physics of the two radio
sources. Finally, using the scalings between radio and X-rays and assuming a
hadronic origin of mini-halos we derive constraints on the magnetic field in
the core of the hosting clusters.
