  The radio mini halos (MH) observed in relaxed clusters probe the presence of
relativistic particles on scales of hundreds of kpc, beyond the scales directly
influenced by the central AGN, but the nature of the mechanism that produces
the relativistic electrons is still debated. In this work we explore the
connection between thermal and non-thermal components of the ICM in a sample of
MH and we study its implications for hadronic models for the origin of the
relativistic electrons. We studied the thermal and non-thermal connection by
carrying out a point-to-point comparison of the radio and the X-ray surface
brightness. We extended the method generally applied to giant radio halos by
considering the effects of a grid randomly generated through a Monte Carlo
chain. Contrary to what is generally observed for giant radio halos, we find
that the mini-halos in our sample have super-linear scaling between radio and
X-rays, which suggests a peaked distribution of relativistic electrons and
magnetic field. We used the radio and X-ray correlation to constrain the
physical parameters of a hadronic model and we compared the model predictions
with current observations. Specifically, we focus on a model where cosmic rays
are injected by the central AGN and they generate secondaries in the ICM, and
we assume that the role of turbulent re-acceleration is negligible. This model
allows us to constrain the AGN cosmic ray luminosity in the range
$\sim10^{44-46}$ erg s$^{-1}$ and the central magnetic field in the range 10-40
$\mu$G. The resulting $\gamma$-ray fluxes calculated assuming these model
parameters do not violate the upper limits on $\gamma$-ray diffuse emission set
by the Fermi-LAT telescope. Further studies are now required to explore the
consistency of these large magnetic fields with Faraday rotation studies and to
study the interplay between the secondary electrons and the ICM turbulence.
