  The origin of the gamma-ray emission of the blazar Mrk 421 is still a matter
of debate. We used 5.5 years of unbiased observing campaign data, obtained
using the FACT telescope and the Fermi LAT detector at TeV and GeV energies,
the longest and densest so far, together with contemporaneous multi-wavelength
observations, to characterise the variability of Mrk 421 and to constrain the
underlying physical mechanisms. We studied and correlated light curves obtained
by ten different instruments and found two significant results. The TeV and
X-ray light curves are very well correlated with a lag of <0.6 days. The GeV
and radio (15 Ghz band) light curves are widely and strongly correlated.
Variations of the GeV light curve lead those in the radio. Lepto-hadronic and
purely hadronic models in the frame of shock acceleration predict proton
acceleration or cooling timescales that are ruled out by the short variability
timescales and delays observed in Mrk 421. Instead the observations match the
predictions of leptonic models.
