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.
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