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

The relentless variability of Mrk 421 from the TeV to the radio