  In 2012 Markarian 421 underwent the largest flare ever observed in this
blazar at radio frequencies. In the present study, we start exploring this
unique event and compare it to a less extreme event in 2013. We use 15 GHz
radio data obtained with the Owens Valley Radio Observatory 40-m telescope, 95
GHz millimeter data from the Combined Array for Research in Millimeter-Wave
Astronomy, and GeV gamma-ray data from the Fermi Gamma-ray Space Telescope. The
radio light curves during the flaring periods in 2012 and 2013 have very
different appearances, both in shape and peak flux density. Assuming that the
radio and gamma-ray flares are physically connected, we attempt to model the
most prominent sub-flares of the 2012 and 2013 activity periods by using the
simplest possible theoretical framework. We first fit a one-zone synchrotron
self-Compton (SSC) model to the less extreme 2013 flare and estimate parameters
describing the emission region. We then model the major gamma-ray and radio
flares of 2012 using the same framework. The 2012 gamma-ray flare shows two
distinct spikes of similar amplitude, so we examine scenarios associating the
radio flare with each spike in turn. In the first scenario, we cannot explain
the sharp radio flare with a simple SSC model, but we can accommodate this by
adding plausible time variations to the Doppler beaming factor. In the second
scenario, a varying Doppler factor is not needed, but the SSC model parameters
require fine tuning. Both alternatives indicate that the sharp radio flare, if
physically connected to the preceding gamma-ray flares, can be reproduced only
for a very specific choice of parameters.
