  We use the macroscopic fluctuation theory to determine the statistics of
large currents in the Kipnis-Marchioro-Presutti (KMP) model in a ring geometry.
About 10 years ago this simple setting was instrumental in identifying a
breakdown of the additivity principle in a class of lattice gases at currents
exceeding a critical value. Building on earlier work, we assume that, for
supercritical currents, the optimal density profile, conditioned on the given
current, has the form of a traveling wave (TW). For the KMP model we find this
TW analytically, in terms of elliptic functions, for any supercritical current
$I$. Using this TW solution, we evaluate, up to a pre-exponential factor, the
probability distribution $P(I)$. We obtain simple asymptotics of the TW and of
$P(I)$ for currents close to the critical current, and for currents much larger
than the critical current. In the latter case we show that $-\ln P (I) \sim
I\ln I$, whereas the optimal density profile acquires a soliton-like shape. Our
analytic results are in a very good agreement with Monte-Carlo simulations and
numerical solutions of Hurtado and Garrido (2011).
