  Earth-mass bodies are expected to undergo Type I migration directed either
inward or outward depending on the thermodynamical state of the protoplanetary
disc. Zones of convergent migration exist where the Type I torque cancels out.
We study the evolution of multiple protoplanets of a few Earth masses embedded
in a non-isothermal protoplanetary disc. The protoplanets are located in the
vicinity of a convergence zone located at the transition between two different
opacity regimes. Inside the convergence zone, Type I migration is directed
outward and outside the zone migration is directed inward. We used a grid-based
hydrodynamical code that includes radiative effects. We performed simulations
varying the initial number of embryos and tested the effect of including
stochastic forces to mimic the effects resulting from turbulence. We also
performed N-body runs calibrated on hydrodynamical calculations to follow the
evolution on Myr timescales. For a small number of initial embryos (N = 5-7)
and in the absence of stochastic forcing, the population of protoplanets
migrates convergently toward the zero-torque radius and forms a stable resonant
chain that protects embryos from close encounters. In systems with a larger
initial number of embryos, or in which stochastic forces were included, these
resonant configurations are disrupted. This in turn leads to the growth of
larger cores via a phase of giant impacts, after which the system settles to a
new stable resonant configuration. Giant planets cores with masses of 10 Earth
masses formed in about half of the simulations with initial protoplanet masses
of m_p = 3 Earth masses but in only 15% of simulations with m_p = 1 Earth mass.
This suggests that if ~2-3 Earth mass protoplanets can form in less than ~1
Myr, convergent migration and giant collisions can grow giant planet cores at
Type I migration convergence zones.
