Context : Rapid gas accretion onto gas giants requires the prior formation of \sim 10 M _ { \oplus } cores , and this presents a continuing challenge to planet formation models .
Recent studies of oligarchic growth indicate that in the region around 5 AU growth stalls at \sim 2 M _ { \oplus } .
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 .
These “ convergence zones ” may represent ideal sites for the growth of giant planet cores by giant impacts between Earth-mass embryos .

Aims : 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 .

Methods : 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 disc turbulence .
We also performed N-body runs calibrated on hydrodynamical calculations to follow the evolution on Myr timescales .

Results : 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 between protoplanets , after which the system settles to a new stable resonant configuration .
Giant planets cores with masses \geq 10 M _ { \oplus } formed in about half of the simulations with initial protoplanet masses of m _ { p } = 3 M _ { \oplus } but in only 15 % of simulations with m _ { p } = 1 M _ { \oplus } , even with the same total solid mass .

Conclusions : If 2 - 3 M _ { \oplus } protoplanets can form in less than \sim 1 Myr , convergent migration and giant collisions can grow giant planet cores at Type I migration convergence zones .
This process can happen fast enough to allow for a subsequent phase of rapid gas accretion during the disc ’ s lifetime .