  We investigate how very large populations are able to reach a global
consensus, out of local "microscopic" interaction rules, in the framework of a
recently introduced class of models of semiotic dynamics, the so-called Naming
Game. We compare in particular the convergence mechanism for interacting agents
embedded in a low-dimensional lattice with respect to the mean-field case. We
highlight that in low-dimensions consensus is reached through a coarsening
process which requires less cognitive effort of the agents, with respect to the
mean-field case, but takes longer to complete. In 1-d the dynamics of the
boundaries is mapped onto a truncated Markov process from which we analytically
computed the diffusion coefficient. More generally we show that the convergence
process requires a memory per agent scaling as N and lasts a time N^{1+2/d} in
dimension d<5 (d=4 being the upper critical dimension), while in mean-field
both memory and time scale as N^{3/2}, for a population of N agents. We present
analytical and numerical evidences supporting this picture.
