  Migration phenomena and all the related issues, like integration of different
social groups, are intrinsically complex problems since they strongly depend on
several competitive mechanisms as economic factors, cultural differences and
many others. By identifying a few essential assumptions, and using the
statistical mechanics of complex systems, we propose a novel quantitative
approach that provides a minimal theory for those phenomena. We show that the
competitive interactions in decision making among a population of $N$ host
citizens and $P$ immigrants, a bi-partite spin-glass, give rise to a "social
consciousness" inside the host community in the sense of the associative memory
of neural networks. The theory leads to a natural quantitative definition of
migrant's "integration" inside the community. From the technical point of view
this minimal picture assumes, as control parameters, only general notions like
strength of the random interactions, the ratio among the two party sizes and
the cultural influence. Few steps forward, toward more refined models, which
include some structure on the random interaction topology (as dilution to avoid
the plain mean field approach) and correlations of experiences felt among the
two parties (biasing the distribution of the coupling) are discussed at the
end, where we show the robustness of our approach.
