Concentration phenomena in the geometry of Bell correlations

Bell's theorem shows that local measurements on entangled states give rise to correlations incompatible with local hidden variable models. The degree of quantum nonlocality is not maximal though, as there are even more nonlocal theories beyond quantum theory still compatible with the nonsignalling principle. In spite of decades of research, we still have a very fragmented picture of the whole geometry of these different sets of correlations. Here we employ both analytical and numerical tools to ameliorate that. First, we identify two different classes of Bell scenarios where the nonsignalling correlations can behave very differently: in one case, the correlations are generically quantum and nonlocal while on the other quite the opposite happens as the correlations are generically classical and local. Second, by randomly sampling over nonsignalling correlations, we compute the distribution of a nonlocality quantifier based on the trace distance to the local set. With that, we conclude that the nonlocal correlations can show concentration phenomena: their distribution is peaked at a distance from the local set that increases both with the number of parts or measurements being performed.
Comment: Minor modifications. Still with 13 pages, 8 figures, 5 tables. Comments are welcome!