Sampling quantum nonlocal correlations with high probability

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form $γ=(\langle u_i,v_j\rangle)_{i,j=1}^n$, where the vectors $u_i$ and $v_j$ are in the unit ball of a real Hilbert space. In this work we study the probability of the nonlocal nature of these correlations as a function of $α=\frac{m}{n}$, where the previous vectors are sampled according to the Haar measure in the unit sphere of $\mathbb R^m$. In particular, we prove the existence of an $α_0>0$ such that if $α\leq α_0$, $γ$ is nonlocal with probability tending to $1$ as $n\rightarrow \infty$, while for $α> 2$, $γ$ is local with probability tending to $1$ as $n\rightarrow \infty$.

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