Sampling quantum nonlocal correlations with high probability

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form $\gamma =(⟨u_i,v_j⟩{)}_{i,j=1}^n$, 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 $\alpha =\frac{m}{n}$, 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 ${\alpha }_0>0$ such that if $\alpha \le {\alpha }_0$, $\gamma$ is nonlocal with probability tending to $1$ as $n\to \mathrm{\infty }$, while for $\alpha >2$, $\gamma$ is local with probability tending to $1$ as $n\to \mathrm{\infty }$.