  The question of how Bell nonlocality behaves in bipartite systems of higher
dimensions is addressed. By employing the probability of violation of local
realism under random measurements as the figure of merit, we investigate the
nonlocality of entangled qudits with dimensions ranging from $d=2$ to $d=7$. We
proceed in two complementary directions. First, we study the specific Bell
scenario defined by the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality.
Second, we consider the nonlocality of the same states under a more general
perspective, by directly addressing the space of joint probabilities (computing
the frequencies of behaviours outside the local polytope). In both approaches
we find that the nonlocality decreases as the dimension $d$ grows, but in quite
distinct ways. While the drop in the probability of violation is exponential in
the CGLMP scenario, it presents, at most, a linear decay in the space of
behaviours. Furthermore, in both cases the states that produce maximal numeric
violations in the CGLMP inequality present low probabilities of violation in
comparison with maximally entangled states, so, no anomaly is observed.
Finally, the nonlocality of states with non-maximal Schmidt rank is
investigated.
