  We consider the expected violations of Bell inequalities from random pure
states. More precisely, we focus on a slightly generalised version of the CGLMP
inequality, which concerns Bell experiments of two parties, two measurement
options and N outcomes and analyse their expected quantum violations from
random pure states for varying N, assuming the conjectured optimal measurement
operators. It is seen that for small N the Bell inequality is not violated on
average, while for larger N it is. Both ensembles of unstructured as well as
structured random pure states are considered. Using techniques from random
matrix theory this is obtained analytically for small and large N and
numerically for intermediate N. The results show a beautiful interplay of
different aspects of random matrix theory, ranging from the Marchenko-Pastur
distribution and fixed-trace ensembles to the O(n) model.
