Small violations of Bell inequalities for multipartite pure random states*

For any finite number of parts, measurements and outcomes in a Bell scenario we estimate the probability of random N-qudit pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the local Hilbert space dimension goes to infinity.