Guillaume Cébron - Université Toulouse III (Paul Sabatier) The large-N behaviour of conditional expectation on random matrices is related to the transition kernels given by the addition of noncommutative random variables in free probability theory. I will discuss different aspects of this relation, with applications including large-N limit of processes on Lie groups, and the study (joint work with Ching Wei Ho) of the q-deformation of the Segal-Bargmann transform.