The smooth degree d complex curves of are Riemann surfaces of the same genus . If we equip them with the restriction of the ambient metric and choose them at random, what can be say about the length of their systole? I will explain that the probability that the systole is of the order is bounded from below by a uniform positive constant. This gives an partial analogous result to Mirzakhani's theorem on random hyperbolic curves. If I have time, I will explain that in higher dimensions, these probabilistic arguments provide a new deterministic result about Lagrangian submanifolds and the topology of complex projective hypersurfaces.