Minimal surfaces in spheres, which are invariant by a group of symmetries, are closely related to group theory, hyperbolic geometry and geometric topology. In this talk, I will discuss a new connection with random matrices. As we will explain, from two permutations, one can associate a 2d minimal surface in a Euclidean sphere. The main property is a probabilistic rigidity phenomenon: if the two permutations are chosen uniformly at random, then with high probability, the minimal surface has a geometry close to that of the hyperbolic plane.