Minimal surfaces are surfaces which locally minimize the area, and are a classical topic in Differential Geometry. This talk is going to be about minimal surfaces in Hilbert spheres. I will motivate their study by explaining how these objects provide a new way to understand fundamental results like Mostow's rigidity theorem or Thurston-Perelman's hyperbolization theorem. Then, I will explain a surprising connection between minimal surfaces in spheres and Probability: from two permutations, one can naturally associate a closed 2-dimensional minimal surface S in a Euclidean sphere, which we think of as the "shape" of the pair of permutations. The main result is that if those two permutations are chosen uniformly at random, then with high probability, S is asymptotically close to another well-studied shape, the hyperbolic plane.