Minimum weight disk triangulations and fillings

We study the minimum total weight of a disk triangulation using vertices out of {1,..., n}, where the boundary is the triangle (123) and the (ⁿ₃ ) triangles have independent weights, e.g. Exp(1) or U(0, 1). We show that for explicit constants c1, c2 > 0, this minimum is c₁^log √_nⁿ + c₂^log√^log_nⁿ + √^Yn_n, where the random variable Y_n is tight, and it is attained by a triangulation that consists of ¹₄ log n+O_p(√log n) vertices. Moreover, for disk triangulations that are canonical, in that no inner triangle contains all but O(1) of the vertices, the minimum weight has the above form with the law of Y_n converging weakly to a shifted Gumbel. In addition, we prove that, with high probability, the minimum weights of a homological filling and a homotopical filling of the cycle (123) are both attained by the minimum weight disk triangulation.