Minimum weight disk triangulations and fillings

We study the minimum total weight of a disk triangulation using vertices out of { 1 , … , n } \{1,\ldots ,n\} , where the boundary is the triangle ( 123 ) (123) and the ( n 3 ) \binom {n}3 triangles have independent weights, e.g. E x p ( 1 ) \mathrm {Exp}(1) or U ( 0 , 1 ) \mathrm {U}(0,1) . We show that for explicit constants c 1 , c 2 > 0 c_1,c_2>0 , this minimum is c 1 log ⁡ n n + c 2 log ⁡ log ⁡ n n + Y n n c_1 \frac {\log n}{\sqrt n} + c_2 \frac {\log \log n}{\sqrt n} + \frac {Y_n}{\sqrt n} , where the random variable Y n Y_n