We study the minimum total weight of a disk triangulation using vertices out of $\{1,\ldots,n\}$, where the boundary is the triangle $(123)$ and the $\binom{n}3$ triangles have independent weights, e.g. $\mathrm{Exp}(1)$ or $\mathrm{U}(0,1)$. We show that for explicit constants $c_1,c_2>0$, this minimum is $c_1 \frac{\log n}{\sqrt n} + c_2 \frac{\log\log n}{\sqrt n} + \frac{Y_n}{\sqrt n}$ where the random variable $Y_n$ is tight, and it is attained by a triangulation that consists of $\frac14\log n + O_P(\sqrt{\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.

中文翻译：

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最小重量圆盘三角剖分和填充

我们使用 $\{1,\ldots,n\}$ 中的顶点来研究磁盘三角剖分的最小总权重，其中边界是三角形 $(123)$，$\binom{n}3$ 三角形有独立权重，例如 $\mathrm{Exp}(1)$ 或 $\mathrm{U}(0,1)$。我们表明，对于显式常量 $c_1,c_2>0$，这个最小值是 $c_1 \frac{\log n}{\sqrt n} + c_2 \frac{\log\log n}{\sqrt n} + \frac {Y_n}{\sqrt n}$ 其中随机变量 $Y_n$ 是紧的，它是通过由 $\frac14\log n + O_P(\sqrt{\log n}) $ 顶点组成的三角剖分获得的。此外，对于规范的磁盘三角剖分，由于没有内三角形包含所有顶点，除了 $O(1)$ 之外，最小权重具有上述形式，其中 $Y_n$ 定律弱收敛到一个 shift~Gumbel。此外，我们证明，很有可能，