Minimum weight disk triangulations and fillings

TL;DR

This paper analyzes the asymptotic behavior of the minimum total weight of disk triangulations with random triangle weights, revealing precise growth rates and distributional limits, including convergence to a shifted Gumbel law.

Contribution

It provides explicit asymptotic formulas for the minimum weight of disk triangulations with random weights and characterizes the distributional limits, including for canonical triangulations.

Findings

Minimum weight scales as c1 (log n)/√n + c2 (log log n)/√n + Y_n/√n.

The random variable Y_n is tight and converges weakly to a shifted Gumbel distribution.

Minimum weights of homological and homotopical fillings coincide with the minimal disk triangulation with high probability.

Abstract

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.