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- W2983207471 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 $c_1 frac{log n}{sqrt n} + c_2 frac{loglog n}{sqrt n} + frac{Y_n}{sqrt n}$ where the random variable $Y_n$ is tight, and it is attained by a triangulation that consists of $frac14log 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." @default.
- W2983207471 created "2019-11-22" @default.
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- W2983207471 date "2019-11-06" @default.
- W2983207471 modified "2023-10-18" @default.
- W2983207471 title "Minimum weight disk triangulations and fillings" @default.
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- W2983207471 doi "https://doi.org/10.48550/arxiv.1911.02569" @default.
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