# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Minimum weight disk triangulations and fillingsHTML articles powered by AMS MathViewer

by Itai Benjamini, Eyal Lubetzky and Yuval Peled
Trans. Amer. Math. Soc. 374 (2021), 3265-3287 Request permission

## 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 $\frac 14\log n + O_{P}(\sqrt {\log n})$ 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.
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• Itai Benjamini
• Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
• MR Author ID: 311800
• Email: itai.benjamini@weizmann.ac.il
• Eyal Lubetzky
• Affiliation: Courant Institute, New York University, 251 Mercer Street, New York, New York 10012
• MR Author ID: 787713
• ORCID: 0000-0002-2281-3542
• Email: eyal@courant.nyu.edu
• Yuval Peled
• Affiliation: Courant Institute, New York University, 251 Mercer Street, New York, New York 10012
• MR Author ID: 1064288
• Email: yuval.peled@courant.nyu.edu
• Received by editor(s): January 27, 2020
• Received by editor(s) in revised form: July 23, 2020
• Published electronically: February 12, 2021
• Additional Notes: The second author was supported in part by NSF grant DMS-1812095.