Minimum weight disk triangulations and fillings

Itai Benjamini, Eyal Lubetzky, Yuval Peled

Research output: Contribution to journalArticlepeer-review

Abstract

We study the minimum total weight of a disk triangulation using vertices out of {1,..., n}, where the boundary is the triangle (123) and the (n3 ) triangles have independent weights, e.g. Exp(1) or U(0, 1). We show that for explicit constants c1, c2 > 0, this minimum is c1lognn + c2loglognn + √Ynn, where the random variable Yn is tight, and it is attained by a triangulation that consists of 14 log n+Op(√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 Yn 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.

Original languageEnglish (US)
Pages (from-to)3265-3287
Number of pages23
JournalTransactions of the American Mathematical Society
Volume374
Issue number5
DOIs
StatePublished - May 2021

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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