TY - JOUR

T1 - Minimum weight disk triangulations and fillings

AU - Benjamini, Itai

AU - Lubetzky, Eyal

AU - Peled, Yuval

N1 - Funding Information:
Received by the editors January 27, 2020, and, in revised form, July 23, 2020. 2020 Mathematics Subject Classification. Primary 57K20, 60C05. The second author was supported in part by NSF grant DMS-1812095.
Publisher Copyright:
© 2021 American Mathematical Society

PY - 2021/5

Y1 - 2021/5

N2 - 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 c1log √nn + c2log√lognn + √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.

AB - 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 c1log √nn + c2log√lognn + √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.

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U2 - 10.1090/tran/8255

DO - 10.1090/tran/8255

M3 - Article

AN - SCOPUS:85104241913

VL - 374

SP - 3265

EP - 3287

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -