Random triangulations of the d-sphere with minimum volume
Preprint, 2024

We study a higher-dimensional analogue of the Random Travelling Salesman Problem: let the complete d-dimensional simplicial complex K^d_n on n vertices be equipped with i.i.d. volumes on its facets, uniformly random in [0,1]. What is the minimum volume M_{n,d} of a sub-complex homeomorphic to the d-dimensional sphere 𝕊_d, containing all vertices? We determine the growth rate of M_{n,2}, and prove that it is well-concentrated. For d>2 we prove such results to the extent that current knowledge about the number of triangulations of 𝕊d allows.
We remark that this can be thought of as a model of random geometry in the spirit of Angel & Schramm's UIPT, and provide a generalised framework that interpolates between our model and the uniform random triangulation of 𝕊_d.

Författare

Joel Danielsson

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Agelos Georgakopoulos

The University of Warwick

John Haslegrave

Lancaster University

Kategorisering

Ämneskategorier (SSIF 2011)

Geometri

Sannolikhetsteori och statistik

Diskret matematik

Identifikatorer

DOI

10.48550/arXiv.2409.00235

Mer information

Skapat

2024-09-04