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.