Random triangulations of the d-sphere with minimum volume

Abstract

We study a higher-dimensional analogue of the {Random Travelling Salesman Problem}: let the complete $d$-dimensional simplicial complex $K_n^d$ 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 ${\mathbb{S}}^d$, 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 ${\mathbb{S}}^d$ 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 ${\mathbb{S}}^d$.

25 pages, 1 figure