Random triangulations of the d-sphere with minimum volume

TL;DR

This paper investigates the minimal volume of a subcomplex homeomorphic to a d-sphere within a randomly weighted simplicial complex, extending concepts of random geometry and triangulation enumeration.

Contribution

It determines the growth rate and concentration of the minimal volume for 2-spheres and extends partial results to higher dimensions based on triangulation counts.

Findings

Growth rate of minimal volume for d=2 established

Concentration results for d=2 proven

Partial results for higher dimensions based on triangulation counts

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 $\mathbb{S}^d$.