What does a typical metric space look like?

Abstract

The collection $\mathcal{M}_n$ of all metric spaces on $n$ points whose diameter is at most $2$ can naturally be viewed as a compact convex subset of $\mathbb{R}^{\binom{n}{2}}$, known as the metric polytope. In this paper, we study the metric polytope for large $n$ and show that it is close to the cube $[1,2]^{\binom{n}{2}} \subseteq \mathcal{M}_n$ in the following two senses. First, the volume of the polytope is not much larger than that of the cube, with the following quantitative estimates: \[ \left(\tfrac{1}{6}+o(1)\right)n^{3/2} \le \log \mathrm{Vol}(\mathcal{M}_n)\le O(n^{3/2}). \] Second, when sampling a metric space from $\mathcal{M}_n$ uniformly at random, the minimum distance is at least $1 - n^{-c}$ with high probability, for some $c > 0$. Our proof is based on entropy techniques. We discuss alternative approaches to estimating the volume of $\mathcal{M}_n$ using exchangeability, Szemer\'edi's regularity lemma, the hypergraph container method, and the K\H{o}v\'ari--S\'os--Tur\'an theorem.