Hyperbolicity, slimness, and minsize, on average

Abstract

A metric space $(X,d)$ is said to be $\delta$-hyperbolic if $d(x,y)+d(z,w)$ is at most $\max(d(x,z)+d(y,w), d(x,w)+d(y,z))$ by $2 \delta$. A geodesic space is $\delta$-slim if every geodesic triangle $\Delta(x,y,z)$ is $\delta$-slim. It is well-established that the notions of $\delta$-slimness, $\delta$-hyperbolicity, $\delta$-thinness and similar concepts are equivalent up to a constant factor. In this paper, we investigate these properties under an average-case framework and reveal a surprising discrepancy: while $\mathbb{E}\delta$-slimness implies $\mathbb{E}\delta$-hyperbolicity, the converse does not hold. Furthermore, similar asymmetries emerge for other definitions when comparing average-case and worst-case formulations of hyperbolicity. We exploit these differences to analyze the random Gaussian distribution in Euclidean space, random $d$-regular graph, and the random Erd\H{o}s-R\'enyi graph model, illustrating the implications of these average-case deviations.