The small-scale limit of magnitude and the one-point property

TL;DR

The paper investigates the small-scale behavior of magnitude for finite metric spaces and the one-point property, showing that generic finite spaces have the one-point property while providing a five-point counterexample. It develops a formal magnitude framework and analyzes joins of finite homogeneous spaces to control the small-scale limit, proving that the limit $\lim_{t \to 0}|tX|$ can be any real value $\ge 1$. The results illuminate the variability of magnitude at small scales and its stability under Gromov–Hausdorff-type perturbations, with Willerton's example and a broad join-construction illustrating the range of possible limits. This establishes that the failure of the one-point property, while rare, can be arbitrarily severe in terms of the small-scale limit.

Abstract

The magnitude of a metric space is a real-valued function whose parameter controls the scale of the metric. A metric space is said to have the one-point property if its magnitude converges to 1 as the space is scaled down to a point. Not every finite metric space has the one-point property: to date, exactly one example has been found of a finite space for which the property fails. Understanding the failure of the one-point property is of interest in clarifying the interpretation of magnitude and its stability with respect to the Gromov--Hausdorff topology. We prove that the one-point property holds generically for finite metric spaces, but that when it fails, the failure can be arbitrarily bad: the small-scale limit of magnitude can take arbitrary real values greater than 1.

Figures (4)

• Figure 1: The magnitude function of a three-point space (note the logarithmic scale). When $t$ is very small, $|tX|$ is close to 1, as though recognizing that 'from far away' $X$ looks like a one-point space. As $t$ increases, the cluster of two points on the left in $X$ becomes distinguishable from the point on the right and for a while $|tX|$ lingers close to 2. As $t \to \infty$, the magnitude function converges to the cardinality of $X$: 'from close up' it is clear that $X$ is a three-point space. This is Example 6.4.6 in LeinsterEntropy2021, originally due to Willerton.
• Figure 2: A six-point space without the one-point property. Solid black lines represent distances equal to 1. The distance between each non-adjacent pair of vertices is $2$.
• Figure 3: The join $X*Y$. Solid black lines represent distances equal to 1. In principle one can consider joining distances other than 1, but for us this would not offer extra generality, since we always consider $X \ast Y$ along with its rescalings $t(X \ast Y)$ for $t \in [0, \infty)$.
• Figure 4: A five-point space without the one-point property. Solid black lines represent distances equal to 1.

Theorems & Definitions (18)

• Remark 2.1
• Example 2.2
• Theorem 2.3
• proof
• Remark 2.4
• Example 3.1
• Definition 3.2
• Theorem 3.3
• proof
• Example 3.4: A five-point space without the one-point property