The gap between Gromov-vague and Gromov–Hausdorff-vague topology

Abstract

Abstract. In [ALW15] an invariance principle is stated for a class of strong Markov processes on tree-like metric measure spaces. It is shown that if the underlying spaces converge Gromov vaguely, then the processes converge in the sense of finite dimensional distributions. Further, if the underlying spaces converge Gromov-Hausdorff vaguely, then the processes converge weakly in path space. In this paper we systematically introduce and study the Gromov-vague and the Gromov-Hausdorff-vague topology on the space of equivalence classes of metric boundedly finite measure spaces. The latter topology is closely related to the Gromov-Hausdorff-Prohorov metric which is defined on different equivalence classes of metric measure spaces.We explain the necessity of these two topologies via several examples, and close the gap between them. That is, we show that convergence in Gromovvague topology implies convergence in Gromov-Hausdorff-vague topology if and only if the so-called lower mass-bound property is satisfied. Furthermore, we prove and disprove Polishness of several spaces of metric measure spaces in the topologies mentioned above (summarized in Figure 1).As an application, we consider the Galton-Watson tree with critical offspring distribution of finite variance conditioned to not get extinct, and construct the so-called Kallenberg-Kesten tree as the weak limit in GromovHausdorff-vague topology when the edge length are scaled down to go to zero.

Metrization of the Gromov–Hausdorff (-Prokhorov) topology for boundedly-compact metric spaces

“…As mentioned in Remark 3.20, this method gives more quantitative bounds in the arguments. Despite some similarities in the arguments (which are also similar to those of [1] and other literature that use the localization method to generalize the Gromov-Hausdorff metric), the results of [5] do not give a metrization of the Gromov-Hausdorff-Prokhorov topology on M * and do not imply its Polishness. Also, the Strassen-type theorems (Theorems 2.1 and 3.6) and the results based on them are new in the present paper.…”

“…A second proof for Polishness of M ′ * can be given by Alexandrov's theorem by using Polishness of M * and by showing that M ′ * corresponds to a G δ subspace of M * (given n > 0, it can be shown that the set of (X, o, µ) ∈ M * such that ∀x ∈ X : µ(B 1/n (x)) > 0 is open). The method of [5] is different from the present paper. It defines the metric on M ′ * by modifying (3.13) (since (3.13) does not make M ′ * complete), but the definition in the present paper is based on the notion of PMM-subspaces, Lemma 3.12 and (3.11).…”

Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees

“…We first prove convergence of the object of interest in the Gromov‐weak topology, essentially showing that for each fixed k ≥ 2, the distance matrix constructed from k randomly sampled vertices converges in distribution to the distance matrix constructed from k points appropriately sampled from the limiting structure. This result, coupled with a global lower mass bound implies via general theory that convergence occurs in the stronger Gromov‐Hausdorff‐Prokhorov sense. In the context of critical random graphs, this technique was first used in to analyze the so‐called rank‐one critical inhomogeneous random graph.…”

Scaling Limit for the Ant in High‐Dimensional Labyrinths

“…The notion for convergence is the one given by the spatial Gromov-Hausdorff-vague metric. For details, see [7,27].…”