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

DOI10.1016/j.spa.2016.02.009zbMath1384.60016arXiv1407.6309OpenAlexW1697426710MaRDI QIDQ737166

Anita Winter, Wolfgang Löhr, Siva R. Athreya

Publication date: 8 August 2016

Published in: Stochastic Processes and their Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1407.6309

zbMATH Keywords

metric measure spacescoding trees by excursionsfull support assumptionGromov-Hausdorff-vagueGromov-Hausdorff-weakGromov-prohorov metricGromov-vague topologyGromov-weakKallenberg treelower mass-bound property

Mathematics Subject Classification ID

Probability measures on topological spaces (60B05) Random graphs (graph-theoretic aspects) (05C80) Convergence of probability measures (60B10) Branching processes (Galton-Watson, birth-and-death, etc.) (60J80) Probability theory on algebraic and topological structures (60B99)

Related Items

Scaling limits of stochastic processes associated with resistance forms ⋮ Metrization of the Gromov-Hausdorff (-Prokhorov) topology for boundedly-compact metric spaces ⋮ Infinite stable looptrees ⋮ Universality for critical heavy-tailed network models: metric structure of maximal components ⋮ Scaling limit for the ant in a simple high-dimensional labyrinth ⋮ Branching trees I: concatenation and infinite divisibility ⋮ The Brownian Web as a random \(\mathbb{R} \)-tree ⋮ Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees ⋮ Convergence of bi-measure \(\mathbb{R}\)-trees and the pruning process ⋮ A unified framework for generalizing the Gromov-Hausdorff metric ⋮ The GHP scaling limit of uniform spanning trees in high dimensions ⋮ Convergence of continuous stochastic processes on compact metric spaces converging in the Lipschitz distance ⋮ On trees invariant under edge contraction ⋮ The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs ⋮ Pathwise construction of tree-valued Fleming-Viot processes ⋮ The random conductance model with heavy tails on nested fractal graphs ⋮ Curvature-dimension condition meets Gromov's \(n\)-volumic scalar curvature ⋮ Self-similar real trees defined as fixed points and their geometric properties ⋮ Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree ⋮ Spaces of algebraic measure trees and triangulations of the circle ⋮ Global lower mass-bound for critical configuration models in the heavy-tailed regime

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces
• The stable trees are nested
• Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces
• Dirichlet forms and symmetric Markov processes.
• Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees
• Equivalence of Gromov-Prohorov- and Gromov's \(\underline{\square}_\lambda\)-metric on the space of metric measure spaces
• Persistence criteria for a class of critical branching particle systems in continuous time
• Convergence in distribution of random metric measure spaces (\(\Lambda \)-coalescent measure trees)
• Convergence of bi-measure \(\mathbb{R}\)-trees and the pruning process
• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator
• Subdiffusive behavior of random walk on a random cluster
• A limit theorem for the contour process of conditioned Galton-Watson trees
• The scaling limit of the minimum spanning tree of the complete graph
• Tree-valued resampling dynamics martingale problems and applications
• Exit times for an increasing Lévy tree-valued process
• The continuum random tree. III
• The topological structure of scaling limits of large planar maps
• Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry
• Genealogy of catalytic branching models
• Invariance principle for variable speed random walks on trees
• Volume growth and heat kernel estimates for the continuum random tree
• Subtree prune and regraft: a reversible real tree-valued Markov process
• Random walk on the incipient infinite cluster on trees
• Exceptionally small balls in stable trees
• Tessellations of random maps of arbitrary genus
• One-dimensional Brownian motion and the three-dimensional Bessel process
• Stability of Critical Cluster Fields
• Topological pattern recognition for point cloud data
• The CRT is the scaling limit of unordered binary trees
• Optimal Transport
• Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu