The Gromov-weak topology is a topology on the space of equivalence classes of marked metric measure spaces (mmm-spaces) that captures convergence of the metric and measure structure simultaneously. It was introduced by Greven, Pfaffelhuber, and Winter (2009) and is central to the study of scaling limits of random combinatorial structures.
A marked metric measure space (mmm-space) is a triple X = (X, d, mu) where (X, d) is a separable complete metric space and mu is a finite measure on X x E, where E is a fixed Polish “mark space.” Two mmm-spaces are equivalent if there is a measure-and-mark-preserving isometry between their supports.
Convergence in the Gromov-weak topology is defined via monomials: functionals Phi of order k >= 1 of the form
Phi(X) = integral phi(d(x), e) mu^k(dx, de)
where phi is bounded continuous on R_+^{k x k} x E^k. A sequence (X_n) converges Gromov-weakly to X if Phi(X_n) → Phi(X) for all monomials Phi.
The Gromov-weak topology is metrised by the Gromov-Prokhorov distance:
d_GP(X, X’) = inf_{Z, iota, iota’} d_P^Z(iota_* mu, iota’_* mu’)
where the infimum is over all complete separable metric spaces (Z, d_Z) and isometric embeddings iota: X → Z, iota’: X’ → Z.
Foutel-Rodier (2024) introduced vague convergence in this setting by restricting to test functionals whose support is bounded away from the zero mmm-space 0 (the trivial space with null sampling measure). This is particularly natural for:
- Branching processes: where the population can go extinct (reaching 0) and one wants to study the genealogy conditional on survival
- Infinite measures: like the excursion measure of the Brownian continuum random tree
- Point processes: where locally finite point measures form the natural state space
Key Details
- Underlying space: equivalence classes X of mmm-spaces (X, d, mu)
- Convergence: via monomials encoding k-point metric/mark correlations
- Metrisation: Gromov-Prokhorov distance d_GP
- Vague version: test functionals with support bounded away from 0 (Foutel-Rodier 2024)
- Method of moments: convergence of moments of order k >= 1 + Carleman condition ⇒ vague convergence (Foutel-Rodier, Theorem 4.2)
- Applications: scaling limits of random graphs, trees, genealogies of branching processes, continuum random tree