Benjamini–Schramm Convergence

Updated 31 August 2025

Benjamini–Schramm convergence is a mode of local convergence that rigorously connects small-scale structures in graphs, manifolds, and quantum models to their global properties.
It enables the estimation of key invariants such as graph polynomials, spectral measures, and entropy by analyzing local neighborhood statistics.
The framework extends to diverse settings—including hyperbolic surfaces, random matrices, and simplicial complexes—bridging local geometric data with global analytic and topological outcomes.

Benjamini–Schramm convergence is a foundational concept in modern probability, combinatorics, geometry, and mathematical physics, formalizing a mode of "local convergence" for sequences of graphs, manifolds, or spaces with additional structures. It rigorously connects local geometric statistics to global analytic, spectral, topological, and probabilistic properties, enabling local-to-global transfer principles in graph theory, geometry, arithmetic groups, random matrices, and random geometric models.

1. Foundational Definitions and General Framework

A rooted combinatorial or geometric object (graph, manifold, complex, etc.) is considered in which a "root" (typically a vertex or point) is distinguished. Let $(G_n)$ be a sequence of finite graphs with uniformly bounded degree. The sequence is said to be Benjamini–Schramm (BS) convergent if, for every finite rooted graph $\alpha$ and radius $R > 0$ , the probability $P(G_n, \alpha, R)$ that the $R$ -ball about a uniformly chosen vertex is isomorphic to $\alpha$ converges as $n\to\infty$ (Abért et al., 2012).

In geometric contexts (e.g., compact hyperbolic surfaces $X_n$ ), BS convergence to the simply connected model (the hyperbolic plane $\mathbb{H}$ ) is defined via the measure of "thin regions":

$\lim_{n\to\infty} \frac{\mathrm{Vol}(\{ z \in X_n : \mathrm{InjRad}_{X_n}(z) < R\})}{\mathrm{Vol}(X_n)} = 0$

for every $R > 0$ , so that, outside a vanishing volume, small metric balls in $X_n$ are isometric to balls in $\mathbb{H}$ (Masson et al., 2016).

For quantum graphs and complexes, the definition is analogously extended: the sequence is BS convergent if, for any radius $r$ , the rooted $r$ -ball (with all combinatorial and metric, potential, and boundary data in the quantum case) about uniformly chosen basepoints converges in law to a random rooted limit object (Anantharaman et al., 2020).

An essential feature is focus on "local" (finite-radius) statistics, viewing large objects "from the root," yielding convergence in rooted isomorphism classes for graphs, or in local pointed isometry/Gromov–Hausdorff–Prokhorov sense for geometric structures.

2. Applications to Graph Polynomials and Combinatorics

BS convergence underpins estimability of combinatorial graph invariants in sparse graphs, especially via graph polynomials:

The chromatic measure $\mu_G$ is defined as the uniform measure on the chromatic roots of a graph $G$ . For a BS-convergent sequence $(G_n)$ , the chromatic measures $\mu_{G_n}$ converge in holomorphic moments:

$\lim_{n\to\infty} \int_D f(z) \, d\mu_{G_n}(z)$

for any holomorphic $f$ on a neighborhood $D$ of the roots, and $D$ is a disc bounded by Sokal's result (Abért et al., 2012).

The normalized logarithm of the chromatic polynomial, $t_{G_n}(z) = \frac{1}{|V(G_n)|} \ln \mathrm{ch}_{G_n}(z)$ , converges (locally uniformly) to a real analytic function on $\mathbb{C} \setminus D$ (Abért et al., 2012).
These principles generalize to broad families: any multiplicative, isomorphism-invariant, exponential-type graph polynomial $f$ has holomorphic root moment convergence, and the normalized log converges to a harmonic function outside the root locus (Csikvári et al., 2012).
In matching theory, the matching measure (uniform on roots of the matching polynomial) also converges under BS convergence, and the normalized logarithm of the number of matchings is an estimable parameter; this does not extend in general to perfect matchings except for sequences converging to regular trees, in which case the limit equals Schrijver's exponent (Abért et al., 2014).
The normalized entropy of Eulerian orientations, $\lim_{n\to\infty} \frac{1}{v(G_n)} \ln \varepsilon(G_n)$ , for BS-convergent Eulerian graph sequences, always exists and is determined by local statistics, as proved using Lee–Yang type zero distribution techniques (Bencs et al., 26 Sep 2024).

A plausible implication is that many global combinatorial invariants expressible via local statistics or subgraph counts are estimable parameters in the sense of Elek and Lippner.

3. Generalizations: Trees, Complexes, and Nonstandard Limits

BS convergence encompasses models beyond bounded-degree graphs:

For random trees of increasing size with edge weights proportional to $\prod_{v \in V(T)} d(v)!$ , the local structure (studied via the rooted $k$ -ball) converges to that of a random infinite rooted tree, with explicit description of the limiting law in terms of automorphism counts and factorial degree weight (Deák, 2014).
For subcritical random graphs (e.g., outerplanar/cacti), local limits are infinite chains of connected 2-ended links; compactification of the rooted graph space is introduced to handle high-degree vertices and excluded minors, yielding limits with possibly infinite-degree vertices (Georgakopoulos et al., 2015).
BS convergence admits equivariant generalizations for simplicial complexes with group actions, enabling definitions of random rooted simplicial $G$ -complexes, and continuity of $\ell^2$ -multiplicities in equivariant homology under BS convergence (Kionke et al., 2019).

4. Connections with Geometry, Manifolds, and Symmetry

BS convergence is naturally extended to Riemannian manifolds, locally symmetric spaces, and quantum graphs:

For hyperbolic surfaces, as genus grows, "typical" (Weil–Petersson random) surfaces BS-converge to the hyperbolic plane. Quantitative forms of this are exploited using the Selberg trace formula to control spectral measures (Monk, 2020).
In compact or finite-volume locally symmetric spaces (excluding $\mathbb{H}^3$ ), normalized Betti numbers $b_k(M_n)/\mathrm{vol}(M_n)$ converge along any BS-convergent sequence, and, in the higher-rank case, the limits are the $L^2$ -Betti numbers of the universal cover (Abert et al., 2018).
For translation surfaces of genus $g$ with area $g$ , BS convergence holds: for any radius $r > 0$ , the $r$ -ball neighborhood around a uniformly random point converges in law to that in a Poisson translation plane—an infinite planar surface with singularities forming a Poisson point process of intensity 4 (Bowen et al., 7 Jan 2025).
For genus-0 hyperbolic surfaces with many punctures, local Benjamini–Schramm convergence gives an infinite-volume hyperbolic surface homeomorphic to $\mathbb{R}^2\setminus\mathbb{Z}^2$ (Budd et al., 26 Aug 2025).
In quantum graphs (networks with edge lengths, operators, and boundary conditions), BS convergence is defined via distributions of rooted $r$ -balls; empirical spectral measures of such graphs then converge to expected local spectral measures derived from the BS limit, calculated in terms of Green’s kernels at the root (Anantharaman et al., 2020).
Benjamini–Schramm convergence has been related to the appearance of special substructures: for example, the volume of a Coxeter polytope in hyperbolic space is mostly in a thin neighborhood of the boundary, and limits of such polytopes maintain reflection group structure under BS convergence (Raimbault, 2022).

5. Spectral Theory, Operator Limits, and Group Actions

BS convergence provides a bridge between geometric structure and spectral theory:

In lattices $\Gamma_n$ of locally compact groups $G$ , BS convergence (vanishing measure of points with small stabilizer, or "injectivity radius") is equivalent to spectral convergence (Plancherel measure, normalized traces of invariant operators) under uniform discreteness assumptions (Deitmar, 2018, Deitmar, 24 Jul 2024).
The key correspondence is that, for any algebra $A_2$ of invariant operators (e.g., bounded integral kernels with finite propagation), and under uniform lower bound on injectivity radius ("uniform discreteness"), the normalized traces converge:

$\lim_{n\to\infty} \frac{\mathrm{tr}(A_{\Gamma_n})}{\mathrm{vol}(\Gamma_n \backslash X)} = T(A)$

where $T(A)$ is computed from the kernel’s diagonal in the universal cover (Deitmar, 24 Jul 2024).

Open questions concern the necessity of uniform discreteness, with partial results showing spectral convergence $\implies$ BS convergence in broader settings, but the reverse implication failing generally when local anomalies with vanishing injectivity radius persist in the limit (Deitmar, 24 Jul 2024).

6. Random Matrices, Eigenvalue Distribution, and Random Waves

BS convergence can be employed to understand global spectral properties via local statistics:

Certain random matrix ensembles can be viewed as adjacency operators of weighted graphs. Under BS convergence (of the graph), the empirical spectral measure converges to the local spectral measure of the limit operator at the root. Explicitly, averaging over the limiting rooted graph yields classical distributions, such as the Wigner semicircle and Marchenko–Pastur laws for Hermite and Laguerre β-ensembles, respectively (Andraus, 2017).
In the context of quantum ergodicity, BS convergence for sequences of hyperbolic surfaces with growing volume yields equidistribution of Laplacian eigenfunctions (in fixed spectral windows), as established by wave-propagation approaches using renormalized disk averaging operators and ergodic theorems of Nevo (Masson et al., 2016). This extends concepts of quantum ergodicity from the high-eigenvalue to "large geometry" regime with mild requirements on observables.
The BS formalism also supports a rigorous formulation of Berry’s conjecture and quantum unique ergodicity (QUE) type results for Laplace eigenfunctions via "decorated" spaces. In this framework, local statistics of eigenfunctions on a sequence of manifolds converge to those of Gaussian random waves on the universal cover (Abert et al., 2018).

7. Technical Extensions, Compactification, and Limit Objects

In minor-closed or non-locally-finite settings, BS convergence requires compactification of the space of rooted connected induced subgraphs (RCIS), yielding a compact ultrametric on isomorphism classes. This allows well-defined limits even when infinite-degree vertices or other local anomalies occur (Georgakopoulos et al., 2015).
For periodic orbits of Lie groups, BS convergence is established via explicit test functions $f_R$ detecting large local injectivity radius, revealing that weak- $*$ convergence of orbit measures together with non-collapsing stabilizer conditions suffices for BS convergence in homogeneous and nonhomogeneous settings (Mohammadi et al., 2021).

Summary Table: Key Manifestations and Objects in Benjamini–Schramm Convergence

Domain	Limit Object/Measure	Key Invariant Convergence
Bounded-degree graphs	Prob. rooted infinite graph	Graph polynomials: holomorphic moments, entropy, etc.
Hyperbolic surfaces	Hyperbolic plane, $\mathbb{H}$	Laplacian spectrum, spectral measure
Quantum graphs	Random rooted quantum graph	Empirical spectral measure
Lattices in groups	Universal cover (trivial subgroup)	Plancherel/trace convergence
Simplicial $G$ -complexes	Random rooted $G$ -complex	$\ell^2$ -Betti numbers, $\ell^2$ -multiplicities

References

(Abért et al., 2012): Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs
(Csikvári et al., 2012): Benjamini–Schramm continuity of root moments of graph polynomials
(Deák, 2014): Limits of Random Trees
(Abért et al., 2014): Matchings in Benjamini-Schramm convergent graph sequences
(Georgakopoulos et al., 2015): Limits of subcritical random graphs and random graphs with excluded minors
(Masson et al., 2016): Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces
(Andraus, 2017): Benjamini-Schramm convergence and limiting eigenvalue density of random matrices
(Deitmar, 2018): Benjamini-Schramm and spectral convergence
(Abert et al., 2018): Eigenfunctions and Random Waves in the Benjamini-Schramm limit
(Abert et al., 2018): Convergence of normalized Betti numbers in nonpositive curvature
(Kionke et al., 2019): Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and $\ell^2$ -Multiplicities
(Cerbo et al., 2019): Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence
(Monk, 2020): Benjamini-Schramm convergence and spectrum of random hyperbolic surfaces of high genus
(Anantharaman et al., 2020): Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit
(Mohammadi et al., 2021): Benjamini-Schramm convergence of periodic orbits
(Raimbault, 2022): Coxeter polytopes and Benjamini--Schramm convergence
(Deitmar, 24 Jul 2024): Benjamini-Schramm and spectral convergence II. The non-homogeneous case
(Bencs et al., 26 Sep 2024): Number of Eulerian orientations for Benjamini--Schramm convergent graph sequences
(Bowen et al., 7 Jan 2025): Benjamini-Schramm limits of high genus translation surfaces: research announcement
(Budd et al., 26 Aug 2025): Random punctured hyperbolic surfaces & the Brownian sphere

Conclusion

Benjamini–Schramm convergence provides a robust analytic and probabilistic language for understanding how local statistics determine global phenomena in large combinatorial and geometric structures. Its implications range from combinatorial enumeration and spectral theory to topology, geometry, and mathematical physics, with ongoing developments in nonhomogeneous, quantum, and higher-rank settings. The theory enables explicit computation of asymptotic invariants, provides tools for analyzing phase transitions and universality, and continues to evolve through new applications in geometric group theory, moduli spaces, and random geometry.