All spatial random graphs with weak long-range effects have chemical distance comparable to Euclidean distance

This note provides a sufficient condition for linear lower bounds on chemical distances (compared to the Euclidean distance) in general spatial random graphs. The condition is based on the scarceness of long edges in the graph and weak correlations at large distances and is valid for all translation invariant and locally finite graphs that fulfil these conditions. The proof is based on a renormalisation scheme introduced by Berger [arXiv: 0409021 (2004)].

💡 Research Summary

This paper establishes a general sufficient condition under which the graph (or chemical) distance between two vertices in a spatial random graph grows at least linearly with their Euclidean separation. The setting is deliberately broad: the vertex set is generated by a translation‑invariant simple point process on ℝ^d (for example a homogeneous Poisson process or a Bernoulli‑percolated lattice), the edge formation rule is translation‑invariant, and each vertex has finite degree almost surely. These three baseline assumptions (G1–G3) encompass most models studied in continuum percolation, long‑range percolation, random connection models, and Boolean models.

The core of the contribution lies in two probabilistic properties that the graph must satisfy. The first, called polynomial mixing (PM_ξ), requires that for any two disjoint boxes of side length m, the covariance of any two local events defined inside the boxes decays as C_mix·m^ξ with an exponent ξ<0. In other words, correlations between distant regions are at most polynomially weak. The second, called the “no long edges” property (PL_μ), demands that the probability of finding an edge longer than n inside a box of side length m is bounded by C_L·m^d·n^μ with μ<−d. This quantifies the scarcity of long edges: the larger the absolute value of μ, the rarer such edges become.

The main result (Theorem 1) shows that if a spatial random graph satisfies both PM_ξ and PL_μ for some ξ<0 and μ<−d, then there exists a deterministic constant η>0 (depending only on model parameters) such that for every fixed inner box Λ_L and for all sufficiently large outer boxes Λ_{c m}, the event
D_η^L(m) = { d_G(x,y) ≥ η |x−y| for all x∈Λ_L, y∈Λ_{c m} }
holds with probability at least 1−O(m^{ξ∨(d+μ)}). Equivalently, the probability that a path connecting a vertex near the origin to a far‑away vertex uses fewer than a constant multiple of the Euclidean distance decays at the same polynomial rate as the probability of observing a single long edge of comparable length. Consequently, in regimes where long edges are sufficiently rare, almost all long‑range connections must be built from many short edges, and the chemical distance is forced to be linear in the Euclidean distance.

The theorem recovers and extends earlier results by Berger (2004) on long‑range percolation, where the exponent δ>2 guarantees linear lower bounds, and by Biskup (2004) for the case δ<2 where logarithmic scaling appears. The novelty here is that the proof does not rely on specific lattice structure or independence; it only needs the two abstract properties PM and PL, which can be verified in a wide variety of models.

The paper proceeds to illustrate the applicability of the theorem in several concrete settings.

1. Weight‑dependent random connection model – Vertices carry independent marks u∈(0,1) that influence connection probabilities via a kernel ρ((u∧v)^γ (u∨v)^{γ’}|x−y|^{-d}). Independence of marks yields PM_{−∞} automatically. The scarcity of long edges is controlled by a “downward boundary exponent” ζ, which measures how many vertices can connect to a much stronger vertex at distance m. When ζ<0, long edges are rare, and the authors prove PL_{d(ζ−1)} by estimating the probability that a vertex with a sufficiently small mark appears or that two weak vertices manage to connect over a large distance. This leads to explicit parameter regimes (δ>2, γ<1−1/δ, γ′<1−γ) where the linear lower bound holds.

2. Classical long‑range percolation on ℤ^d – Edges between lattice sites at distance r appear with probability proportional to r^{-dδ}. For δ>2, the probability of a long edge of length n decays as n^{-d(δ−2)}, satisfying PL with μ=−d(δ−2). Independence of edges gives PM_{−∞}. Hence the theorem yields the known linear lower bound for δ>2 in a unified framework.

3. Heavy‑tailed Boolean model – Points of a Poisson process are equipped with random radii having a heavy tail. Two points are connected if their balls intersect. Existing literature conjectured linear scaling for tail exponent >2. By bounding the probability that a ball of radius larger than n appears in a box of size m, the authors verify PL with an appropriate μ, while the Boolean model’s local independence provides PM. The theorem thus confirms the conjectured linear lower bound.

The authors also discuss the complementary upper bound. While Theorem 1 only gives a lower bound, they argue that under the same weak‑correlation assumptions an analogous linear upper bound is expected. In models with independent edge formation (e.g., Poisson‑based random connection models) recent work on continuity of the critical parameter allows one to truncate edges longer than a suitable threshold and apply known results for finite‑range percolation, thereby obtaining matching upper bounds. For correlated models, establishing such an upper bound remains technically challenging, but the authors point to recent progress in related settings.

In summary, the paper provides a robust, model‑independent criterion—scarcity of long edges together with polynomially weak long‑range correlations—that guarantees linear scaling of chemical distance in a broad class of spatial random graphs. This unifies and extends several earlier results, offers a clear diagnostic for new models, and opens the way for further investigations into matching upper bounds and finer asymptotics.