The preprint identifies conditions under which random graphs with long-distance connections are recurrent in one and two dimensions. In the paper's mathematical language, recurrence is defined through an electrical-network test: every edge has unit conductance, and a component is recurrent when its effective resistance from a vertex to infinity is infinite. Finite components count as recurrent by convention.
The models are random geometric graphs, written as G=(V,E), in Euclidean space. The study asks which graph laws and parameter regimes are enough to make connected components recurrent, then applies those criteria to inhomogeneous long-range percolation and interpolation-kernel WDRCMs.
On the line, a small condition goes a long way
On the line, the main theorem uses a condition about a single reference cut. If the graph is locally finite, its law is unchanged by integer translations and is ergodic, and the cut has only finitely many crossing edges with positive probability, then every connected component is recurrent almost surely. The threshold is notable because the finite-cut event need not be certain: the theorem requires its probability to be greater than zero.
A second one-dimensional result turns this into a strong-decay test. The summability condition labelled (2.1) is sufficient for almost-sure recurrence of every component on the line. Here, strong decay means the model's long edges become sufficiently scarce for the relevant sum to converge, but the statement remains a sufficient criterion under the theorem's assumptions, not a verdict on every possible long-range graph.
For the one-dimensional interpolation-kernel WDRCM, the recurrent region is specified by δ>2, γ<1−1/δ and α<1−γ. With nearest-neighbour augmentation, the graph, identified in the statement with its unique infinite component, is recurrent almost surely throughout that range.
In two dimensions, the proof builds barriers
Two dimensions require a more geometric argument. The planar assumption applies to a stationary, ergodic, locally finite graph with a polynomial mixing exponent ξ<0, polynomial long-edge decay exponent μ<−2 and finite degree-measure intensity. The first two requirements concern decay of dependence and long edges, while the last is a finiteness requirement.
The proof then searches for good annuli, or rings, in which graph distance grows linearly with the annulus's Euclidean width while the number of edges is of area order. Each graph-distance layer supplies an edge-disjoint cutset, a separate barrier in the network. Nash-Williams' inequality combines the layers: their repeated positive resistance contributions add up to divergent effective resistance.
Under Assumption 2.4, the planar theorem concludes that every connected component is recurrent almost surely. The same conclusion is obtained for the two-dimensional interpolation-kernel WDRCM throughout the displayed strong-decay range: δ>2, γ<1−1/δ and α<1−γ.
For that interpolation model, the proof verifies the needed mixing and long-edge conditions in the strong-decay phase. It uses independence between disjoint local configurations and checks both Poisson and lattice formulations. A separate expected-degree calculation is finite when γ<1 and α+γ<1, giving finite degree-measure intensity and almost-sure local finiteness.
A result with clear edges
The paper's results fill in a recurrent part of the model's phase diagram, but leave its borders unresolved. It states that phase-boundary behavior remains open; the recurrent region newly established here sits alongside other phases that were already established in cited work.
The authors describe the recurrence regimes as essentially optimal because transience has been established slightly outside them. That assessment is limited to the displayed regimes and the assumptions behind the theorems, so it does not settle behavior on the boundaries or in graph laws that fail those assumptions.
The work is presented as a version-one arXiv preprint dated 25 August 2026.
Paper data and sources
Original title: Recurrence of strong-decay inhomogeneous long-range percolation clusters
Authors: Johannes Bäumler, Lukas Lüchtrath, Christian Mönch
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text