Subcritical annulus crossing in spatial random graphs

TL;DR

The paper develops a general framework for continuum percolation on spatial graphs built from stationary point sets, introducing the subcritical annulus-crossing intensity $\widehat{\lambda}_{\textsf{c}}$ and linking its positivity to the absence of long edges via a multiscale renormalisation. It shows that, under weak long-range correlations and a no-long-edges condition, annuli are subcritically crossed (so $\widehat{\lambda}_{\textsf{c}}>0$), while abundant long edges drive $\widehat{\lambda}_{\textsf{c}}$ to zero, with a polynomial decay rate for crossing probabilities governed by the effective decay exponent $\zeta$ and the mixing exponent $\xi$. The results are applied to the weight-dependent random connection model and several correlated variants, including interpolation models and soft Boolean models with local interferences, yielding concrete, easy-to-verify criteria for the presence or absence of long edges. The work clarifies how long-range effects influence subcritical behavior and diameter tails, and connects these phenomena to broader questions about the equality (or lack thereof) between $\widehat{\lambda}_{\textsf{c}}$ and the classical percolation threshold $\lambda_{\textsf{c}}$. Overall, it provides a versatile toolkit for assessing subcritical connectivity in a wide class of spatial random graphs with long-range interactions.

Abstract

We consider general continuum percolation models obeying sparseness, translation invariance, and spatial decorrelation. In particular, this includes models constructed on general point sets other than the standard Poisson point process or the Bernoulli-percolated lattice. Moreover, in our setting the existence of an edge may depend not only on the two end vertices but also on a surrounding vertex set and models are included that are not monotone in some of their parameters. We study the critical annulus-crossing intensity $\widehatλ_{c}$, which is smaller or equal to the classical critical percolation intensity $λ_{c}$ and derive a condition for $\widehatλ_{c}>0$ by relating the crossing of annuli to the occurrence of long edges. This condition is sharp for models that have a modicum of independence. In a nutshell, our result states that annuli are either not crossed for small intensities or crossed by a single edge. Our proof rests on a multiscale argument that further allows us to directly describe the decay of the annulus-crossing probability with the decay of long edges probabilities. We apply our result to a number of examples from the literature. Most importantly, we extensively discuss the weight-dependent random connection model in a generalised version, for which we derive sufficient conditions for the presence or absence of long edges that are typically easy to check. These conditions are built on a decay coefficient $ζ$ that has recently seen some attention due to its importance for various proofs of global graph properties.

Figures (4)

• Figure 1: Phase diagram in $\gamma$ and $\alpha$ for the interpolation model constructed with a long-range profile with $\delta>2$ in (a) and $\delta\in(1,2)$ in (b). The $\zeta<0$ phase in (a) is shaded in grey while the $\zeta>0$ phases are not shaded. The values of $\zeta$ in the corresponding parameter regimes are shown. The solid line in (a) marks the phase transition $\zeta=0$. Dashed lines represent no change of behaviour.
• Figure 2: Examples for the soft Boolean model, Figure \ref{['fig:softVsInterSoft']}, and the soft Boolean model with local interference, Figure \ref{['fig:softVsInterInter']}, on the same $150$ vertices sampled from a Poisson point process of intensity $\lambda=0.04$. For the edge probabilities, the parameters $\gamma=0.65, \delta=2.7$, and $\beta=0.3$ are used. Hence, $\zeta>0$ for the soft Boolean model but $\zeta<0$ for the model with local interference.
• Figure 3: Phase diagram for $\gamma$ and $\delta$ for the soft Boolean model with local interference. Represented from left to right the phase transition for $\zeta<0$ for $\xi=0,0.3,0.6,0.9$.
• Figure 4: Sketch of \ref{['eq:FactoriseP(G)']} in $d=2$. A path starting inside $B(10r)$ and leaving $B(20r)$, where no edge longer than $r$ is used. The red vertex on the path is close to the center of the red ball of the covering of $\partial B(10r)$ and it is the starting point for the event $\mathscr{G}($red balls$)$. Further, the covering of the annulus $B(10.5 r)\setminus B(9.5r)$ is indicated. The same applies to the blue vertex on the path, which lies close to the center of the blue ball of the covering of $\partial B(20r)$.

Theorems & Definitions (29)

• Remark 1.1
• Definition 1.2: Mixing
• Definition 1.3: Independent setting
• Definition 1.4
• Lemma 1.5: Properties of \ref{['G:noLongEdge']}$_\lambda$
• proof
• Theorem 1.6: Existence of subcritical annulus-crossing phase
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9: Existence of a subcritical percolation phase