2.1. Verifying the no-long-edges condition using the effective decay exponent

In order to apply our results to WDRCMs, we need to verify the Properties $\mathscr{L}$ $_\lambda$ and $\overline{\mathscr{L}}$ $_\lambda$ . In this context, we can provide relatively simple sufficient conditions for verifying or excluding these properties, based on natural exponents associated with the model. We begin with a heuristic explanation. Consider the two sets $\mathscr{B}_1$ and $\mathscr{B}_2$ of vertices, where $\mathscr{B}_1$ consists of all vertices located within B(r) and $\mathscr{B}_2$ consists of those located in the annulus $B(3r)\setminus B(2r)$ . We compute the expected number of edges between these sets using Campbell’s formula as

\begin{equation*} \int_{B(r)}\mathrm{d} x \int_{B(3r)\setminus B(2r)} \mathrm{d} y \int_0^1 \mathrm{d} u \int_0^1 \mathrm{d} v \, \varphi(u,v,|x-y|^d). \end{equation*}

Note that all vertices in $\mathscr{B}_1$ and $\mathscr{B}_2$ are roughly distance r apart, so we may approximate $|x-y|$ by r, resulting in the order

\begin{equation*} r^{2d}\int_0^1 \mathrm{d} u \int_0^1 \mathrm{d} v \, \varphi(u,v,r^d). \end{equation*}

A first nave approach would be to assume a Poisson structure of the edges and, as a result, estimating the probability of finding an edge between the two regions as

\begin{equation*} 1-\exp\bigg({-}r^{2d}\int_0^1 \mathrm{d} u \int_0^1 \mathrm{d} v \, \varphi(u,v,r^d)\bigg). \end{equation*}

However, an important contribution to this expectation may come from the strongest vertices, those with lowest marks, which are rare but, when present, may connect to many long edges. Put differently, the full expectation may not capture the likelihood of long edges adequately. We refer the reader to [Reference Gracar, Lüchtrath and Mönch22] for a more detailed discussion on this matter.

To address this, we introduce a free parameter $\mu\in ({-}\infty,1)$ and instead count:

1. (i) the number of edges between ‘weak’ vertices with marks at least $r^{-d+d\mu}$ ; and

2. (ii) the number of ‘strong’ vertices with marks at most $r^{-d+d\mu}$ .

On average, there are of order

\begin{equation*} I(\mu,r)\,:\!=\,r^{2d}\int_{r^{-d+d\mu}}^1 \mathrm{d} u \int_{r^{-d+d\mu}}^1 \mathrm{d} v \, \varphi(u,v,r^d) \end{equation*}

such edges between weak vertices, and of order $r^{d\mu}$ such strong vertices. We then select a $\mu$ that minimizes the combined contributions. More precisely, we define

(4) \begin{equation} \psi(\mu) \,:\!=\, \limsup_{r\to\infty} \frac{\log \bigg(\int_{r^{-d+d\mu}}^1 \mathrm{d} u \int_{r^{-d+d\mu}}^1 \mathrm{d} v \, \varphi(u, v, r^d)\bigg)}{d\log r} \end{equation}

and

(5) \begin{equation} \zeta \,:\!=\,\sup\{\mu\in ({-}\infty,1)\,:\, \mu \lt 2+\psi(\mu)\}. \end{equation}

It is worth noting here that $\psi$ is a non-increasing function with values in $[{-}\infty,-1]$ . Thus, $\zeta$ is well-defined, and $I(\zeta, r)+r^{d\zeta}$ yields the smallest order in r. Furthermore, for any $\mu\gt \zeta$ , we have $r^d/(r^{-d+d\mu}-r^{-d+d\zeta})=r^{2d-d\mu}/(1-r^{d(\zeta-\mu)})$ , and therefore by monotonicity of $\varphi$ in the mark arguments

(6) \begin{equation}\begin{aligned} r^{d}\!\!\int\limits^1_{r^{-d+d\mu}}\!\! \mathrm{d} v \, \varphi(r^{-d+d\mu},v,r^d) \leq \tfrac{r^{2d-d\mu}}{1-r^{d(\zeta-\mu)}}\!\! \int\limits_{r^{-d+d\mu}}^1\!\! \mathrm{d} v\!\! \int\limits_{r^{-d+d\zeta}}^{r^{-d+d\mu}}\!\! \mathrm{d} u \,\varphi(u,v,r^d) \leq \tfrac{1}{1-r^{d(\zeta-\mu)}} r^{-d\mu}I(\mu,r) \lt 1, \end{aligned}\end{equation}

for large r. This indicates that a vertex in $\mathscr{B}_1$ with mark $u=r^{-d+d\mu}$ will, on average, only connect by a long edge to fewer than one weaker vertex in $\mathscr{B}_2$ with mark at least u. Conversely, a vertex with mark $u=r^{-d+d\mu}$ with $\mu\lt \zeta$ will typically be incident to more than one long edge with weaker target vertex. This characteristic naturally motivates the definition of $\zeta$ as it essentially estimates the expected number of vertices in $\mathscr{B}_1$ incident to at least one long edge with weaker target vertex. This very definition is actually used in [Reference Jorritsma, Komjáthy and Mitsche37, Reference Jorritsma, Komjáthy and Mitsche38] to introduce the parameter $\zeta_{{\rm long}}$ , which quantifies the cluster-size decay of finite components in supercritical WDRCMs in the regimes where long edges appear frequently ( $\zeta_{\rm long}\gt 0$ ). It can be verified that we always have $\zeta_{{\rm long}}=\zeta$ . However, our alternative definition of $\zeta$ is more transparent for our approach and heuristics, particularly facilitating the generalizations to more correlated models in Section 2.2.

We further elaborate on the heuristic link between $\zeta$ and $\mathbb{P}_\lambda(\mathcal{L}(r,1))$ . Suppose first that $\zeta\lt 0$ . In this case, the expectation above is approximately $r^{d\zeta}\ll 1$ , implying that the probability of finding a long edge is bounded by $r^{d\zeta}$ , yielding Property $\mathscr{L}$ $_\lambda$ .

Suppose now that $\zeta\gt 0$ . Then, the expectation above is approximately $r^{d\zeta}\gg 1$ , so we typically expect to find many long edges. Assuming sufficient continuity to have $\zeta=2+\psi(\zeta)$ for simplicity, then long edges should only be absent if either:

1. (i) no vertex has mark smaller than $r^{-d+d\zeta}$ , which occurs with probability roughly $\exp({-}r^{d\zeta})$ ; or

2. (ii) there is no edge between ‘weak’ vertices with larger marks.

If we accept that $r^d {\unicode{x1D7D9}}_{[r^{-d+d\zeta},1]}$ is well enough approximated by the empirical distributions of these weaker vertices marks, then the number of such edges should be given by a Poisson random variable with mean $I(\zeta,r)$ . Thus, the probability of finding no such edge is roughly $\exp({-}r^{d(2 +\psi(\zeta))})=\exp({-}r^{d\zeta})$ . Formalizing these heuristics, we obtain the following theorem.

We give the proof of Theorem 4 in Section 4.2.

Figure 1.

Phase diagram in $\gamma$ and $\alpha$ for the interpolation model constructed with a long-range profile with $\delta\gt 2$ in (a) and $\delta\in(1,2)$ in (b). The $\zeta\lt 0$ phase in (a) is shaded in grey whereas the $\zeta\gt 0$ 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.

The theorem highlights the importance of determining the value of $\zeta$ and, in particular, whether $\zeta$ is positive or negative. Let us define

\begin{equation*} \delta_{\rm eff}^+\,:\!=\,-\psi(0{+})\quad\text{and}\quad \delta_{\rm eff}^-\,:\!=\,-\psi(0{-}), \end{equation*}

the limit from the right and left, respectively, of the absolute value of $\psi$ at zero. Then $\delta_{\rm eff}^-\gt 2$ implies $\zeta\lt 0$ , whereas $\delta_{\rm eff}^+\lt 2$ implies $\zeta\gt 0$ . If $\psi$ is continuous at zero, we can simply define

\begin{equation*} \delta_{\rm eff}\,:\!=\,\delta_{\rm eff}^+=\delta_{\rm eff}^-=-\psi(0)=- \limsup_{r\to\infty} \frac{\log \big(\int_{r^{-d}}^1 \mathrm{d} u \int^1_{r^{-d}} \mathrm{d} v \, \varphi(u, v, r^d)\big)}{d\log r}. \end{equation*}

The parameter $\delta_{\rm eff}$ was first introduced in [Reference Gracar, Lüchtrath and Mönch22] as the effective decay exponent in order to derive the existence or absence of supercritical percolation phases in one-dimensional WDRCMs. It was further used in [Reference Mönch43] to prove continuity of the percolation function in the $\delta_{\rm eff}\lt 2$ regime. The motivation for introducing $\delta_{\rm eff}$ comes from the observation that the smallest vertex marks among the vertices in $\mathscr B_1$ and $\mathscr B_2$ (defined as above as ball and annulus, respectively) are typically of order $r^{-d}$ , suggesting that the probability of two randomly picked vertices in $\mathscr B_1$ and $\mathscr B_2$ being connected by an edge scales like $r^{-d\delta_{\rm eff}}$ .

This leads to a natural analogy with a classical long-range percolation model, where each pair of vertices $\boldsymbol{{x}},\boldsymbol{{y}}$ is independently connected by an edge with probability proportional to $|x-y|^{-d\delta_{\rm eff}}$ , which justifies the term effective decay exponent for $\delta_{\rm eff}$ . Our results demonstrates that this simple exponent $\delta_{\rm eff}$ largely dictates the principal behaviour of the model and distinguishes the two universality classes, analogous to classical long-range percolation [Reference Penrose45, Reference Schulman48]. However, while $\delta_{\rm eff}$ captures the model’s overall behaviour, it may lack precision for finer results such as explicit tail bounds. In particular, the specific manner in which vertex marks contribute can be too nuanced to be fully characterized by $\delta_{\rm eff}$ alone leading us to the finer exponent $\zeta$ .

It is worth noting that the two universality classes according to strong ( $\zeta\gt 0$ ) or weak ( $\zeta\lt 0$ ) long-range effects are insufficient to fully describe all global properties accurately. Indeed, while $\zeta$ effectively captures the overall occurrence of long edges, it does not capture their precise connectivity. Specifically, in some models the $\zeta\gt 0$ phase includes parameter regimes with $\lambda_{\textsf{c}}\gt 0$ as well as those with $\lambda_{\textsf{c}}=0$ [Reference Gracar, Lüchtrath and Mörters23]. Detecting such a change of behaviour requires a deeper understanding and control over the inhomogeneities arising from the marks.

Finally, let us note that although we have exclusively assumed vertex locations as governed by a Poisson point process throughout the section, all results remain applicable to a Bernoulli site-percolated lattice. In this case, the integrals over vertex locations should be replaced by sums.

2.1.1. The interpolation model.

We apply the previously established results and, in particular, derive values of $\zeta$ for a specific yet general instance of the WDRCM. More precisely, we discuss the interpolation model first introduced in [Reference Gracar, Lüchtrath and Mönch22] as a model that contains and interpolates between many important and well-established models from the literature. Recall that $\mathscr{V}$ is given by a standard Poisson point process on $\mathbb{R}^d\times(0,1)$ and that we denote vertices by $\boldsymbol{{x}}=(x,u_x)$ . In the interpolation model, edges are drawn independently given $\mathscr{V}$ with probability

(7) \begin{equation} \mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}})=\varphi(u_x,u_y,|x-y|^d)=\rho(g(u_x,u_y)|x-y|^d), \end{equation}

where $\rho\,:\,(0,\infty)\to[0,1]$ is an integrable and non-increasing profile function and

(8) \begin{equation} g\,:\, (0,1)\times (0,1)\to (1,\infty), \quad (s,t) \mapsto (s\wedge t)^{\gamma}(s\vee t)^{\alpha}, \end{equation}

for $\gamma\in[0,1)$ , $\alpha\in[0,2-\gamma)$ , is the interpolation kernel. Note that the interpolation kernel g is non-increasing and symmetric in both arguments. It is straightforward to deduce that the interpolation model is a WDRCM with all required properties. Further, regardless of the profile function, the degree distribution of the interpolation model follows a power law if either $\gamma\gt 0$ or $\alpha\gt 1$ , see [Reference Lüchtrath41].

Let us now examine the profile function more closely. Two types have been established in the literature: The long-range profile function $\rho(x)\,:\!=\,p(1\wedge |x|^{-\delta})$ for $\delta\gt 1$ or the short-range profile function $\rho(x)\,:\!=\,p{\unicode{x1D7D9}}\{0\le x\le 1\}$ . Here, $p\in(0,1]$ can be used to simultaneously introduce Bernoulli bond percolation. However, we mainly assume $p=1$ as it does not qualitatively affect our results. As summarized in Table 1, the interpolation model can be used to describe versions of many well-established models. This model has the huge advantage that it describes these models with only four real parameters and allows for easy comparisons and couplings. A similar, yet slightly different, parameterization has also been studied, for instance, in [Reference van der Hofstad, van der Hoorn and Maitra30, Reference Jorritsma, Komjáthy and Mitsche37, Reference Jorritsma, Komjáthy and Mitsche38]; in Table 2 we present a translation between the two parameterizations.

Table 2.

Translation of parameters in kernel-based spatial random graphs and the classical WDRCM.

The following proposition summarizes the values $\zeta$ for the interpolation model with a long-range profile function. However, the according results for a short-range profile can simply be derived by sending $\delta\to\infty$ . Its proof is given in Section 4.2.

Proposition 1. Consider the interpolation model with a profile function with $\delta\in(1,\infty)$ .

1. (i) If $\delta\gt 2$ , $\gamma\lt 1-1/\delta$ , and $\alpha\lt 1-\gamma$ then

   \begin{equation*} \zeta = \max\left\{2-\delta, 1-\tfrac{\delta-1}{\gamma\delta}, \tfrac{\alpha+\gamma-1}{\gamma},\tfrac{2(\alpha+\gamma-1)}{\alpha+\gamma}\right\} \lt 0. \end{equation*}

2. (ii) If $\delta=2$ , $\gamma\lt 1-1/2$ , and $\alpha\lt 1-\gamma$ , or if $\delta\geq 2$ and either $\gamma \lt 1-1/\delta$ and $\alpha=1-\gamma$ , or $\gamma=1-1/\delta$ and $\alpha\lt 1-\gamma$ , then $\zeta=0$ .

3. (iii) If $\delta\lt 2$ , or $\gamma\gt 1-1/\delta$ , or $\alpha\gt 1-\gamma$ then

   \begin{equation*} \zeta= \max\left\{2-\delta, 1-\tfrac{\delta-1}{\gamma\delta}, \tfrac{\delta(\alpha+\gamma-1)}{\delta(\alpha+\gamma)-1}\right\}\gt 0. \end{equation*}

Our findings are summarized in Figure 1, and we further apply part (i) to various models from the literature in Table 1.

Let us further discuss the possible subcritical annulus-crossing phase, depending on the value of $\zeta$ . First, when $\zeta\gt 0$ , by Theorem 4 the long-edge probability $\mathbb{P}_1(\mathcal{L}(r,1))$ does not decay to zero but actually approaches one. Observing that the WDRCM is a model in the independent setting, by Theorem 2 this is equivalent to saying that $\widehat{\lambda}_{\textsf{c}}=0$ , and there is no subcritical annulus-crossing phase.

When $\zeta=0$ , as outlined in Remark 2 (b), whether $\widehat{\lambda}_{\textsf{c}}=0$ or $\widehat{\lambda}_{\textsf{c}}\gt 0$ depends on fine details of the model. In fact, an additional slowly varying correction term may influence that substantially. However, in the strict Pareto case of our parameterization of the interpolation model, it is easily seen that we still have $\widehat{\lambda}_{\textsf{c}}=0$ , as was first observed in [Reference Jacob, Kerriou, Linker and Mörters31] for the special instance of scale-free percolation.

Proposition 2. Consider the interpolation model with $\delta=2$ or with $\delta\gt 2$ and either $\gamma=1-1/\delta$ and $\alpha \leq 1-\gamma$ or $\gamma\lt 1-1/\delta$ and $\alpha=1-\gamma$ , then $\mathbb{P}_1(\mathcal{L}(r,1))\gt 0$ and, consequently, $\widehat{\lambda}_{\textsf{c}}=0$ .

We present the proof in Section 4.2. Let us finally comment on the more interesting $\zeta\lt 0$ regime of the interpolation model. We can again use Theorems 4 and 2 to deduce that we have a non-trivial subcritical annulus-crossing phase, namely $\widehat{\lambda}_{\textsf{c}}\gt 0$ , in that case. Moreover, it is easy to see that the $\limsup$ defining $\psi$ is a true limit, so $\mathbb{P}_1(\mathcal{L}(r,1))= r^{d\zeta +o(1)}\to 0$ by Remark 2(a). In the whole subcritical annulus-crossing phase $\lambda\in(0,\widehat{\lambda}_{\textsf{c}})$ , the annulus-crossing probabilities also decay at the same speed, and one might thus expect that the subcritical behaviour is driven by $\mathbb{P}_\lambda(\mathcal{L}(r,1))$ and explained by $\zeta$ alone.

As already mentioned below Theorem 3, the probability of the annulus-crossing event also determines the largest Euclidean diameter of the clusters in B(r). In the classical WDRCM this further applies to the tail of the Euclidean diameter of the typical cluster of the origin when the origin is added with a uniformly distributed vertex mark to the graph. Hence,

\begin{equation*} \mathbb{P}^o_\lambda(\exists \boldsymbol{{x}}\,:\, |x|\gt r, \mathbf{o}\xleftrightarrow[]{}\boldsymbol{{x}}) \leq \mathbb{P}_\lambda \big(B(r)\xleftrightarrow[]{} B(2r)^\textsf{c}\big) = r^{d\zeta+o(1)}. \end{equation*}

Furthermore, the tail of the Euclidean diameter can provide bounds on the cardinality of the cluster of the origin as the Poisson point process is well-concentrated. This relation becomes particularly strong for the standard Boolean model ( $\alpha=0, \delta\to \infty$ ), for which it is shown in [Reference Gouéré16] that for all $\lambda\lt \widehat{\lambda}_{\textsf{c}}$ ,

\begin{equation*} \mathbb{P}_\lambda\big(\sharp\{\boldsymbol{{x}}\,:\, \mathbf{o}\xleftrightarrow[]{}\boldsymbol{{x}}\}\gt r^d\big)\asymp \mathbb{P}_\lambda\big(\exists \boldsymbol{{x}}\,:\, |x|\gt r, \mathbf{o}\xleftrightarrow[]{}\boldsymbol{{x}}\big) \asymp \mathbb{P}_\lambda(\mathcal{L}(r,1)). \end{equation*}

Further, it was recently shown that $\lambda_{\textsf{c}}=\widehat{\lambda}_{\textsf{c}}$ in the standard Boolean model for all but at most countably many values of $\gamma$ (see [Reference Dembin and Tassion10]). The potential exclusion of countably many values of $\gamma$ results from the techniques applied in the proof, though it is widely believed that both phase transitions are always the same. These findings combined appear to strengthen the above idea and suggest that even the whole subcritical behaviour of the Boolean model can be described solely by the tail of the long-edges event. However, this is not the case in general. For instance, consider long-range percolation ( $\gamma=\alpha=0$ and $\delta\lt \infty$ ). Thus, $\widehat{\lambda}_{\textsf{c}} = 0$ whenever $\delta\lt 2$ according to Proposition 1. However, it is well known that $\lambda_{\textsf{c}}\gt 0$ for all $\delta\gt 1$ , see [Reference Meester and Roy42], implying $0=\widehat{\lambda}_{\textsf{c}} \lt \lambda_{\textsf{c}}$ . For $\delta\gt 2$ and $\lambda\lt \widehat{\lambda}_{\textsf{c}}$ our results yield the polynomial decay with exponent $2-\delta$ as a bound for the Euclidean diameter and the cardinality of the origin’s component. However, the actual tail exponent of the Euclidean diameter is $1-\delta$ due to [Reference Jahnel, Lüchtrath and Ortgiese33] while the cardinality is light-tailed [Reference Meester and Roy42]. Thus, describing the subcritical phase solely via annulus crossings is no longer accurate due to the long-range effects inherent in the model. The same can further be observed in the soft Boolean model, a model that interpolates between the classical Boolean model and long-range percolation. For a detailed discussion on this model, we refer the reader to [Reference Jahnel, Lüchtrath and Ortgiese33]. This of course does not give insights into the already mentioned question as to whether $0\lt \widehat{\lambda}_{\textsf{c}}\lt \lambda_{\textsf{c}}$ is possible. It is also of interest to determine whether the subcritical tail distribution of diameter and cardinality as mentioned previously is generally governed by $\mathbb{P}_1(\mathcal{L}(r,1))$ for any kernel combined with a short-range profile or whether this is specific to the Boolean model, as well as the correct type of connection for an accurate description of the subcritical phase if this is not the case.

2.2. Generalizations

The previous results of this section and particularly Theorem 4 apply our main results to the classical WDRCM. The results get particularly strong as the classical model takes place in the independent setting of Definition 2. Our main results however allow for generalizations towards more correlated models. There are two natural ways to implement correlations into the framework of the WDRCM: either by replacing the underlying Poisson point process with a correlated point process or by extending the connection mechanism in a way that the presence of an edge additionally depends on vertices surrounding its end vertices. Here, we focus on the latter approach and stick with the Poisson process for vertex placement, but we also provide examples of correlated vertex sets in Sections 3.2 and 3.3. Both effects could, of course, be combined, though we have chosen not to, focusing on presenting the essential concepts in a concise way.

The setting of the generalized WDRCM is as follows. The vertex set $\mathscr{V}$ remains a standard Poisson point process on $\mathbb{R}^d\times(0,1)$ . Given $\mathscr{V}$ , a pair $\boldsymbol{{x}}=(x,u_x),\boldsymbol{{y}}=(y,u_y)\in\mathscr{V}$ of vertices is connected by an edge with probability

\begin{equation*} \mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}},\mathscr{V}\,\setminus\{\boldsymbol{{x}},\boldsymbol{{y}}\})=\mathbf{p}\big(\boldsymbol{{y}},\boldsymbol{{x}},\mathscr{V}\,\setminus\{\boldsymbol{{x}},\boldsymbol{{y}}\}\big), \end{equation*}

where we assume that $\mathbf{p}$ is invariant under translations. We further impose the following homogeneity condition when $\mathbf{p}$ is integrated with respect to the underlying Poisson point process: For two deterministically given vertices $\boldsymbol{{x}}$ and $\boldsymbol{{y}}$ , we have

(9) \begin{equation} \mathbb{E}_\lambda\big[\mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}},\mathscr{V}\,)\big]= \varphi_\lambda(u_x, u_y, |x-y|^d), \end{equation}

where $\varphi_\lambda\,:\, (0,1)\times(0,1)\times(0,\infty)\to[0,1]$ is an integrable function that is symmetric in the vertex marks arguments. Note that these deterministically given vertices are no elements of the Poisson point process almost surely. Put differently, the occurrence of an edge between any two vertices does additionally depend on (potentially) the entire vertex set. However, the connection probability between two given vertices $\boldsymbol{{x}}$ and $\boldsymbol{{y}}$ resembles, on average, the connection probability of the same fixed vertices in a classical WDRCM. Nevertheless, the annealed connection function $\varphi_\lambda$ now depends on $\lambda$ and we no longer assume monotonicity in the vertex marks arguments. It is thus clear that our basic assumptions (G1)–(G3) are always satisfied but one further has to verify one kind of mixing condition and monotonicity (if needed) separately for each connection rule $\mathbf{p}$ .

In this paragraph, we focus on identifying instances of the models, where long edges are absent. To this end, we demonstrate that the results derived previously can be adapted in a straightforward manner. We define, in the same way as in (4) and (5),

\begin{equation*} \psi_\lambda(\mu)\,:\!=\, \limsup_{r\to\infty}\frac{\log \Big(\int_{r^{-d+d\mu}}^1 \mathrm{d} u \int_{r^{-d+d\mu}}^1 \mathrm{d} v \, \varphi_\lambda(u, v, r^d)\Big)}{d\log r}, \end{equation*}

as well as

\begin{equation*} \zeta_\lambda \,:\!=\, \sup\big\{\mu\in({-}\infty,1)\,:\,\mu\lt 2+\psi_\lambda(\mu)\big\}, \end{equation*}

noting that both quantities now depend on $\lambda$ . The following results shows that our heuristic explained above remains valid when long edges are rare.

Let us mention that the upper bounds in Theorem 4 are special cases of those in Theorem 5. Specifically, with the previous notation of $\zeta$ in (5), we have $\overline{\zeta}_\lambda\equiv\zeta$ . One of the reasons why we only derive upper but not lower bounds is due to not assuming any monotonicity in the vertex marks arguments of $\varphi_\lambda$ . Nonetheless, the results imply some kind of monotonicity. Indeed, $\zeta_\lambda\lt 0$ suggests that either vertices with smaller marks are more powerful or the marks play no influential role on the connection probability.

Let us finally emphasize that throughout Sections 2, 2.1, and 2.2 we have explicitly used the Euclidean distance in the connection functions and in the derivation of the tail exponent  $\zeta$ . This essentially makes the model rotationally invariant. However, the rotation invariance is never used in our proofs and we could easily replace the Euclidean norm by any other norm and adapt our results. In fact, to ensure translation invariance it suffices to assume connection probabilities of the type $\varphi(u_x,u_y,x-y)$ and our results still remain true as long as there is a way to translate x–y into some form of distance that allows for meaningful definitions of $\psi$ and $\zeta$ .

2.2.1. The soft Boolean model with local interferences.

We provide an example of a graph that is a generalized WDRCM but no classical WDRCM, which satisfies polynomial mixing and $\overline{\zeta}_\lambda\lt 0$ . The idea is to combine the soft Boolean model [Reference Gracar, Grauer and Mörters18, Reference Jahnel, Lüchtrath and Ortgiese33] with local interference and noise in the spirit of signal-to-interference plus noise ratio (SINR) percolation [Reference Dousse, Franceschetti, Macris, Meester and Thiran12, Reference Jahnel and Tóbiás34, Reference Tóbiás51]. The model has three parameters, $\gamma\in(0,1)$ , $\delta\gt 1$ , and $\beta\in[0,1)$ , and the construction is as follows. Each vertex $\boldsymbol{{x}}=(x,u_x)$ has a sphere of influence with radius $u_x^{-\gamma/d}$ . The larger the sphere of influence, the more attractive the vertex is to other vertices. Further, the sphere of influence may be enlarged by additional long-range effects with exponent $\delta\gt 1$ . This setup so far is precisely the soft Boolean model contained in the classical WDRCM discussed above, cf. Table 1 and [Reference Gracar, Grauer and Mörters18, Reference Jahnel, Lüchtrath and Ortgiese33]. In addition, each vertex now has a sphere of interference with radius $u_x^{-\beta/d}$ . The vertices within this sphere distract the vertex $\boldsymbol{{x}}$ and complicate its edge creation, cf. Figure 2.

Figure 2.

Examples for (a) the soft Boolean model and (b) the soft Boolean model with local interference 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, $\zeta\gt 0$ for the soft Boolean model but $\zeta\lt 0$ for the model with local interference.

More precisely, for a given vertex $\boldsymbol{{x}}=(x,u_x)$ , let us introduce the random variable

\begin{equation*} \mathscr{N\,\,}^\beta(\boldsymbol{{x}},\mathscr{V}\,)\,:\!=\, \sharp \big\{\boldsymbol{{y}}\in\mathscr{V} \,:\, |x-y|^d\leq u_x^{-\beta}\big\}.\end{equation*}

The graph $\mathscr{G}$ is then generated, given $\mathscr{V}$ , by connecting $\boldsymbol{{x}}=(x,u_x)$ and $\boldsymbol{{y}}=(y,u_y)$ with probability

(10) \begin{align} \mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}},\mathscr{V}\,\setminus\{\boldsymbol{{x}},\boldsymbol{{y}}\}) = {\unicode{x1D7D9}}_{\{u_x\lt u_y\}}\frac{1\wedge u_x^{-\gamma\delta}|x-y|^{-d\delta}}{1+\mathscr{N\,\,}^{\beta}(\boldsymbol{{x}},\mathscr{V}\,\setminus \{\boldsymbol{{x}},\boldsymbol{{y}}\})}+{\unicode{x1D7D9}}_{\{u_x\geq u_y\}}\frac{1\wedge u_y^{-\gamma\delta}|x-y|^{-d\delta}}{1+\mathscr{N\,\,}^{\beta}(\boldsymbol{{y}},\mathscr{V}\,\setminus \{\boldsymbol{{x}},\boldsymbol{{y}}\})}.\end{align}

Corollary 2. (Soft Boolean model with local interference.) Let $\gamma\lt 1$ , $\delta\gt 1$ , and $\beta\in[0,1)$ . The soft Boolean model with local interference (10) has Property ${{\mathscr{P}}\!\!{\mathscr{M}}}$ $_\lambda^{\xi}$ with parameter $\xi=d(1-1/\beta)$ for all $\lambda$ . Further,

\begin{equation*} \overline{\zeta}_\lambda \equiv \zeta = (2-\delta)\vee \tfrac{1-\delta(1-\gamma)-\beta}{\gamma\delta-\beta}. \end{equation*}

In particular, $\overline{\zeta}_\lambda\lt 0$ if $\delta\gt 2$ and $\gamma\lt (\delta+\beta-1)/\delta$ . Thus, we have $\widehat{\lambda}_{\textsf{c}}\gt 0$ in this case, and for all $\lambda\lt \widehat{\lambda}_{\textsf{c}}$ ,

\begin{equation*} \limsup_{r\to \infty}\frac{\log \mathbb{P}_\lambda\big(B(r)\xleftrightarrow[]{}B(2r)^\textsf{c}\big)}{\log r} \leq d\big((\delta-2)\vee \tfrac{1-\delta(1-\gamma)-\beta}{\gamma\delta-\beta} \vee (1-1/\beta)\big). \end{equation*}

If we choose $\beta=0$ , each sphere of interference has radius one. Hence, the model reduces to a version of the soft Boolean model with additional yet ultimately insignificant fluctuations on large scales. Indeed, for $\beta=0$ the identified $\zeta\lt 0$ phase aligns with the results found for the soft Boolean model. For a phase diagram illustrating the $\zeta\lt 0$ phase of the model with local interference see Figure 3.

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 $\zeta\lt 0$ for $\xi=0,0.3,0.6,0.9$ .

3.1. Ellipses percolation

Ellipses percolation is a percolation model first introduced in 2017 [Reference Teixeira and Ungaretti50] as a generalization of the classical Boolean model with random radii. The vertices are given through a standard Poisson point process of intensity $\lambda\gt 0$ in the plane. Each vertex is centre of an ellipse with a uniform direction, minor axis equal to one, and a heavy-tailed major axis. More precisely, each vertex $\boldsymbol{{x}}$ is assigned an independent direction $\pi_x$ , distributed uniformly on $[{-}\pi/2,\pi/2)$ , and another independent vertex mark $R_x$ , distributed according to a Pareto distribution with parameter $2/\gamma\gt 0$ . Here, the parameterization is chosen to be consistent with that above. Each vertex $\boldsymbol{{x}}$ it thus associated with the ellipse, centred in $x\in\mathbb{R}^2$ with minor axis of length 1, major axis of length $R_x$ , and orientation $\pi_x$ . We connect any pair of vertices by an edge if their associated ellipses intersect. This model can be seen as a generalization of the standard Boolean model where the circles around each vertex are replaced by the described ellipses. However, it is worth noting that using ellipses introduces additional correlations into the model. These additional correlations have interesting effects, some of which we comment on in the following. Note that the model is translation invariant and takes place in the independent setting of Definition 2. The model is locally finite for all $\gamma\lt 2$ (see [Reference Teixeira and Ungaretti50]), and monotone in the sense of (Mon).

The proof can be found in Section 4.2. We briefly compare our result with the findings in [Reference Teixeira and Ungaretti50]. It is shown in the Theorems 1.1.2 and 1.2.2 therein that the model is locally finite but contains no subcritical percolation phase for $1\lt \gamma\lt 2$ . This aligns with our result, as $\lambda_{\textsf{c}}=0$ implies $\widehat{\lambda}_{\textsf{c}}=0$ . For the standard Boolean model it is impossible to be locally finite while having $\lambda_{\textsf{c}}=0$ at the same time. Only if additional long-range edges are introduced such a behaviour can be achieved [Reference Gracar, Lüchtrath and Mörters23]. This implies that some sort of long-range behaviour is entailed in ellipses percolation. Indeed, in [Reference Hilário and Ungaretti26] a comparison between ellipses percolation and long-range percolation is used to derive chemical distances in the $1\lt \gamma\lt 2$ regime. Conversely, our result states $\widehat{\lambda}_{\textsf{c}}\gt 0$ and thus implies $\lambda_{\textsf{c}}\gt 0$ for $\gamma\lt 1$ , which also aligns with the findings in [Reference Teixeira and Ungaretti50]. Hence, except potentially for the boundary regime $\gamma=1$ , we find that the two critical intensities are either both positive or both zero. This is unlike the other long-range models studied in the previous section, all of which contained a parameter regime with $\widehat{\lambda}_{\textsf{c}}=0\lt \lambda_{\textsf{c}}$ . The question of whether a Boolean model with arbitrary convex grains can have $\lambda_{\textsf{c}}=0$ while still being locally finite is explored in detail in the preprints [Reference Gracar and Korfhage20, Reference Gracar, Korfhage and Mörters21]. The results therein include but are not limited to generalizations of ellipses percolation to higher dimensions together with additional correlations between the axes. Whether $\lambda_{\textsf{c}}\gt 0$ implies $\widehat{\lambda}_{\textsf{c}}\gt 0$ in all these models remains an interesting open question that we leave for future work.

3.2. Models based on Cox processes

So far we have exclusively presented applications of our results towards models where the underlying point cloud is given by Poisson point processes. However, our mixing Condition $\overline{\mathscr{M}}$ $_\lambda$ not only allows us to cover models where the edge-drawing mechanism is not strictly local, as shown in Section 2.2.1, but models based on correlated point processes as well. For simplicity, we focus here on correlations introduced by the vertices, assuming independent uniform markings as well as translation-invariant connection probabilities of the form $\mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}})$ with

(11) \begin{equation} \int \mathrm{d} u_o\int\mathrm{d} x\int \mathrm{d} u_x\,\mathbf{p}((o,u_o),(x,u_x))\lt \infty.\end{equation}

That is, only the end vertices of a potential edge are relevant in a translation-invariant and integrable way. Put differently, we consider here the WDRCM based on more general point clouds. Let us however note that of course also mixed versions can be considered where correlations are present in both the vertex positions and the edge drawing mechanism.

To enable explicit computations, we further narrow our attention here to Cox point processes, i.e. Poisson point processes with a random intensity. Let us mention however that also other classes of point processes such as many Gibbs point processes satisfy condition $\overline{\mathscr{M}}$ $_\lambda$ , as well. Their analysis in this context is left for future work and we refer the reader to the associated literature, respectively. In this section, $\mathscr{V}$ is a (independent and uniformly marked) Cox point process, meaning a Poisson point process on $\mathbb{R}^d$ with intensity measure $\lambda\Lambda$ , where $\lambda\gt 0$ and $\Lambda$ is a random element of the space of Borel measures on $\mathbb{R}^d$ , equipped with the evaluation $\sigma$ -algebra. We assume that the distribution of $\Lambda$ is invariant under spatial shifts and that $\mathbb{E}[\Lambda([0,1]^d)]=1$ . The annealed process (with respect to the random environment $\Lambda$ ) is then stationary with intensity $\lambda$ and simple if $\Lambda$ contains no atoms almost surely.

An interesting class of examples can be constructed for $d=2$ , where the directing measure $\Lambda$ is given by the edges of a random line process. More precisely, let $S\subset \mathbb{R}^2$ be a random segment process, then $\Lambda(A)=\nu_1(S\cap A)$ , for all Borel sets $A\subset\mathbb{R}^d$ , where $\nu_1$ denotes the one-dimensional Hausdorff measure. The segment process can be given, for example, by the edges of an independent Poisson–Voronoi tessellation or the rectangular line process, where lines are drawn perpendicular to two independent Poisson processes on the x- and y-axis in $\mathbb{R}^2$ . Let us highlight that, due to the local nature of the construction of the Poisson–Voronoi tessellation, the associated Cox point process features (exponential) decorrelations, whereas the Cox point process based on the rectangular line process does not possess this property (at least not along the axes). Alternatively, a large class of examples can be constructed, where the directing measure is given by a random density with respect to the Lebesgue measure, i.e. $\Lambda(\mathrm{d} x)=\ell_x\,\mathrm{d} x$ . For instance, the density $(\ell_x)_{x\in \mathbb{R}^d}$ can be defined as $\ell_x=\phi_1{\unicode{x1D7D9}}\{x\in \Xi\}+\phi_2{\unicode{x1D7D9}}\{x\not\in \Xi\}$ , where $\Xi\subset\mathbb{R}^d$ is a random closed set and $\phi_1,\phi_2\ge 0$ . In turn, $\Xi$ could be, for example, given by an independent Poisson–Boolean model, with or without random radii (viewed as a subset of $\mathbb{R}^d$ , not as a graph).

Let us mention that the percolation analysis for Boolean models based on Cox point processes has seen some interest in recent years [Reference Hirsch, Jahnel and Cali28, Reference Hirsch, Jahnel and Muirhead29, Reference Jahnel, Jhawar and Vu32, Reference Jahnel, Tóbiás and Cali35], with results that essentially recover the non-trivial percolation behaviour seen in the classical Poisson case. In this work, we extend this analysis by considering not only Gilbert graphs based on Cox point processes, but also more general weight-dependent graph structures, at least with respect to the subcritical annulus-crossing phase. We do not consider concrete connection mechanisms or deal with the Property $\mathscr{L}$ $_\lambda$ here but solely focus on the mixing condition as many possibilities for $\mathbf{p}$ have already been discussed in Section 2.

We now present an example of a non-trivial Cox point process featuring polynomial mixing in $\mathbb{R}^2$ . For this, consider $\Lambda(\mathrm{d} x)=\ell_x\,\mathrm{d} x$ with

(12) \begin{align}\ell_x=\phi\sum_{i\in I}{\unicode{x1D7D9}}\{|Y_i-x|\lt \rho_i\},\end{align}

where $\phi\gt 0$ and $\boldsymbol{{Y}}=((Y_i,\rho_i)\,:\,i\in\mathbb{N})$ is an independent homogeneous Poisson point process with intensity one and i.i.d. marks $\rho_i\ge 0$ , distributed according to a Pareto distribution with tail index $\beta\gt 2$ . In words, $\ell_x$ is the number of Poisson–Boolean discs that overlap a location x and, since $\beta\gt 2$ , the parameter $\phi=\phi(\beta)$ can be tuned in order for $\Lambda$ to be normalized in the sense $\mathbb{E}[\ell_o]=1$ . Then, we have the following result, the proof of which can be found in Section 4.2:

3.3. Models based on worm percolation

After primarily focusing on continuum models, we now apply our results to an example of a lattice-based model. While the case of independent Bernoulli site-percolated lattices is similar to the Poisson process (cf. Remark 3), the objective here is to discuss a correlated lattice model based on percolation of worms [Reference Ráth and Rokob47].

Before diving deeper into this model and its mixing properties, let us comment on another important model for a correlated lattice, namely random interlacements (RI) [Reference Sznitman49]; we refer the reader to [Reference Drewitz, Ráth and Sapozhnikov13] for an introduction to RI. In a nutshell, this model is a Poisson soup of doubly infinite random walk trajectories in $\mathbb{Z}^d$ , $d\geq 3$ , with a parameter $u\gt 0$ determining the intensity of these trajectories. The vertex set $\mathscr{V}$ consists of all sites visited by at least one trajectory. When equipped with nearest-neighbour edges, it is obvious that a single trajectory already forms an infinite component and crosses all annuli by transience of the simple random walk in dimensions three and higher. Moreover, it can even be shown that the corresponding graph is connected for all intensities [Reference Černý and Popov7, Reference Sznitman49]. Consequently, our results cannot apply for RI. Indeed, our mixing assumption is never fulfilled as any random-walk trajectory that crosses an annulus with radii r and 2r has a significant chance to also cross another such annulus at distance r. Note, that this is a result of the two annuli and their distance growing at the same rate. In a more standard notion, however, RI is mixing as, for two given vertices, their covariance is determined by the Green function which decays polynomially with exponent $2-d$ in dimensions $d\geq 3 $ . This shows the importance of our stronger mixing assumption in our results.

However, the RI model can be made mixing in our sense if the length of the random walks is limited, which brings us to percolation of worms. The model is defined as follows. Fix a step-length distribution $\ell$ and an intensity parameter $\nu\gt 0$ . We independently assign each site of $\mathbb{Z}^d$ a Poisson number of worms with mean $\nu$ . Each worm placed at $x\in\mathbb{Z}^d$ draws an independent copy of $\ell$ and performs the corresponding number of simple random walk steps, starting from x. The vertex set $\mathscr{V}$ then consists of all sites of $\mathbb{Z}^d$ visited by at least one worm. Classically, the graph $\mathscr{G}$ is then constructed by adding all nearest-neighbour edges between vertices of $\mathscr{V}$ . Just like in Section 3.2, we stick with this choice and focus primarily on the mixing properties but, in principle, other connection mechanisms, e.g. those of Section 2, could be considered. Observe that $\mathscr{V}$ is stationary and the associated nearest-neighbour graph is translation invariant and locally finite.

An important special case for the step-length distribution $\ell$ is the geometric distribution with mean $T\geq 1$ [Reference Lawler40]. If we further restrict to $d\geq 3$ , we obtain finitary random interlacements (FRI) with intensity $u\,:\!=\,(T+1)/(2d)$ and walk length T, as introduced in [Reference Bowen4]. Notably, FRI converge in distribution to RI with intensity 2du as $T\to\infty$ , giving the model its name. While RI always contain a unique infinite component for all $u\gt 0$ , this holds no longer true for FRI. Intriguingly, there are parameter regimes where there are infinitely many infinite components [Reference Bowen4, Reference Procaccia, Ye and Zhang46].

In our setting, we are not restricted to geometric step-length distributions but may consider general distributions $\ell$ and study the positivity of $\widehat{\lambda}_{\textsf{c}}$ . Fix such a distribution $\ell$ and recall from Remark 1 that the intensity parameter our results apply for is defined by $\lambda=\mathbb{P}(o\in\mathscr{V}\,)$ . That is, $\lambda$ is given by the probability that the origin is visited by at least one worm. Note that the expected number of walks that traverse o is given by $\nu\mathbb{E}[\ell] \mathbb{P}_o^{\ell}(\tilde{H}_o=\infty)$ , where $\mathbb{P}_o^{\ell}(\tilde{H}_o=\infty)$ denotes the probability that a worm with step-length distribution $\ell$ starting in o never returns, see [Reference Bowen4, Reference Drewitz, Ráth and Sapozhnikov13]. Further, $\lambda$ is bounded from below by $1-\mathrm{e}^{-\nu}$ , the probability that at least one worm has been assigned to o in the beginning. Therefore, $\nu\sim 1-\mathrm{e}^{-\nu}\leq \lambda\leq C\nu$ and $\lambda$ can be appropriately varied by adjusting $\nu$ whenever $\ell$ has finite expectation, which we assume in the following.

It was shown in [Reference Ráth and Rokob47] that $\lambda_{\textsf{c}}\gt 0$ if $\ell$ has finite variance in all dimensions and $\lambda_{\textsf{c}}=0$ in dimensions $\mathrm{d}\geq 5$ if $\ell$ has infinite second moment and a slightly stronger tail assumption on $\ell$ is satisfied. For annulus crossings, however, the situation is quite different. While the results of [Reference Ráth and Rokob47] are independent of the dimension (provided $d\geq 5$ ), our findings depend on the dimension; a characteristic that has not previously be shown in our analysis.

3.4. k-nearest-neighbour percolation and other $\boldsymbol{\lambda}$ -independent models

One may question whether the assumption $\overline{\mathscr{L}}$ $_\lambda$ (respectively, $\overline{\mathscr{M}}$ $_\lambda$ ) could be relaxed and replaced by the assumption $\{\forall \lambda'\leq \lambda,$ $\mathscr{L}$ $_\lambda\}$ (respectively, $\mathscr{M}$ $_\lambda$ ) in the statement of Theorem 1. We show that this is not possible by discussing a simple class of models where the $\lambda$ -dependence of the model is only superficial. Consider any graph $\mathscr{G}_1=(\mathscr{V}_1,\mathscr{E}_1)$ on $\mathbb{R}^d$ obeying our settings, but defined only for $\lambda=1$ . Then, one may (re)define the model for $\lambda\ne 1$ by keeping the exact same graph structure, but scaling all vertex locations by a factor $\lambda^{-1/d}$ . That is,

\begin{equation*}\mathscr{V}_\lambda\,:\!=\, \{\lambda^{-1/d} x\,:\, x\in \mathscr{V}_1 \}, \quad \mathscr{E}_\lambda\,:\!=\,\{\{\lambda^{-1/d} x, \lambda^{-1/d}y\}\,:\, \{x,y\}\in \mathscr{E}_1\}.\end{equation*}

This produces a model $(\mathscr{G}_\lambda\,:\,\lambda\gt 0)$ that is perfectly legitimate in our general settings. However, a phase-transition in $\lambda$ is clearly impossible. For example, let $\mathscr{G}_1$ be a classical WDRCM with no long edges (i.e. $\mathscr{L}$ $_1$ is satisfied) but for which the annuli are crossed (still for $\lambda=1$ ). With our definition of $\mathscr{G}_\lambda$ , we immediately infer the properties $\mathscr{L}$ $_\lambda$ and $\overline{\mathscr{M}}$ $_\lambda$ for all values of $\lambda$ (for mixing, note the model satisfies the independent settings condition and, thus, has no relevant correlations), but still $\widehat{\lambda}_{\textsf{c}}=0$ . It is clear that this model does neither satisfy monotonicity (Mon) nor $\overline{\mathscr{L}}$ $_\lambda$ . Another interesting example in this class k-nearest-neighbour percolation [Reference Balister and Bollobás2], which contains an infinite outgoing component if k is large enough, independently of $\lambda$ . This model also satisfies the properties $\mathscr{L}$ $_\lambda$ and $\mathscr{M}$ $_\lambda$ for every $\lambda\gt 0$ , yet it neither satisfies $\overline{\mathscr{L}}$ $_\lambda$ nor $\overline{\mathscr{M}}$ $_\lambda$ for any $\lambda\gt 0$ .

4.1. Proofs of the results in Section 1.3

We give the proofs of our main Theorems 1, 2, and 3 in this section. We start with the proof of Theorem 1 as the other results rest on either its proof or its statement. To this end, we adapt a renormalization technique developed by Gouéré for the Poisson–Boolean model [Reference Gouéré16].

Proof of Theorem 1. Let us start with the proof of part (ii). Fix any $\lambda\gt 0$ . Then, as $\mathscr{L}$ $_\lambda$ is not satisfied, we have $\mathbb{P}(\mathcal{L}(r,3))\gt c$ for all $r\geq 1$ . The event $\mathcal{L}(r,3)$ implies the existence of an edge connecting a vertex in B(r) to some vertex at distance at least 3r. The latter vertex is located outside B(2r) by necessity. Therefore

\begin{equation*} \mathbb{P}_{\lambda}\big(B(r)\xleftrightarrow[]{}B(2r)^\textsf{c}\big)\geq \mathbb{P}_\lambda(\mathcal{L}(r,3))\gt c \end{equation*}

and, $\lambda\gt \widehat{\lambda}_{\textsf{c}}$ . As $\lambda$ can be chosen arbitrarily small, this yields $\widehat{\lambda}_{\textsf{c}}=0$ , proving part (ii).

In order to prove part (i) of the theorem, we have to show that $\widehat{\lambda}_{\textsf{c}}\gt 0$ whenever $\mathscr{G}$ fulfils the properties $\overline{\mathscr{M}}$ $_{\bar\lambda}$ and $\overline{\mathscr{L}}$ $_{\bar\lambda}$ for some $\bar\lambda\gt 0$ . Let us denote the annulus-crossing event by

(13) \begin{equation} \mathcal{E}(x,r) = \{B(x,r)\xleftrightarrow[]{} B(x,2r)^{\textsf{c}}\} \end{equation}

and abbreviate $\mathcal{E}(r)=\mathcal{E}(o,r)$ . Then the claim is equivalent to showing $\mathbb{P}_\lambda(\mathcal{E}(r))\to 0$ , as $r\to\infty$ , for $\lambda$ sufficiently small. To this end, we adapt the renormalization scheme of [Reference Gouéré16]. Let us introduce two additional events. First, we let

\begin{equation*} \mathcal{F}(r) \,:\!=\, \{\exists \boldsymbol{{x}}\sim \boldsymbol{{y}}\,:\, |x|\lt 2r, |x-y|\gt r/10\} = \mathcal{L}(2r,1/20), \end{equation*}

be the event that there exists an edge longer than $r/10$ incident to some vertex located in B(2r). Secondly, we recall the local annulus-crossing event $\mathcal{G}(x,r)$ with $\mathcal{G}(r)=\mathcal{G}(o,r)$ , introduced in (1). In words, the event $\mathcal{E}(r)$ defines the annulus crossing subject to our main result. The event $\mathcal{F}(r)$ describes the presence of long edges in the considered annulus. In particular, if $\mathcal{F}(r)$ is not fulfilled, any annulus crossing cannot use far apart vertices. The event $\mathcal{G}(r)$ can then be seen as a localized variant of $\mathcal{E}(r)$ when no such far apart vertex is used and any path leaving B(2r) thus ends in $B(3r)\setminus B(2r)$ . Naturally, $\mathcal{G}(r)\subset\mathcal{E}(r)$ . The main step in the renormalization is then to argue that either long edges in the sense of $\mathcal{F}(r)$ are present or the localized annulus crossing event occurs twice almost independently on a smaller scale due to the mixing property. To make this precise, let us denote by $\partial B $ the boundary of a set $B\subset\mathbb{R}^d$ . Further let $I,J\subset\mathbb{R}^d$ be two finite sets satisfying $I\subset\partial B(10)$ and $J\subset\partial B(20)$ such that

\begin{equation*} \partial B(10)\subset\bigcup_{i\in I}B(i,1/2) \quad \text{and} \quad \partial B(20)\subset\bigcup_{j\in J} B(j,1/2). \end{equation*}

Put differently, I and J induce a covering of the spheres $\partial B(10)$ and $\partial B(20)$ , respectively, with balls of radius $1/2$ . Note that this particularly implies that the annulus $B(10.5)\setminus B(9.5)$ is completely covered by $\bigcup_{i\in I} B(i,1)$ , that is, using radius-1 balls instead of radius- $1/2$ balls. This covering property particularly implies that any vertex located in said annulus is at distance at most 1 to some point in I. The same is true for $B(20.5)\setminus B(19.5)$ and the union $\bigcup_{j\in J}B(j,1)$ . The outlined key observation is then

(14) \begin{equation} \mathcal{E}({10 r})\setminus \mathcal{F}(10r)\subset \bigg(\bigcup_{i\in I}\mathcal{G}(r\cdot i,r )\bigg)\cap\bigg(\bigcup_{j\in J}\mathcal{G}(r\cdot j,r)\bigg), \end{equation}

where we use the notation $r\cdot i$ for the product of r and i to put emphasis on the fact that i is an element of $\mathbb{R}^d$ while $r\in\mathbb{R}$ . Equation (14) follows from the fact that on $\mathcal{E}({10 r})\setminus \mathcal{F}(10r)$ , there exists a path from a vertex located in B(10r) to some vertex located outside of B(20r) using no edges longer than r. Therefore, at least one vertex of said path must lie within distance $r/2$ to the sphere $\partial B(10 r)$ and, thus, within distance r to some point in $r\cdot I$ . Put differently, this vertex is located in $B(r\cdot i, r)$ for some $i\in I$ . Restarting the path from this vertex, it realizes the localized annulus-crossing event $\mathcal{G}(r\cdot i, r)$ , as the original path must ultimately reach a vertex located in $B(20 r)^\textsf{c}$ using only edges shorter than r, and, in particular, it must first reach some vertex located in $B(r\cdot i, 3r)\setminus B(r\cdot i, 2r)$ passing only vertices located in $B(r\cdot i, 3r)$ . Again, we have used that all edges are shorter than r. By a similar argumentation also $\mathcal{G}(r\cdot j,r)$ occurs for some $j\in J$ , justifying (14). The above argument is sketched in Figure 4.

Figure 4.

Sketch of (14) in $d=2$ . A path starting inside B(10r) and leaving B(20r), where no edge longer than r is used. The depicted vertex in B(10 r) on the path is close to the centre of the solid ball of the covering of $\partial B(10r)$ and it is the starting point for the event $\mathcal{G}($ solid balls). Further, the covering of the annulus $B(10.5 r)\setminus B(9.5r)$ is indicated. The same applies to the vertex on the path, which lies close to the center of the dashed ball of the covering of $\partial B(20r)$ .

As a result, we obtain

\begin{equation*} \mathbb{P}_\lambda(\mathcal{E}(10 r)\setminus \mathcal{F}(10 r))\leq \sum_{i\in I, \, j\in J}\mathbb{P}_\lambda(\mathcal{G}(r\cdot i,r)\cap \mathcal{G}(r\cdot j,r)). \end{equation*}

Using translation invariance (G2), we infer, writing $C_1=(\sharp I)(\sharp J)$ ,

\begin{equation*} \begin{aligned} \sum_{i\in I, \, j\in J} \mathbb{P}_\lambda(\mathcal{G}(r\cdot i,r)\cap \mathcal{G}(r\cdot j,r)) & \leq C_1 \max_{i\in I, j\in J}\big(\mathbb{P}_\lambda(\mathcal{G}(r\cdot i,r)\cap \mathcal{G}(r\cdot j,r))\big) \\ & \leq C_1\max_{i\in I, j\in J}\big(\!\operatorname{\rm Cov}_\lambda({\unicode{x1D7D9}}_{\mathcal{G}(o,r)}, {\unicode{x1D7D9}}_{\mathcal{G}(r\cdot (j-i), r)})\big) + C_1 \mathbb{P}_\lambda(\mathcal{G}(r))^2. \end{aligned} \end{equation*}

Now, as I and J are finite sets, we can use the mixing property $\overline{\mathscr{M}}$ $_{\bar\lambda}$ and write

(15) \begin{equation} \operatorname{\rm cov}(r) \,:\!=\, C_1\sup_{\lambda\lt \bar\lambda}\max_{i\in I, j\in J}\big(\!\operatorname{\rm Cov}({\unicode{x1D7D9}}_{\mathcal{G}(o,r)},{\unicode{x1D7D9}}_{\mathcal{G}(r\cdot (j-i), r)})\big) = o(1). \end{equation}

Using $\mathcal{G}(r)\subset\mathcal{E}(r)$ , we hence obtain

(16) \begin{equation} \mathbb{P}_\lambda(\mathcal{E}(10r)) \leq \mathbb{P}_{\lambda}(\mathcal{F}(10r)) +\operatorname{\rm cov}(r) + C_1 \mathbb{P}_\lambda(\mathcal{E}(r))^2, \end{equation}

which is key in showing $\mathbb{P}_{\lambda}(\mathcal{E}(r))\to 0$ for sufficiently small $\lambda$ . Recall that we work under the assumption $\overline{\mathscr{L}}$ $_{\bar\lambda}$ for some $\bar\lambda\gt 0$ . Hence, $\mathbb{P}_\lambda(\mathcal{F}(r))\to 0$ uniformly for all $\lambda\lt \bar\lambda$ . Therefore, there exists $r_0\gt 1$ such that

\begin{equation*} C_1\Big(\sup_{\lambda\lt \bar\lambda}\mathbb{P}_\lambda(\mathcal{F}(r))+\operatorname{\rm cov}(r)\Big) \leq 1/4, \quad \text{for all }r\geq r_0. \end{equation*}

Let us define the following two functions on $[r_0,\infty)$ :

(17) \begin{equation} g(r)\,:\!=\, C_1\Big(\sup_{\lambda\lt \bar\lambda}\mathbb{P}_\lambda(\mathcal{F}(r))+\operatorname{\rm cov}(r)\Big) \quad \text{and} \quad f_\lambda(r) = C_1 \mathbb{P}_{\lambda}(\mathcal{E}(r)), \end{equation}

and choose

(18) \begin{equation} \lambda \lt \bar\lambda \wedge \tfrac{1}{2\, C_1 \, (10 r_0)^d \omega_d}, \end{equation}

where $\omega_d$ denotes the volume of the d-dimensional unit ball. Since $\lambda \lt \bar\lambda$ , we have $g(r)\leq 1/4$ . Further, since $\lambda \lt 1/(2C_1(10 r_0)^d \omega_d)$ , we have with Markov’s inequality

(19) \begin{equation} f_\lambda(r) = C_1\mathbb{P}_\lambda(\mathcal{E}(r))\leq C_1\mathbb{E}_\lambda[\sharp(\mathscr{V} \cap B(r))]=C_1 \lambda \omega_d r^d \leq 1/2, \end{equation}

for all $r\in[r_0,10 r_0]$ , where the second to last step follows from the fact that the intensity measure of the underlying point process is the Lebesgue measure multiplied by $\lambda$ . In addition, (16) implies

(20) \begin{equation} f_\lambda(r) \leq g(r) + f_\lambda(r/10)^2 \quad \text{for all }r\geq 10 r_0. \end{equation}

Together with the established bounds on $f_\lambda$ and g, we infer $f_\lambda(r)\leq 1/2$ for all $r\geq r_0$ . This can, for instance, easily be seen by proving inductively $f_\lambda(r)\leq 1/2$ for all $r\in[10^{n-1}r_0,10^n r_0]$ . Applying the bound $f_\lambda\leq 1/2$ as well as (20) iteratively yields for all $n\in\mathbb{N}$ and $r\in[r_0,10 r_0]$ ,

\begin{equation*} f_\lambda(10^{n} r)\leq g(10^{n}r)+ \frac{f_\lambda(10^{n-1}r)}{2} \leq \cdots \leq \sum_{j=1}^n \frac{g(10^{j}r)}{2^{n-j}} + 2^{-n-1}. \end{equation*}

Finally, for $\varepsilon\gt 0$ , we can choose n large enough such that

\begin{equation*} \sum_{j=1}^n \frac{g(10^{j}r)}{2^{n-j}} \leq \frac{1}{4}\sum_{j=1}^{n/2} 2^{-n+j} + \Big(\max_{n/2\leq k\leq n }g(10^k r)\Big)\sum_{j=n/2}^n 2^{-n+j} \leq 2^{-n/2} + 2 \max_{n/2\leq k\leq n}g(10^k r) \leq \varepsilon, \end{equation*}

uniformly for $r\in[r_0, 10r_0]$ . Note here that the $\max g(10^k r)$ term on the right-hand side is small as the function g captures exactly the two assumptions of the theorem as it is defined via the corresponding events, cf. (17). In conclusion, $\mathbb{P}_\lambda(\mathcal{E}(r))=f_\lambda(r)/C_1\to 0$ as $r\to \infty$ , for all $\lambda$ satisfying (18), proving part (i). This finishes the proof of Theorem 1.

We continue with the proof of Theorem 3 as it builds on the arguments derived in the previous proof and postpone the proof of Theorem 2 to the end of the section.

Proof of Theorem 3. Recall that we work under the assumption ${{\mathscr{P}}\!\!{\mathscr{M}}}$ $_\lambda^\xi$ for some $\xi_\lambda\lt 0$ and recall the decay exponent $\zeta_\lambda\lt 0$ describing the polynomial decay of $\mathbb{P}_\lambda(\mathcal{L}(r,1))$ . As $\mathbb{P}_{\lambda}(\mathcal{E}(r))$ is a decreasing function in r, the claim of Theorem 3 is equivalent to showing that $r^{s-1}\mathbb{P}_{\lambda}(\mathcal{E}(r))$ is integrable for all $s\lt |\zeta_\lambda \vee \xi_\lambda|$ , where $\mathcal{E}(r)$ denotes the annulus-crossing event as introduced in (13). Note that the polynomial mixing condition can equivalently be written as

\begin{equation*} \big|\operatorname{\rm Cov}_\lambda\big({\unicode{x1D7D9}}_{\mathcal{G}(r)},{\unicode{x1D7D9}}_{\mathcal{G}(xr,r)}\big)\big| \leq \ell_\lambda(x,r) r^{\xi_\lambda}, \end{equation*}

where $r\mapsto\ell_\lambda(x,r)$ is a slowly varying function [Reference Bingham, Goldie and Teugels3]. Hence, the mixing term in (15) now reads

\begin{equation*} \operatorname{\rm cov}_\lambda(r) \leq \ell_\lambda(r) r^{\xi_\lambda}, \quad \text{where} \ \ell_\lambda(r)= \sum_{i\in I,\, j\in J} \ell_{\lambda}(j-i, r). \end{equation*}

Note that $\ell_{\lambda}(r)$ again is slowly varying. As above there exists $r_0^{(\lambda)}$ such that the, now $\lambda$ -dependent, function $g_\lambda$ satisfies

\begin{equation*} g_\lambda(r)= C_1\big(\mathbb{P}_\lambda(\mathcal{F}(r))+\operatorname{\rm cov}_\lambda(r)\big)\lt 1/4 \end{equation*}

on $[r_0^{(\lambda)},\infty)$ . Recall further the function $f_{\lambda}$ from the previous proof, cf. (17). Essentially, we have reestablished the same setting as above but now for the fixed value of $\lambda$ instead of taking the supremum over $\lambda\lt \bar\lambda$ . By the definition of $g_\lambda(r)$ and as $\mathbb{P}_\lambda(\mathcal{F}(r))\asymp\mathbb{P}_\lambda(\mathcal{L}(r,1))$ , we have that $r^{s-1}g_\lambda(r)$ is integrable for all $s\lt |\zeta_\lambda \vee \xi_\lambda|$ . To prove the claimed integrability of $\mathbb{P}_\lambda(\mathcal{E}(r)) =f_\lambda(r)/C_1$ , observe first that $f_\lambda(r)=o(1)$ for $\lambda\lt \widehat{\lambda}_{\textsf{c}}$ . In particular, $f_\lambda(r)\leq 10^{-s-1}/2$ for all $r\geq R_\lambda/10$ for some sufficiently large $R_\lambda\geq r_0^{(\lambda)}$ . Following the arguments in [Reference Gouéré16], we find for $S\gt 10 R_\lambda$ , using (20),

\begin{align*} \int_{R_\lambda}^{S} r^{s-1} f_\lambda(r)\,\mathrm{d} r & \leq \int_{R_\lambda}^S r^{s-1}f_\lambda(r/10)^2\,\mathrm{d} r +\int_{R_\lambda}^{S} r^{s-1} g_\lambda(r) \,\mathrm{d} r \\ & \leq \tfrac{1}{2} \int_{R_\lambda/10}^{S/10} r^{s-1} f_\lambda(r) \,\mathrm{d} r + \int_{R_\lambda}^{S} r^{s-1} g_\lambda(r) \,\mathrm{d} r \\ & \leq \tfrac{1}{2}\int_{R_\lambda/10}^{R_\lambda} r^{s-1} f_\lambda(r) \,\mathrm{d} r + \tfrac{1}{2}\int_{R_\lambda}^{S} r^{s-1}f_\lambda(r) \,\mathrm{d} r + \int_{R_\lambda}^\infty r^{s-1} g_\lambda(r) \,\mathrm{d} r, \end{align*}

where we used the change of variables $r\mapsto 10r$ together with the bound $f_\lambda(r)\leq 10^{-s-1}/2$ in the second step. Sending $S\to\infty$ , the established integrability of $r^{s-1}g_\lambda(r)$ then yields

\begin{equation*} \tfrac{1}{2}\int_{R_\lambda}^\infty r^{s-1}f_\lambda(r) \leq \tfrac{1}{2}\int_{R_\lambda/10}^R r^{s-1} f_\lambda(r)\, \mathrm{d} r + \int_{R_\lambda}^\infty r^{s-1} g_\lambda(r) \,\mathrm{d} r \lt \infty, \end{equation*}

for all $s\lt |\zeta_\lambda \vee \xi_\lambda|$ . This concludes the proof of Theorem 3.

Next, we give the proof of Theorem 2 stating that the rareness of long edges and the positivity of $\widehat{\lambda}_{\textsf{c}}$ are equivalent in the independent setting.

Proof of Theorem 2. Recall that in the independent setting, and for $\lambda\lt \lambda'$ , the graph $\mathscr{G}_\lambda$ can be constructed from $\mathscr{G}_{\lambda'}$ by performing site percolation with retention parameter $\lambda/\lambda'$ , as explained below Definition 2. Hence, we have

\begin{equation*} \mathbb{P}_{\lambda}(\mathcal{L}(r,1)) \le \mathbb{P}_{\lambda'}(\mathcal{L}(r,1)). \end{equation*}

as well as

\begin{align*} \mathbb{P}_{\lambda}(\mathcal{L}(r,1)) &= \mathbb{P}_{\lambda'}\big(\exists \boldsymbol{{x}}\sim \boldsymbol{{y}}\,:\, |x|\lt r, |x-y|\gt r, \boldsymbol{{x}},\boldsymbol{{y}} \text{ survive percolation}\big) \\ & \geq \mathbb{P}_{\lambda'}\big(\exists \boldsymbol{{x}}\sim \boldsymbol{{y}}\,:\, |x|\lt r, |x-y|\gt r, \ \substack{\text{of these edges the one with endpoint} \\ \text{closest to the origin survives percolation}}\big) \\ & =\big(\tfrac{\lambda}{\lambda'}\big)^2\mathbb{P}_{\lambda'}(\mathcal{L}(r,1)), \end{align*}

where an edge survives percolation if both of its endpoints do. Both inequalities combined prove the independence of $\mathscr{L}$ $\,_\lambda$ and $\overline{\mathscr{L}}$ $\,_{\lambda}$ from $\lambda$ and show that both probabilities have the same tail. The equivalence of $\overline{\mathscr{L}}$ $_{1}$ and $\widehat{\lambda}_{\textsf{c}}\gt 0$ is then an immediate consequence of Theorem 1.

4.2. Proofs of the results in Sections 2 and 3

In this section we prove the results established for the WDRCM and all the other examples.

4.2.1. Proofs for the WDRCM of Section 2.1

Recall first the framework of the WDRCM. In particular, recall that vertices $\boldsymbol{{x}}=(x,u_x)$ are given via a homogeneous Poisson point process on $\mathbb{R}^d\times(0,1)$ and recall the connection mechanism given in (3). Key in proving the results for WDRCMs and, in particular, Theorem 4 is to properly relate the tail behaviour of $\mathbb{P}_{\lambda}(\mathcal{L}(r,1))$ with the exponent $\zeta$ , derived in (5). To this end, we need some auxiliary result about the involved vertex marks. Indeed, when we introduced $\zeta$ heuristically, we claimed that the empirical distribution of the ‘weak’ vertices in a set consisting of $r^d$ vertices should roughly be $r^d {\unicode{x1D7D9}}_{[r^{-d+d\zeta},1]}$ to argue that $\zeta$ describes the occurrence of long edges appropriately. Before giving the proof of Theorem 4, we shall make this claim precise.

Recall that, given the vertices’ locations, the associated vertex marks form an independent collection of uniformly distributed random variables. We are thus interested in some kind of regular behaviour of independent uniforms guaranteeing that the marks are evenly distributed over the interval. To this end, consider a collection of independent random variables $U=(U_1,\ldots,U_n)$ distributed uniformly on (0,1) and denote their joint distribution by P. We follow the arguments of [Reference Gracar, Lüchtrath and Mönch22] and fix a parameter $0\lt a\lt 1$ and define for all $i=1,\ldots,\lfloor n^{1-a}\rfloor$

\begin{equation*} N^a_n(i) = \sum_{j=1}^n {\unicode{x1D7D9}}\left\{U_j\leq\frac{i}{\lfloor n^{1-a}\rfloor}\right\}. \end{equation*}

We say that U is a-regular if

\begin{equation*} N^a_n(i)\geq \frac{i n}{2\lfloor n^{1-a}\rfloor}, \quad \text{for all } i=1,\ldots,\lfloor n^{1-a}\rfloor. \end{equation*}

The following lemma shows that a-regularity provides the desired regularity of the vertex marks.

Lemma 2.

1. (i) For the collection $U=(U_1,\ldots,U_n)$ of independent uniform distributions with joint distribution P and any $a\in(0,1)$ , we have

   \begin{equation*} P(U \text{ is } a\text{-regular}) \leq \mathrm{e}^{-c n^a}, \end{equation*}
   for some constant $c\gt 0$ and $n\in\mathbb{N}$ large enough.

2. (ii) Consider an a-regular realization $u=(u_1,\ldots,u_n)$ with $n\gt 2^{1/(1-a)}$ . Then, for any non-increasing function $\varphi\,:\, [0,1]\to[0,\infty)$ , we have

   \begin{equation*} \sum_{j=1}^n \varphi(u_j)\geq \frac{n}{2}\int_{2n^{a-1}}^1 \varphi(t) \, \mathrm{d} t. \end{equation*}

Proof. A simple calculation yields

\begin{equation*} E [N^a_n(i)] = \frac{in}{\lfloor n^{1-a}\rfloor}, \end{equation*}

so that the standard Chernoff bound for independent Bernoulli random variables implies

\begin{equation*} P\big(N^a_n(i)\lt \tfrac{1}{2}E[N^a_n(i)]\big)\leq \mathrm{e}^{-c' {\rm i} n^a} \end{equation*}

for some constant $c'\gt 0$ . Therefore, a union bound yields

\begin{equation*} \begin{aligned} P(U \text{ is not } a\text{-regular})\leq n^{1-a}\mathrm{e}^{-c' n^a} \leq \mathrm{e}^{-c n^a}, \end{aligned} \end{equation*}

for some $c\lt c'$ and n large enough as $a\lt 1$ ; thus proving (i).

To prove part (ii), we first derive bounds on the empirical distribution function F of an a-regular collection $u=(u_1,\ldots,u_n)$ . Namely, we have on the event of a-regularity, for any $t\in(0,1)$ and $n\geq 2$ ,

\begin{align*} n F(t) & = \sum_{i=1}^n {\unicode{x1D7D9}}\{u_i\leq t\} \geq \sum_{i=1}^{\lfloor n^{1-a}\rfloor}N^a_n(i-1){\unicode{x1D7D9}}\big\{i-1\lt \lfloor n^{1-a}\rfloor t\leq i\big\} = N^a_n(\lfloor t \lfloor n^{1-a}\rfloor\rfloor) \nonumber\\ & \geq \frac{n \lfloor t\lfloor n^{1-a}\rfloor\rfloor}{2\lfloor n^{1-a}\rfloor}\geq \frac{n}{2}\left(t-\frac{1}{\lfloor n^{1-a}\rfloor}\right)\geq \frac{n}{2}(t-2n^{a-1}). \end{align*}

Now observe that the function $G(t)=(t-2n^{a-1})/2$ induces the finite measure $\mathrm{d} t /2$ on $(2n^{a-1},1)$ , which is non-empty as $n\gt 2^{1/(1-a)}$ . This particularly implies for all non-increasing functions $\varphi(t)$ and an a-regular realization $u=(u_1,\ldots,u_n)$

\begin{equation*} \sum_{j=1}^n \varphi(u_j) \geq n \int_{0}^{\varphi(2n^{a-1})} \mathrm{d} x \, F(\varphi^{-1}(x)) \geq \frac{n}{2}\int_{2n^{a-1}}^1 \mathrm{d} t \, \varphi(t), \end{equation*}

where $\varphi^{-1}$ denotes the generalized inverse of $\varphi$ . This concludes the proof.

We have now everything in place in order to prove Theorem 4.

Proof of Theorem 4, subject to Theorem 5. Let us start with the proof of part (i) and assume $\zeta\lt 0$ . Note that Theorem 5(i) (that is independently proved in the following) implies

\begin{equation*} \limsup_{r\to\infty}\frac{\log \mathbb{P}_1(\mathcal{L}(r,1))}{\log r}\leq {d\zeta}. \end{equation*}

As the WDRCM is monotone and takes place in the independent setting, we have $\widehat{\lambda}_{\textsf{c}}\gt 0$ by Theorem 2. Let us assume that we had already proved the corresponding lower bound

(21) \begin{equation} \limsup_{r\to\infty}\frac{\log\mathbb{P}_1(\mathcal{L}(r,1))}{\log r} \ge d\zeta. \end{equation}

Then, for all $\lambda\lt \widehat{\lambda}_{\textsf{c}}$ by Theorem 5 and Lemma 1

\begin{equation*} \limsup_{r\to\infty}\frac{\log\mathbb{P}_\lambda(\mathcal{E}(r))}{\log\mathbb{P}_1(\mathcal{L}(r,1))}\leq 1, \end{equation*}

where $\mathcal{E}(r)$ again denotes the annulus-crossing event as introduced in (13). As $\mathbb{P}_1(\mathcal{L}(r,1))\asymp\mathbb{P}_\lambda(\mathcal{L}(r,3))\leq \mathbb{P}_\lambda(\mathcal{E}(r))$ by Lemma 1, we find the corresponding lower bound for the $\limsup$ of the long edge probability and have, thus, established the claimed polynomial decay of the annulus-crossing probability. It therefore remains to verify the lower bound (21). To this end, we fix the intensity $\lambda=1$ and consider the event

\begin{equation*} \bar{\mathcal{L}}(r) = \big\{\exists \boldsymbol{{x}}\sim\boldsymbol{{y}}\,:\, x\in B(r), y\in B(3r)\setminus B(2r)\big\} \subset \mathcal{L}(r,1). \end{equation*}

We hence aim to show that for all $\mu\lt \zeta$ arbitrarily close to $\zeta$ and sufficiently large r, we have

\begin{equation*} \log\mathbb{P}_1(\bar{\mathcal{L}}(r)) \geq d\mu\log r. \end{equation*}

Fix such $\mu\lt \zeta$ . We provide the claimed lower bound for the probability of $\bar{\mathcal{L}}(r)$ by only considering connections between two vertices with marks greater than $r^{-d+d\mu}$ . Recall the integral

\begin{equation*} I(\mu, r)= r^{2d}\int_{r^{-d+d\mu}}^1 \mathrm{d} u \int_{r^{-d+d\mu}}^1 \mathrm{d} v \, \varphi(u,v,r^d). \end{equation*}

By our definition of $\psi$ in (4) and our assumption on $\mu$ , we have

\begin{equation*} \limsup_{r\to\infty} \frac{\log I(\mu,r)}{d\log r}= 2+\psi(\mu)\gt \mu. \end{equation*}

We let $\varepsilon\,:\!=\, (2+\psi(\mu)-\mu)/3\gt 0$ and split the integral according to the domain of integration into

\begin{equation*} I(\mu,r)=I^{(1)}(\mu,r)+ I^{(2)}(\mu,r)+ I^{(3)}(r), \end{equation*}

where the integral $I^{(1)}$ runs over the domain $\{r^{-d+d\mu}\lt u\lt r^{-d+d\varepsilon}, u\lt v\lt 1\}$ , $I^{(2)}$ over the domain $\{r^{-d+d\mu}\lt v\lt r^{-d+d\varepsilon}, v\lt u\lt 1\}$ , and $I^{(3)}$ over the domain $\{u,v\in [r^{-d+d\varepsilon},1]\}$ . Let us recall here that $\mu\lt 0$ while $\varepsilon\gt 0$ to avoid confusion. Hence, by the Laplace principle, for at least one $j\in\{1,2,3\}$ , we must have

(22) \begin{equation} \limsup_{r\to\infty} \frac {\log I^{(j)}(\mu,r)}{d\log r}= 2+\psi(\mu). \end{equation}

We have to consider the three cases associated to which j satisfies equation (22). Let us first assume that it holds true for $j=1$ . We can then fix a sequence $r_n$ diverging to infinity such that $I^{(1)}(\mu,r_n)\geq r_n^{d\mu+2d\varepsilon}$ . We define

\begin{equation*} \Phi(u) \,:\!=\, r_n^d\int_{r_n^{-d+d\mu}}^1 \mathrm{d} v \, \varphi\big(u,v,r_n^d\big) \end{equation*}

and infer

(23) \begin{equation} r_n^{d} \int_{r_n^{-d+d\mu}}^{r_n^{-d+d\varepsilon}}\mathrm{d} u \,\Phi(u) \geq I^{(1)}(\mu,r_n)\geq r_n^{d\mu+2d\varepsilon}. \end{equation}

Hence, there must exists $u_0\in[r_n^{-d+d\mu},r_n^{-d}]$ such that $u_0 \Phi(u_0)\geq r_n^{-d+d\mu}$ . Indeed, if this was not the case, we would infer

\begin{equation*} I^{(1)}(\mu,r_n) \leq r_n^{d\mu} \int_{r_n^{-d+d\mu}}^{r_n^{-d}} \frac{\mathrm{d} u}{u} + r_n^{d\varepsilon}\Phi\big(r_n^{-d}\big) \leq r_n^{d\mu}\log r_n^{-d\mu} + r_n^{d\mu+d\varepsilon}, \end{equation*}

where we used monotonicity of $\varphi$ in the vertex mark arguments and the assumed bound on $u\Phi(u)$ . This, however, contradicts (23) and, thus, yields the existence of $u_0$ . If there is more than one such $u_0$ , fix any and observe that the probability of finding a vertex located in $B(r_n)$ with mark at most $u_0$ is, by standard Poisson process properties together with $\lambda=1$ ,

\begin{equation*} 1- \exp\big({-}\omega_d r_n^d u_0\big) \geq \tfrac{\omega_d }{2} r_n^d u_0, \end{equation*}

for large enough n (and, thus, $r_n$ ). Conditioned on the existence of such a vertex, the number of neighbours it has located in $B(3r_n)\setminus B(2r_n)$ is a Poisson random variable with parameter at least $\Phi(u_0)$ . This is immediate from the Poisson structure of neighbourhoods in the WDRCM [Reference Lüchtrath41] and monotonicity of $\varphi$ in the vertex marks arguments. Hence, the probability of forming at least one connection from the given vertex to $B(3r_n)\setminus B(2r_n)$ is lower bounded by

\begin{equation*} 1-\exp({-}\Phi(u_0)) = \Omega(\Phi(u_0)\wedge 1). \end{equation*}

Hence, we find a constant $c\gt 0$ , depending on model parameters and the dimension only, such that, for all $\mu\lt \zeta$ and sufficiently large $r_n$ , we have

\begin{equation*} \mathbb{P}_1(\bar{\mathcal{L}}(r_n))\geq c r_n^d u_0(\Phi(u_0)\wedge 1) = c\big(r_n^d u_0 \Phi(u_0)\big)\wedge \big(r_n^d u_0\big)\geq c r_n^{d\mu}, \end{equation*}

as desired. Note that the case (22) for $j=2$ follows from the exact same arguments by reversing the roles of u and v.

It thus remains to consider the case where (22) is fulfilled for $j=3$ . To this end, recall $\varepsilon=(2+\psi(\mu)-\mu)/3 \in(0,1)$ introduced to divide the integration domain, where $\varepsilon\lt 1$ is a result of only considering $\mu\lt \zeta$ that are sufficiently close to $\zeta$ . Again, we want to establish the existence of an edge between a vertex in B(r) and one in $B(3r)\setminus B(2r)$ . For this, let us denote by $\mathscr{V}_1=\{\boldsymbol{{x}}_1=(x_1,u_1),\ldots,\boldsymbol{{x}}_{N_1}=(x_{N_1},u_{N_1})\}$ the vertices of $\mathscr{V}$ restricted to locations in B(r) and by $\mathscr{V}_2=\{\boldsymbol{{y}}_1=(y_1,v_1),\ldots,\boldsymbol{{y}}_{N_2}=(y_{N_2},v_{N_2})\}$ those vertices with locations in $B(3r)\setminus B(2r)$ . Here, $N_1$ and $N_2$ are Poisson random variables with mean of order $r^d$ . Applying the Chernoff bound for Poisson random variables, there are constants $c',c\gt 0$ , such that

(24) \begin{equation} \mathbb{P}_{1}(N_1\lt c' r^d) + \mathbb{P}_1(N_2\lt c' r^d) \leq \mathrm{e}^{-c r^d}. \end{equation}

To ease notation, we treat $c'=1$ and additionally $r^d\in\mathbb{N}$ in the following and denote by $c\gt 0$ and $C\gt 0$ constants that may change from line to line. Further, we may assume without loss of generality that the vertices in $\mathscr{V}_1$ and $\mathscr{V}_2$ may be indexed according to their lexicographic order. We now make use of the previously introduced regularity of uniform random variables applied to the vertex marks in $\mathscr{V}_1$ and $\mathscr{V}_2$ . For the parameter $a=\varepsilon$ , we say that $\mathscr{V}_1$ is $\varepsilon$ -regular, if $N_1\geq r^d$ and if the collection of vertex marks $u_1,\ldots,u_{r^d}$ is $\varepsilon$ -regular. Put differently, $\mathscr{V}_1$ is $\varepsilon$ -regular, if it contains sufficiently many vertices such that the first $r^d$ indexed vertex marks form a $\varepsilon$ -regular collection in the above sense. The same definition applies to $\mathscr{V}_2$ by replacing $N_1$ with $N_2$ and the marks $u_i$ with the marks $v_i$ . Applying Lemma 2 and (24) yields

(25) \begin{equation}\begin{aligned} \mathbb{P}_1(\neg\bar{\mathcal{L}}(r)) & \leq \mathbb{P}_{1}\big(\neg\bar{\mathcal{L}}(r)\cap \{\mathscr{V}_1,\mathscr{V}_2 \text{ are }\varepsilon\text{-regular}\}\big)+ \mathrm{e}^{-c r^d} +2 \mathrm{e}^{-c r^{d\varepsilon}} \\ & \leq \mathbb{P}_{1}\big(\neg\bar{\mathcal{L}}(r)\cap \{\mathscr{V}_1,\mathscr{V}_2 \text{ are }\varepsilon\text{-regular}\}\big)+ \mathrm{e}^{-c r^{d\varepsilon}}, \end{aligned}\end{equation}

for all large enough r, where the last step follows from $\varepsilon\lt 1$ . It remains to bound the probability on the right-hand side. To this end, let us write $\mathcal{A}=\{\mathscr{V}_1,\mathscr{V}_2 \text{ are }\varepsilon\text{-regular}\}$ . Note that the maximal distance between any two vertices in $\mathscr{V}_1$ and $\mathscr{V}_2$ is bounded by 6r. As $\varphi$ is non-increasing in the third argument, we infer

(26) \begin{equation}\begin{aligned}\mathbb{P}_1(\neg \bar{\mathcal{L}}(r)\cap\mathcal{A}) & \leq \mathbb{P}_1\Bigg(\bigcap_{i,j=1}^{r^d}\{\boldsymbol{{x}}_i\not\sim\boldsymbol{{y}}_j\}\cap \mathcal{A}\Bigg) \leq \mathbb{E}_1\Bigg[{\unicode{x1D7D9}}_{\mathcal{A}} \prod_{\substack{i,\,j=1}}^{r^d} \mathrm{e}^{-\varphi\big(u_i,v_j,(6r)^d\big)}\Bigg]\\& =\mathbb{E}_1\Bigg[{\unicode{x1D7D9}}_{\mathcal{A}} \exp\Bigg({-}\sum_{i,j=1}^{r^d} \varphi\big(u_i,v_j,(6r)^d\big)\Bigg)\Bigg] \\& \leq \mathbb{P}_1(\mathcal{A})\exp\Bigg({-}C r^{2d}\int_{2r^{-d+d\varepsilon}}^{1} \mathrm{d} u \int_{2r^{-d+d\varepsilon}}^{1} \mathrm{d} v \,\varphi\big(u,v,(6r)^d\big) \Bigg)\\ & \leq \mathbb{P}_1(\mathcal{A})\exp\big({-}C I^{(3)}(r)\big),\end{aligned}\end{equation}

where we used Lemma 2 in the second to last step. To see the last step, recall that $I^{(3)}(r)$ is of polynomial order with exponent $2+\psi(\mu)$ by assumption. Therefore, $I^{(3)}(r)\asymp I^{(3)}(c r)$ for all constants $c\gt 0$ . We ultimately infer,

\begin{equation*} \mathbb{P}_1(\bar{\mathcal{L}}(r,1)) \geq \mathbb{P}_1(\bar{\mathcal{L}}(r)\cap \mathcal{A}) \ge \mathbb{P}_1(\mathcal{A})\big( 1- \exp({-}C I^{(3)}(r))\big) \end{equation*}

for sufficiently large r. By taking the logarithms on both sides, using our assumption on $I^{(3)}$ , as well as $\mathbb{P}_1(\mathcal{A})\gt 1/2$ for large r, we obtain the desired result. This finishes the proof of part (i).

In order to prove part (ii), we only have to show the claimed tail for $\mathbb{P}_1(\mathcal{L}(r,1))$ as the other statements are then immediate consequences of Theorem 2. Recall that we now have $\zeta\gt 0$ . The proof works with similar argumentation as that used in the $j=3$ case above. Indeed, fix $0\lt \mu\lt \zeta\leq 1$ but arbitrarily close to $\zeta$ and replace the $\varepsilon$ -regularity in the event $\mathcal{A}$ by $\mu$ -regularity. Then, for any $\varepsilon\gt 0$ small enough such that $d(2+\psi(\mu))-\varepsilon\gt 0$ , we have for large enough r that $I(\mu,r)\geq r^{d(2+\psi(\mu))-\varepsilon}$ . We infer with the same calculations performed in (29) for large enough r

\begin{align*} \mathbb{P}_1(\neg \bar{\mathcal{L}}(r)\cap\mathcal{A}) & \leq \mathbb{P}_1(\mathcal{A})\exp\bigg({-}C r^{2d} \int_{2r^{d\mu-d}}^{1} \mathrm{d} u \int_{2r^{d\mu-d}}^{1} \mathrm{d} v \, \varphi\big(u,v,(6r)^d\big) \bigg) \\ & \leq\mathbb{P}_1(\mathcal{A}) \exp\big({-}C r^{d(2+\psi(\mu))-\varepsilon}\big). \end{align*}

As $\varepsilon$ can be chosen arbitrarily small, we can find for each $\mu\lt \zeta$ some $R\geq 1$ and a constant $C\gt 0$ such that for all $r\gt R$ , by combining the final calculation with (25) (but now with respect to $\mu$ -regularity),

\begin{equation*} \mathbb{P}_1(\mathcal{L}(r,1))\geq \mathbb{P}_1(\bar{\mathcal{L}}(r))\geq 1 - \exp({-}C r^{d\mu}). \end{equation*}

This proves the claimed tail and, thus, Theorem 4.

We continue with the proof of Proposition 1 providing the values of $\zeta$ for the interpolation model, defined in (7) and (8).

Proof of Proposition 1. Recall that the connection probability for two given vertices $\boldsymbol{{x}}=(x,u_x)$ and $\boldsymbol{{y}}=(y,u_y)$ is given by $1\wedge (g(u_x,u_y) |x-y|^d)^{-\delta}$ , with $g(u_x,u_y)=(u_x\wedge u_y)^\gamma (u_x\vee u_y)^\alpha $ , in the interpolation model. We aim to derive the values of $\zeta$ . To this end, we calculate the defining integral of $\psi(\mu)$ in (4), namely $I(\mu,r)/r^{2d}$ . By the symmetry of g, this integral is twice the value obtained on the restricted domain in which $u_x\le u_y$ . After a change of variables with $t_x=-\log(u_x)/d\log r$ and $t_y=-\log(u_y)/d\log r$ , we obtain

\begin{align*} \frac {I(\mu,r)}{2r^{2d}} &= (d\log r)^2 \int_R \big(r^{d({-}\delta+(\delta \gamma-1) t_x + (\delta \alpha-1)t_y)}\wedge r^{d ({-}t_x-t_y)}\big) \, \mathrm{d} t_x \,\mathrm{d} t_y \\ & = (d\log r)^2 \int_R r^{d \chi(t_x,t_y)} \, \mathrm{d} t_x \,\mathrm{d} t_y, \end{align*}

where the integral runs over the domain $R\,:\!=\,\{(t_x,t_y)\in \mathbb{R}^2, 0\le t_y\le t_x\le 1-\mu\},$ and

\begin{equation*} \chi(t_x,t_y)\,:\!=\, \min\big\{{-}\delta+(\delta \gamma-1) t_x + (\delta \alpha-1)t_y, -t_x-t_y\big\}. \end{equation*}

Note that $\chi(t_x,t_y)=-t_x-t_y$ on the domain $\gamma t_x+\alpha t_y\ge 1$ , corresponding to the connection probability between the vertices $\boldsymbol{{x}}$ and $\boldsymbol{{y}}$ being equal to 1.

We first consider the case $\mu\ge 1-1/(\alpha+\gamma)$ , which yields $\chi(t_x,t_y)=-\delta+(\delta \gamma-1) t_x + (\delta \alpha-1)t_y$ in the whole domain R. If we further suppose that the terms $\delta \alpha-1$ , $\delta(\alpha+\gamma)-2$ , and $\delta \gamma-1$ are all non-zero, then the integral is easily computed to be equal to

\begin{equation*} \frac {I(\mu,r)}{2r^{2d}} =\frac {r^{-d\delta}}{\delta \alpha-1}\left( \frac {r^{d(\delta(\alpha+\gamma)-2)(1-\mu)}-1}{\delta(\alpha+\gamma)-2} - \frac {r^{d(\delta\gamma-1)(1-\mu)}-1}{\delta\gamma-1}\right). \end{equation*}

The largest exponent in r yields the leading term, so we find

\begin{equation*} \frac {I(\mu,r)}{r^{2d}}\asymp r^{d\psi(\mu)}, \end{equation*}

where

\begin{equation*} \psi(\mu) =\max\big\{{-}\delta, -\delta+ ((\alpha+\gamma)\delta-2)(1-\mu),-\delta+(\delta \gamma-1)(1-\mu)\big\}. \end{equation*}

This formula for $\psi(\mu)$ remains true when either of the three denominators $\delta \alpha-1$ , $\delta(\alpha+\gamma)-2$ , or $\delta \gamma-1$ is zero. Indeed, in those cases the asymptotic for $I(\mu,r)$ also possibly contains a logarithmic correction, but this does not change the exponent $\psi(\mu)$ . It can also be observed that we have

(27) \begin{equation} \psi(\mu)= \sup\big\{\chi(t_x,t_y)\,:\, (t_x,t_y)\in R\big\}, \end{equation}

and our computations above should also provide a good explanation of this equality.

The case $\mu\lt 1-1/(\alpha+\gamma)$ requires slightly more involved computations but (27) remains true and provides a quick way to compute the exponent $\psi$ . If $1-1/\gamma\lt \mu\lt 1-1/(\alpha+\gamma)$ , we obtain

\begin{align*} \psi(\mu) &= \max\left\{\chi(0,0),\chi\left(\tfrac{1}{\alpha+\gamma},\tfrac{1}{\alpha+\gamma}\right),\chi(1-\mu,0),\chi\left(1-\mu,\tfrac {1-\gamma(1-\mu)}{\alpha}\right)\right\} \\& =\max\left\{-\delta, -\tfrac{2}{\alpha+\gamma},-\delta+(\delta \gamma-1)(1-\mu), -\tfrac{1}{\alpha} + \left(\tfrac{\gamma}{\alpha}-1\right)(1-\mu)\right\}. \end{align*}

Finally, if $\mu\lt 1-1/\gamma$ , we obtain

\begin{equation*}\psi(\mu)=\max\left\{\chi(0,0),\chi\left(\tfrac{1}{\alpha+\gamma},\tfrac{1}{\alpha+\gamma}\right),\chi\left(\tfrac{1}{\gamma},0\right)\right\}=\max\left\{-\delta,-\tfrac{2}{\alpha+\gamma}, -\tfrac{1}{\gamma}\right\}. \end{equation*}

We conclude by summarizing our results and computing the value of $\zeta$ depending on the parameters $\alpha$ , $\gamma$ , and $\delta$ . In each case, $\psi$ is continuous in $\mu$ and $\zeta$ is the unique solution in $({-}\infty,1]$ of the equation $\zeta=2+\psi(\zeta)$ . We distinguish the various cases as shown in Figure 1.

1. Case $\boldsymbol{\gamma\lt 1/\delta}$ and $\boldsymbol{\alpha+\gamma\lt 2/\delta.}$ In this case, we have $\psi(\mu)= -\delta$ for all $\mu$ . Further solving $\zeta=2+\psi(\zeta)$ yields $\zeta=2-\delta$ , and therefore $\zeta\gt 0$ if $\delta\lt 2$ , $\zeta=0$ if $\delta=2$ , and $\zeta\lt 0$ if $\delta\gt 2$ .

2. Case $\boldsymbol{\gamma\gt 1/\delta}$ and $\boldsymbol{\alpha\lt 1/\delta.}$ Here, we have

   \begin{equation*} \psi(\mu)= \begin{cases} -\frac{1}{\gamma}, & \text{if } \mu\lt 1-\frac 1 {\gamma}, \\[8pt] -\delta+(\delta\gamma-1)(1-\mu), & \text{if } \mu\gt 1-\frac{1}{\gamma}. \end{cases} \end{equation*}
   Solving $\zeta=2+\psi(\zeta)$ , yields $\zeta=1-(\delta-1)/\gamma\delta$ , which is always larger than $1-1/{\gamma}$ . Further, $\zeta\gt 0$ if $\gamma\gt 1-1/\delta,$ $\zeta=0$ if $\gamma=1-1/\delta$ , and $\zeta\lt 0$ if $\gamma\lt 1-1/\delta$ .

3. Case $\boldsymbol{1/\delta\lt \alpha\lt \gamma}.$ We infer in this case

   \begin{equation*} \psi(\mu)= \begin{cases} -\frac{1}{\gamma}, & \text{if } \mu\lt 1-\frac 1 {\gamma}, \\[8pt] -\frac{1}{\alpha} +\left(\frac{\gamma}{\alpha}-1\right)(1-\mu), & \text{if } 1-\frac 1 {\gamma}\lt \mu \lt 1-\frac 1 {\alpha+\gamma}, \\[4pt] -\delta+(\delta(\alpha+\gamma)-2)(1-\mu), & \text{if } 1-\frac 1 {\alpha+\gamma}\lt \mu. \end{cases} \end{equation*}
   Note that the solution of $\zeta=2+\psi(\zeta)$ is always larger than $1-1/\gamma$ . We obtain
   \begin{equation*} \zeta = \begin{cases} \frac{\alpha+\gamma-1}{\gamma}\lt 0, & \text{if } \alpha+\gamma\lt 1, \\[8pt] 1-\frac{1}{\alpha+\gamma}=0, & \text{if } \alpha+\gamma=1, \\[12pt] \frac{\delta(\alpha+\gamma-1)}{\delta(\alpha+\gamma)-1}\gt 0, & \text{if } \alpha+\gamma\gt 1. \end{cases} \end{equation*}

4. Case $\boldsymbol{\gamma\lt \alpha}$ and $\boldsymbol{\alpha+\gamma\gt 2/\delta.}$ In this final case, we obtain

   \begin{equation*} \psi(\mu)= \begin{cases} -\frac{2}{\alpha+\gamma}, & \text{if } \mu \lt 1-\frac 1 {\alpha+\gamma}, \\[9pt] -\delta+(\delta(\alpha+\gamma)-2)(1-\mu), & \text{if } 1-\frac 1 {\alpha+\gamma}\lt \mu. \end{cases} \end{equation*}
   Again solving the usual equation, we infer
   \begin{equation*} \zeta = \begin{cases} \frac {2(\alpha+\gamma-1)}{\alpha+\gamma}\lt 0, & \text{if } \alpha+\gamma\lt 1, \\[10pt] 1-\frac 1{\alpha+\gamma}=0, & \text{if } \alpha+\gamma=1, \\[10pt] \frac{\delta(\alpha+\gamma-1)}{\delta(\alpha+\gamma)-1}\gt 0, & \text{if } \alpha+\gamma\gt 1. \end{cases} \end{equation*}

Reorganizing these results according to the sign of $\zeta$ , we obtain the expression of $\zeta$ provided in the statement of Proposition 1.

Finally, we give the proof of Proposition 2, dealing with the $\zeta=0$ phase of the interpolation model.

Proof of Proposition 2. Consider first $\delta=2$ . Since the interpolation model naturally dominates long-range percolation, we have $\mathbb{P}_1(\mathcal{L}(r,1))\asymp 1$ by the argument mentioned in Remark 2(b). For the remaining cases, observe first that, with probability $1-\mathrm{e}^{\omega_d}\gt 0$ , there exists a vertex in B(r) with mark at most $r^{-d}$ . Given such a vertex, its neighbours in $B(2r)\setminus B(r)$ with marks at least $r^{-d}$ form a Poisson process. In the case $\gamma=1-1/\delta$ , we may assume $\alpha=0$ and observe that the mean number of such neighbours is bounded from below by order $r^{d}r^{-d\delta(1-\gamma)}\asymp 1$ and for $\gamma\lt 1-1/\gamma$ and $\alpha=1-\gamma$ , the same mean is bounded by

\begin{equation*} r^d r^{-d\delta(1-\gamma)} \int_{r^{-d}}^{1} \mathrm{d} u \ u^{-\delta(1-\gamma)} \asymp 1. \end{equation*}

In both cases, with constant probability, there exists a vertex of mark at most $r^{-d}$ that is incident to a long edge. This concludes the proof.

4.2.2. Proofs for the generalized WDRCM of Section 2.2

We turn now to the proofs for the generalized version of the WDRCM introduced in Section 2.2. Recall the framework and assumptions explained around (9) as well as the generalizations of $\psi_\lambda$ and $\zeta_\lambda$ .

Proof of Theorem 5. Let us start with the proof of part (i). Thus, we assume $\zeta_\lambda\lt 0$ for some $\lambda\gt 0$ . Choose some $\zeta_\lambda/2\gt \mu\gt \zeta_\lambda$ arbitrarily close to $\zeta_\lambda$ . We aim for an appropriate bound for the probability of $\mathcal{L}(r,3)$ . To this end, define

\begin{equation*} \widetilde{\mathcal{L}}(r)\,:\!=\,\widetilde{\mathcal{L}}(r,\mu) =\big\{\exists \ \boldsymbol{{x}}\sim \boldsymbol{{y}}\,:\, x\in B(r)\text{ and }y\in B(2r)^\textsf{c}, u_x\wedge u_y\geq |y|^{-d(1+\mu)}\big\} \end{equation*}

and note that

\begin{equation*} \mathcal{L}(r,3)\subset \widetilde{\mathcal{L}}(r)\cup\big\{\exists \boldsymbol{{y}}\in\mathscr{V}\,:\, y\in B(r)^\textsf{c}, u_y\lt |y|^{-d(1+\mu)}\big\} \cup \big\{\exists \boldsymbol{{x}}\in\mathscr{V}\,:\, x\in B(r), u_x\lt r^{-d(1+\mu)} \big\}. \end{equation*}

By standard properties of the Poisson point process, the latter two events on the right-hand side have probabilities of order

\begin{align*} \lambda\! \int_{|y|\gt r}\mathrm{d} y \, |y|^{-d(1+\mu)} & = \tfrac{\omega_d}{|\mu|} \lambda r^{d\mu} \leq \tfrac{\omega_d}{|\mu|\wedge 1}\lambda r^{d\mu}\quad \! \text{and}\! \quad \lambda\!\int_{|x|\lt r} \mathrm{d} x \, r^{d(1+\mu)} = \omega_d \lambda r^{d\mu}\leq \tfrac{\omega_d}{|\mu|\wedge 1}\lambda r^{d\mu}. \end{align*}

Let us bound next the probability of $\widetilde{\mathcal{L}}(r)$ . Applying Mecke’s equation [Reference Last and Penrose39] and (9), we obtain

\begin{align*} \mathbb{P}_\lambda(\widetilde{\mathcal{L}}(r)) & \leq \mathbb{E}_\lambda\bigg[\sum_{\boldsymbol{{x}}\,:\, x\in B(r)}\sum_{\boldsymbol{{y}}\,:\, y\in B(2r)^\textsf{c}} {\unicode{x1D7D9}}_{\{u_x\wedge u_y \gt |y|^{-d(1+\mu)}\}}\mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}},\mathscr{V}\,\setminus\{\boldsymbol{{x}},\boldsymbol{{y}}\})\bigg] \\ & = \lambda^2 \int_{|y|\gt 2r}\mathrm{d} y \int_{|x|\lt r}\mathrm{d} x \int_{|y|^{-d+d\mu}}^1 \mathrm{d} u_x \int_{|y|^{-d+d\mu}}^1 \mathrm{d} u_y \,\mathbb{E}_\lambda \mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}},\mathscr{V}\,) \\ & = \lambda^2 \int_{|y|\gt 2r}\mathrm{d} y \int_{|x|\lt r}\mathrm{d} x \int_{|y|^{-d+d\mu}}^1 \mathrm{d} u_x \int_{|y|^{-d+d\mu}}^1 \mathrm{d} u_y \,\varphi_\lambda(u_x,u_y,|x-y|^d). \end{align*}

Now observe that $|x|$ and $|x-y|$ are of the same order for large r. Hence, we find a constant $C\gt 1$ such that for $\varepsilon\gt 0$ and all sufficiently large r, by definition of $\psi_\lambda(\mu)$ ,

\begin{align*} \mathbb{P}_\lambda(\widetilde{\mathcal{L}}(r)) & \leq C \lambda^2 \omega_d r^d \int_{|x|\gt r}\mathrm{d} x \int_{|x|^{-d+d\mu}}^1 \mathrm{d} u_x \int_{|x|^{-d+d\mu}}^1 \mathrm{d} u_y \,\varphi_\lambda(u_x,u_y,|x|^d) \\ & \leq C \lambda^2 \omega_d r^d \int_{|x|\gt r} \mathrm{d} x \, |x|^{d(\psi(\mu)+\varepsilon)} \\ & \leq C \omega_d^2 \lambda^2 r^{d(2+\psi(\mu)+\varepsilon)}. \end{align*}

As $\varepsilon$ can be chosen arbitrarily small, we can find for all $\zeta_\lambda/2\gt \mu\gt \zeta_\lambda$ some $R\gt 1$ such that for all $r\gt R$

(28) \begin{equation} \mathbb{P}_\lambda(\mathcal{L}(r,2)) \leq (\lambda\vee \lambda^2)C\left(1 +\tfrac{2}{|\zeta_\lambda|\wedge 1}\right) r^{d\mu}. \end{equation}

This proves part (i). In order to prove part (ii), observe that in the case of $\overline{\zeta}_{\lambda'}\lt 0$ , we can take the supremum $\sup_{\lambda\lt \lambda'}$ on both sides of (28) and note that the $\lambda$ -dependent constant remains finite. This yields the claimed decay of the long event and, in particular, implies property $\overline{\mathscr{L}}$ $_{\lambda'}$ . As further $\overline{\mathscr{M}}$ $_{\lambda'}$ by assumption, $\widehat{\lambda}_{\textsf{c}}\gt 0$ by Theorem 1. This proves part (ii). Finally, part (iii) is a consequence of part (i) and Theorem 3.

Before we turn to the proof of Corollary 2, recall the soft Boolean model with local interferences, introduced in Section 2.2.1.

Proof of Corollary 2. We start by proving that the soft Boolean model with local interferences is a generalized WDRCM. That is, we show that the connection rule (10) satisfies the integration condition (9). First, we observe by the translation invariance of the Poisson point process that $\mathscr{N\,\,}^\beta((x,u),\mathscr{V}\,)$ and $\mathscr{N\,\,}^\beta((o,u),\mathscr{V}\,)$ have the same law for all given coordinates $(x,u)\in\mathbb{R}^d\times(0,1)$ . Second, since $\mathscr{N\,\,}^\beta((o,u),\mathscr{V}\,)$ is Poisson distributed with parameter $\lambda \omega_d u^{-\beta}$ , it is straightforward to deduce

(29) \begin{equation} \mathbb{E}_\lambda \left[\frac{1}{1+\mathscr{N\,\,}^{\beta}((o,u),\mathscr{V}\,)}\right] = \frac{1-\mathrm{e}^{-\lambda \omega_d u^{-\beta}}}{\lambda \omega_d u^{-\beta}} \asymp u^{\beta}. \end{equation}

Therefore, for two given vertices $\boldsymbol{{x}}=(x,u_x)$ and $\boldsymbol{{y}}=(y,u_y)$ with $|x-y|=r$ , Condition (9) is satisfied with

\begin{equation*} \varphi(u_x,u_y,r^d) = {\unicode{x1D7D9}}_{\{u_x\lt u_y\}} \frac{(1{-}\mathrm{e}^{-\lambda \omega_d u_x^{-\beta}})(1\wedge u_x^{-\gamma\delta}r^{-d\delta})}{\lambda \omega_d u_x^{-\beta}}+{\unicode{x1D7D9}}_{\{u_x\geq u_y\}} \frac{(1{-}\mathrm{e}^{-\lambda \omega_d u_y^{-\beta}})(1\wedge u_y^{-\gamma\delta}r^{-d\delta})}{\lambda \omega_d u_y^{-\beta}}. \end{equation*}

We show next that the model has ${{\mathscr{P}}\!\!{\mathscr{M}}}$ $_\lambda^\xi$ with $\xi = d(1-1/\beta)$ for all $\lambda\gt 0$ . We note initially that the realizations of the underlying Poisson point process in the two disjoint balls are independent. Moreover, a vertex located in B(3r) can only interfere an internal edge in B(xr, 3r) subject to $\mathcal{G}(xr,r)$ if the sphere of interference of said vertex intersects B(xr, 3r). As the two balls are at distance cr, this requires a mark smaller than $(cr)^{-d/\beta}$ (by the definition of $\mathscr{N\,\,}^\beta$ ). Thus, by symmetry and Markov’s inequality,

\begin{equation*} \operatorname{\rm Cov}_\lambda({\unicode{x1D7D9}}_{\mathcal{G}(r)}, {\unicode{x1D7D9}}_{\mathcal{G}(xr,r)}) \leq 2\mathbb{P}_\lambda(\exists \boldsymbol{{x}}\in B(r)\cap\mathscr{V}\,:\, u_x\lt (cr)^{-1/\beta}) \leq 2\omega_d \lambda (cr)^{d(1-1/\beta)}. \end{equation*}

Note that $1-1/\beta-1\lt 0$ as $\beta\lt 1$ . To calculate $\zeta$ and identify the phase it is negative in, we calculate by making use of (29)

\begin{equation*} n^{-d\delta}\int\limits_{n^{-d+d\mu}}^1 \mathrm{d} u \, u^{-\gamma\delta+\beta} \asymp n^{-d\delta} \vee n^{d({-}\delta-(1-\mu)(1-\gamma\delta+\beta))}. \end{equation*}

Hence, $\zeta=2-\delta$ and $\zeta\lt 0$ for $\delta\gt 2$ if $\gamma\lt (1+\beta)/\delta$ , as well as

\begin{equation*} \zeta = \frac{1-\beta-\delta(1-\gamma)}{\gamma\delta-\beta} \end{equation*}

otherwise. In the latter case, we have $\zeta\lt 0$ if $\delta\gt 2$ and $\gamma\lt (\delta+\beta-1)/\delta$ . The proof concludes by applying Theorem 5.

4.2.3. Proofs of Section 3

Next, we prove the results about the examples in Section 3. We start with Corollary 3 stated in Section 3.1 about ellipses percolation. We invite the readers to familiarize themselves with the model again.

Proof of Corollary 3. As $\lambda_{\textsf{c}}=0$ implies $\widehat{\lambda}_{\textsf{c}}=0$ , the second part of the statement is a direct consequence of [Reference Sznitman49, Theorem 1.2] where $\lambda_{\textsf{c}}=0$ is shown for $\gamma\gt 1$ . For the first part, we can make use of [Reference Hall25, Lemma 2], which states

\begin{equation*}\mathbb{P}_\lambda\big(\exists x\sim y\,:\, x\in B(r), y\in B(2r)^\textsf{c}\big)\leq 1-\exp\big({-}\lambda c r^{2(1-1/\gamma)}\big) \end{equation*}

for some constant $c\gt 0$ and $r\geq 1$ . As $\mathcal{L}(r,2)$ implies the event on the left-hand side and the right-hand side converges to zero for $\gamma\lt 1$ , we have $\widehat{\lambda}_{\textsf{c}}\gt 0$ by Theorem 2, and the claimed decay of the annulus-crossing probability by Theorem 3.

We now turn to the proof of Proposition 3 showing that Cox processes introduced in Section 3.2 can be suitable candidates for correlated underlying point clouds.

Proof of Proposition 3. The fact that the conditions (G1) and (G2) are satisfied is clear from the construction. First, we verify mixing or more precisely ${{\mathscr{P}}\!\!{\mathscr{M}}}$ $_\lambda^\xi$ for all $\lambda\gt 0$ for $\xi=2-\beta$ . To this end, note that we can write

\begin{equation*} \mathbb{P}_\lambda\big(\mathcal{G}(r)\cap \mathcal{G}(xr,r)\big)=\mathbb{E}[f_r(\Lambda_{B(r)})g_r(\Lambda_{B(xr,r)})], \end{equation*}

where $f_r(\Lambda_{B(r)})=\mathbb{P}_\lambda(\mathcal{G}(r)|\Lambda)$ and $g_r(\Lambda_{B(xr,r)})=\mathbb{P}_\lambda(\mathcal{G}(xr,r)|\Lambda)$ , and where we used the independence of the marked Poisson points as well as the fact that the edge drawing is assumed to be local by (11). Here, the notation $\Lambda_{B(x,r)}$ represents the fact that $\Lambda$ is evaluated only in the area B(x, r). Now, in order to control the correlation coming from the environment, consider the three events

\begin{align*} \mathcal{A}_r &=\{\forall Y_i\in B((1+\varepsilon)r)\,:\, \rho_i\lt \varepsilon r\} \\ \mathcal{B}_r&=\{\forall Y_i\in B(xr,(1+\varepsilon)r)\,:\, \rho_i\lt \varepsilon r\} \\ \mathcal{C}_r&=\big\{\forall Y_i\in \big(B((1+\varepsilon)r)\cup B((1+\varepsilon)r)\big)^{\textsf{c}} \,:\, \rho_i\lt (|Y_i|\wedge |Y_i-yr|)-r\big\}, \end{align*}

with $\varepsilon=(|x|-2)/2$ . Note that, since $\boldsymbol{{Y}}=((Y_i,\rho_i)\,:\,i\in\mathbb{N})$ is an i.i.d. marked Poisson point process, the three events are independent as they refer to points in disjoint regions of $\mathbb{R}^2$ . Moreover, under the event $\mathcal{A}_r\cap \mathcal{B}_r\cap \mathcal{C}_r$ , the random variables $f_r(\Lambda_{B(r)})$ and $g_r(\Lambda_{B(xr,r)})$ are independent. Hence, by translation invariance, Markov’s inequality, and Mecke’s formula,

\begin{equation*} \big|\operatorname{\rm Cov}_\lambda\big({\unicode{x1D7D9}}_{\mathcal{G}(r)}, {\unicode{x1D7D9}}_{\mathcal{G}(xr,r)}\big)\big|\leq c_1(\mathbb{P}_\lambda(\neg\mathcal{A}_r)+\mathbb{P}_\lambda(\neg\mathcal{B}_r)+\mathbb{P}_\lambda(\neg\mathcal{C}_r))\leq c_2r^{2-\beta}, \end{equation*}

as desired. For the condition (G3), we use Mecke’s formula and Markov’s inequality to infer

\begin{align*} \mathbb{P}\lambda(\exists \boldsymbol{{x}}\,:\, \deg(\boldsymbol{{x}})=\infty) & =\lim_{n\uparrow\infty}\mathbb{P}_\lambda(\exists \boldsymbol{{x}}\in B(n)\,:\, \deg(\boldsymbol{{x}})=\infty) \\& \leq \lim_{n\uparrow\infty}\mathbb{E}_\lambda\bigg[\sum_{ X_i\in \boldsymbol{{X}}\cap B(n)}{\unicode{x1D7D9}}\{\deg(\boldsymbol{{x}})=\infty\}\bigg] \\& = \lambda\lim_{n\uparrow\infty}\mathbb{E} \bigg[\int_{B(n)}\Lambda(\mathrm{d} x)\int_0^1 \mathrm{d} u_x \, \mathbb{P}_\lambda^{x,u_x}(\deg(\boldsymbol{{x}})=\infty|\Lambda)\bigg] \\& \leq \lambda\lim_{n\uparrow\infty}\lim_{m\uparrow\infty}\frac{1}{m}\mathbb{E}\bigg[\int_{B(n)}\Lambda(\mathrm{d} x)\int \mathrm{d} u_x \, \mathbb{E}_\lambda^{x,r_x}\bigg[\sum_{\boldsymbol{{X}} \in \mathscr{V}}{\unicode{x1D7D9}}\{\boldsymbol{{X}}\sim \boldsymbol{{x}}\}\bigg|\lambda\bigg]\bigg] \\& = \lambda^2\lim_{n\uparrow\infty}\lim_{m\uparrow\infty}\frac{1}{m}\mathbb{E} \bigg[\int_{B(n)}\Lambda(\mathrm{d} x)\int \mathrm{d} u_x \int\Lambda(\mathrm{d} y)\int \mathrm{d} u_y \, \mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}})\bigg] \\& = \lambda^2\lim_{n\uparrow\infty}\lim_{m\uparrow\infty}\frac{1}{m}\int_{B(n)}\mathrm{d} x\int \mathrm{d} u_x \,\int \mathrm{d} y\int \mathrm{d} u_y \, \mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}})\mathbb{E}[\ell_x\ell_y]. \end{align*}

Another application of Mecke’s formula as well as the normalization assumption shows $\mathbb{E}[\ell_x\ell_y]\leq 2$ and, thus, translation-invariance of $\mathbf{p}$ yields

\begin{align*} \int_{B(n)}&\mathrm{d} x\int \mathrm{d} u_x \int\mathrm{d} y\int \mathrm{d} u_y \, \mathbf{p}(\boldsymbol{{x}},\boldsymbol{{y}})\mathbb{E}[\ell_x\ell_y]\\ &\le 2\omega_2 n^2\int \mathrm{d} u_o \int\mathrm{d} x\int \mathrm{d} u_x \, \mathbf{p}((o,r_o),(x,r_x)), \end{align*}

which is finite by assumption. Hence, $\mathbb{P}(\exists \boldsymbol{{x}}\,:\, \deg(\boldsymbol{{x}})=\infty)=0$ . Finally, the condition (Mon) directly follows from the standard monotone coupling for Poisson point processes once we have conditioned on the environment.

Finally, we prove our result about worm percolation stated in Corollary 4.

Proof of Corollary 4. Recall percolation of worms introduced in Section 3.3. Clearly, the model is translation invariant, locally finite, monotone, and always has property $\overline{\mathscr{L}}$ $_\lambda$ for all $\lambda\gt 0$ as we restricted ourselves to nearest-neighbour edges only. It therefore remains to proof mixing. In order to do so, we argue similar as for the Boolean model with local interference and infer

\begin{equation*} \operatorname{\rm Cov}_\lambda({\unicode{x1D7D9}}_{\mathcal{G}(r)}, {\unicode{x1D7D9}}_{\mathcal{G}(Bxr,r)}) \leq 2\mathbb{P}_\lambda(\exists \boldsymbol{{x}}\in B(r)\,:\, \text{a worm starting in }x \text{ reaches }B(xr,r)) \leq 2\lambda r^d \ell[r,\infty), \end{equation*}

which tends to zero as $r\to\infty$ uniformly in $\lambda$ whenever $\ell$ has finite dth moment. Thus, the model has $\overline{\mathscr{M}}$ $_\lambda$ for all $\lambda$ in this case, and we find $\widehat{\lambda}_{\textsf{c}}\gt 0$ by Theorem 1. Let us now assume that the dth moment of $\ell$ is infinite. We aim to show that at least one worm starting in B(r) manages to reach $B(2r)^\textsf{c}$ and an annulus-crossing event thus occurs. To this end, consider the following thinning process. First, we remove each site of B(r) with probability $1-\exp({-}\nu)$ . That is, each remaining site has assigned at least one worm. Exactly one independent worm is started in each of these sites and we keep only those sites whose worms manage to reach $B(2r)^\textsf{c}$ . As the probability of this to happen is decreasing in the distance of the starting site to the target boundary, said probability is bounded from below by $\mathbb{P}_o^\ell(o\xleftrightarrow[]{} B(2r)^\textsf{c})$ . By independence, the number of sites in B(r) surviving the thinning process is bounded from below by a Binomial random variable with mean of order

\begin{equation*} r^d(1-\mathrm{e}^{-\nu})\mathbb{P}_o^\ell(o\xleftrightarrow[]{} B(2r)^\textsf{c}) \asymp r^d \sum_{n\geq 2r} \mathbb{P}_o^{\ell=n}\Big(\sup_{k\leq n}|S_n|\geq 2r \Big)\ell\{n\} \asymp r^d \ell[2r,\infty). \end{equation*}

Here, $S_n$ denotes a simple random walk and $\mathbb{P}^{\ell=n}_o$ the measure of a random walk, starting in the origin and performing exactly n steps. The last step follows from the fact that for each fixed r, the probability $\mathbb{P}_o^{\ell=n} (\sup_{k\leq n}|S_n|\geq 2r )$ is bounded from zero and tends to one as $n\to\infty$ . As the right-hand side diverges to infinity as $r\to\infty$ , at least one vertex in B(r) is going to survive the described thinning procedure with probability approaching one. This concludes the proof.