We consider random graphs $\mathcal{G}$ built on a homogeneous Poisson point process on ${\mathbb{R}}^d$, $d\ge 2$, with points $x$ marked by i.i.d. random variables $E_x$. Fixed a symmetric function $h(\cdot ,\cdot )$, the vertexes of $\mathcal{G}$ are given by points of the Poisson point process, while the edges are given by pairs $\{x,y\}$ with $x\ne y$ and $|x-y|\le h(E_x,E_y)$. We call $\mathcal{G}$ Poisson $h$-generalized Boolean model, as one recovers the standard Poisson Boolean model by taking $h(a,b)≔a+b$ and $E_x\ge 0$. Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left–right crossings in a box of size $n$ is lower bounded by $C{n}^{d-1}$ apart from an event of exponentially small probability. As special applications, when the marks are non-negative, we consider the Poisson Boolean model and its generalization to $h(a,b)={(a+b)}^{\gamma }$ with $\gamma >0$, the weight-dependent random connection models with max-kernel and with min-kernel and the graph obtained from the Miller–Abrahams random resistor network in which only filaments with conductivity lower bounded by a fixed positive constant are kept.

我们考虑随机图 $\mathcal{G}$ 建立在齐次泊松点过程上 ${\mathbb{[R}}^d$， $d\ge \mathrm{2个}$，含分 $X$ 由iid随机变量标记 $E_X$。修复了对称函数$H（\cdot ，\cdot ）$，的顶点 $\mathcal{G}$ 由泊松点过程的点给定，而边由对给出 $\{X，ÿ\}$ 和 $X\ne ÿ$ 和 $|X-ÿ|\le H（E_X，E_{ÿ}）$。我们称之为$\mathcal{G}$ 泊松 $H$广义布尔模型，因为可以通过以下方法恢复标准的泊松布尔模型$H（\mathrm{一种}，b）≔\mathrm{一种}+b$ 和 $E_X\ge 0$。在一般条件下，我们表明，在超临界阶段，大小为1的盒子中顶点不相交的左右交点的最大数量$ñ$ 下界 $C{ñ}^{d-\mathrm{1个}}$除了几率很小的事件。作为特殊应用，当标记为非负数时，我们考虑使用Poisson布尔模型并将其推广为$H（\mathrm{一种}，b）={（\mathrm{一种}+b）}^{\gamma }$ 和 $\gamma >0$，具有最大内核和最小内核的与重量有关的随机连接模型，以及从Miller-Abrahams随机电阻器网络获得的图表，其中仅保留电导率较低且由固定正常数限制的灯丝。