proof of part (iii).

Chemical distances {#chemDistSec}
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In the present section we investigate the behavior of chemical distances (i.e., shortest-path lengths) on scale-free Gilbert graphs. We have already mentioned in the introduction that the regime, where $s<d$ might be of limited interest, as the diameter is at most $2$ whp.

Regime $s>d$ {#subCritReg}
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In the present subsection, we consider the regime $s>d$ whose connectivity properties turn out to be rather similar to those of the Boolean model with light-tailed radii. For instance, we note that isolated vertices may occur with positive probability.

\[nonConnLem\] Assume that $s>d$. Then $$\lim_{n\to\infty}\P(o\text{ is isolated in }G(X^{(n)}\cup\{(o,R^*)\}))>0.$$

First, we note that similar arguments as in Theorems \[outEdgeThm\] and \[inEdgeThm\] can be used to show that the probabilities $\P(o\text{ is isolated in }G(X^{(n)}\cup\{(o,R^*)\}))$ converge as $n\to\infty$. , the events $\big\{D^{(0)}_{\inn,n}=0\big\}$ and $\big\{D^{(0)}_{\out,n}=0\big\}$ are both decreasing events, the FKG inequality (see, e.g. [@fock Theorem 1.4]) implies that they are positively correlated. Hence, $$\begin{aligned}
\P(o\text{ is isolated in }G(X^{(n)}\cup\{(o,R^*)\}))&\ge\P\big(D^{(0)}_{\inn,n}=0\big)\P\big(D^{(0)}_{\out,n}=0\big),\end{aligned}$$ using Theorems \[outEdgeThm\] and \[inEdgeThm\] completes the proof.

Next, we prove Theorem \[subCrit\], which shows that allowing random radii with tail index $s>d$ does not reduce substantially chemical distances in comparison to the case of constant radii. 