Spatial preferential attachment models: robustness revisited

We investigate robustness under percolation in the degree-driven spatial preferential attachment model. Consider a profile that is regularly varying with exponent $-\delta$, where $\delta >1$, and an attachment rule with asymptotic slope $\gamma \in (0,1)$. Jacob and Mörters (Ann. Probab. 45 (2017)) proved robustness when $$\delta (\frac{1}{\gamma }-1)<1.$$ We prove the converse in the strict regime: if $$\delta (\frac{1}{\gamma }-1)>1,$$ then sufficiently strong Bernoulli vertex thinning destroys the giant component. In particular, the strict robustness boundary coincides with that of the age-dependent random connection model.