Decay of connection probability in high-dimensional continuum percolation

We study a percolation model on ${\mathbb{R}}^d$ called the random connection model. For $d$ large, we use the lace expansion to prove that the critical two-point connection probability decays like $\mathrm{\mid }x{\mathrm{\mid }}^{-(d-2)}$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, with possible anisotropic decay. Our proof also applies to nearest-neighbour Bernoulli percolation on ${\mathbb{Z}}^d$ in $d\ge 11$ and simplifies considerably the proof given by Hara in 2008. The method is based on the recent deconvolution strategy of Liu and Slade and uses an $L^p$ version of Hara’s induction argument.