Decay of connection probability in high-dimensional continuum percolation

TL;DR

This paper analyzes the decay of connection probabilities in high-dimensional continuum percolation models, establishing asymptotic behavior using advanced mathematical techniques.

Contribution

It introduces a new proof method for decay estimates in high-dimensional percolation, simplifying previous approaches and extending results to continuum models.

Findings

Connection probability decays like |x|^{-(d-2)} for large |x| in high dimensions.

The proof applies to both continuum and lattice percolation models in high dimensions.

Simplifies and extends previous decay estimates in percolation theory.

Abstract

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 $|x|^{-(d-2)}$ as $|x| \to \infty$, 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.