Essential enhancements of site percolation in every dimension and on periodic lattices

We prove Conjecture 1 of Balister, Bollobás and Riordan for site percolation on Z^d in every dimension d ≥ 2: for every essential enhancement, activated at any positive density, the enhanced configuration percolates for all p < p_c close enough to the critical probability p_c. Aizenman and Grimmett gave an argument for this in 1991, but Balister, Bollobás and Riordan found that its combinatorial step can fail. They proved the cases d = 2, 3, outlined proofs for bond percolation on Z^d and for site percolation on the triangular lattice, and left the cases d ≥ 4 open. Near a site that is pivotal for a long connection, our proof reroutes the open path through a finite configuration in which the enhancement creates a connection, inside a ball that need not be centred at the pivotal site. A theorem valid in every dimension makes this possible unless the first and last sites of the path in the ball lie at a corner of the ball, and a mass-transport argument finds a ball where they do not. The method extends to site percolation on fourteen further periodic lattices, among them all Archimedean lattices other than the square lattice, and to bond percolation on the honeycomb lattice. The proof for Z^d uses no computer, parts of most lattice proofs are finite computer checks of local configurations, and the main theorems for Z^d and for nine of the lattices have been checked in the Lean proof assistant.

Files: the paper (essential_enhancements_v6.pdf); the supplementary proofs (essential_enhancements_v4_supplementary_proofs.pdf), whose Part Z describes every computation used in a proof; and the archive of programs and logs described there (essential_enhancements_v4_certificates.tar.gz), both unchanged since version 4. The Lean formalization is in the record https://doi.org/10.5281/zenodo.23692474.

Mathematics Subject Classification (2020): Primary 60K35; Secondary 82B43.