bibkey: campaninorusso1985cubic authors: M. Campanino; L. Russo year: 1985 title: An Upper Bound on the Critical Percolation Probability for the Three-Dimensional Cubic Lattice doi: 10.1214/aop/1176993004 url: https://doi.org/10.1214/aop/1176993004 claim: The strict one-half upper bound for independent site percolation on the three-dimensional simple cubic lattice is the external threshold input to the conditional two-colour comparison. strata_touched: [] license: citation-only triage: anchor

Cubic site percolation and complementary phases

Campanino and Russo, An Upper Bound on the Critical Percolation Probability for the Three-Dimensional Cubic Lattice, Annals of Probability 13(2), 478–491. Crossref establishes the title, authors, volume, issue, publication year and DOI. Pages and the site-percolation scope are corroborated by the reference and discussion below.

Geoffrey R. Grimmett, Alexander E. Holroyd and Gady Kozma, Percolation of finite clusters and infinite surfaces, https://arxiv.org/pdf/1303.1657v2, introduction p. 2 and reference [8], explicitly attribute to Campanino–Russo the coexistence of infinite open and closed clusters for site percolation on Z3 at p = 1/2. This is an authoritative corroboration of the site model and central coexistence consequence, distinct from that paper’s bond-percolation complement-of-the-infinite-cluster problem.

The original Campanino–Russo PDF was not obtained: the publisher download returns an access challenge rather than article bytes. The retrieved corroboration states coexistence at one half; it is not an independently read proof or exact theorem locator for the stronger strict threshold inequality. The FIB boundary volume §11.6 uses that strict inequality as an explicitly attributed external premise when deriving the open interval (pc, 1-pc). Its finite interface expectation is proved separately. The source boundary must not be upgraded to a fresh verification of the original proof or an exact numeric threshold.

The comparison concerns infinite-volume independent site percolation with ordinary cubic adjacency. Existence of a cluster almost surely is different from a specified root’s survival probability, and neither attests finite-window reliability, the FIB pedigree threshold, or an actual native FIB transport graph.