bibkey: carstens2011noncoexistence authors: Sebastian Carstens year: 2011 title: Non-coexistence of Infinite Clusters in Two-Dimensional Dependent Site Percolation doi: 10.1007/s10955-011-0329-1 url: https://arxiv.org/abs/1103.0901 claim: Corollary 1 excludes coexistence of an infinite ordinary zero cluster and an infinite star-adjacent one cluster under translation ergodicity, positive association and finite energy. strata_touched: [] license: citation-only triage: anchor

Planar non-coexistence with explicit adjacency and energy

Carstens, Non-coexistence of Infinite Clusters in Two-Dimensional Dependent Site Percolation, Journal of Statistical Physics 144(6), 1223–1237. Crossref confirms the publication metadata and DOI. Primary text: https://arxiv.org/pdf/1103.0901, also https://arxiv.org/html/1103.0901.

Corollary 1 assumes a probability measure on {0,1}^Z2 that is ergodic for translations, positively associated, and satisfies finite energy. The introduction’s footnote specifies that ergodicity includes translation invariance. Definition 2 requires positive conditional probability for every finite local pattern given the outside configuration, almost surely. Ordinary adjacency has Euclidean distance one; star adjacency additionally permits distance sqrt(2). The conclusion rules out coexistence of an infinite ordinary 0 cluster and an infinite 1-star cluster.

Theorem 1 has a different list, including bounded energy, uniqueness assumptions and a uniform positive probability bound; those must not be substituted for Corollary 1’s hypotheses. The FIB boundary volume uses Corollary 1 for independent square-lattice colouring and the implication from an ordinary 1 cluster to a star 1 cluster. It does not claim non-coexistence on every dependent planar model, long-range graph or nonstationary environment.