{"paper":{"title":"Long-range percolation on the hierarchical lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Pieter Trapman, Ronald Meester, Vyacheslav Koval","submitted_at":"2010-04-08T06:28:18Z","abstract_excerpt":"We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\\exp(-\\beta^{-k} \\alpha)$, independently of all other edges. For fixed $\\beta$, we show that the critical value $\\alpha_c(\\beta)$ is non-trivial if and only if $N < \\beta < N^2$. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of $\\alpha_c(\\beta)$ as a function of $\\beta$. This means that the phase diagram of this model is well understood."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.1251","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}