{"paper":{"title":"The Lightning Model","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexandra Deane, Anthony Quas, James Campbell","submitted_at":"2021-01-23T00:09:31Z","abstract_excerpt":"We introduce a non-standard model for percolation on the integer lattice $\\mathbb Z^2$. Randomly assign to each vertex $a \\in \\mathbb Z^2$ a potential, denoted $\\phi_a$, chosen independently and uniformly from the interval $[0, 1]$. For fixed $\\epsilon \\in [0,1]$, draw a directed edge from vertex $a$ to a nearest-neighbor vertex $b$ if $\\phi_b < \\phi_a + \\epsilon$, yielding a directed subgraph of the infinite directed graph $\\overrightarrow{G}$ whose vertex set is $\\mathbb Z^2$, with nearest-neighbor edge set. We define notions of weak and strong percolation for our model, and observe that whe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.09383","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2101.09383/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}