{"paper":{"title":"On one-dimensional Cluster cluster model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Daniel Sharon, Eviatar B. Procaccia, Noam Berger","submitted_at":"2025-07-04T13:02:24Z","abstract_excerpt":"The Cluster-cluster model was introduced by Meakin et al in 1984. Each $x\\in \\mathbb{Z}^d$ starts with a cluster of size 1 with probability $p \\in (0,1]$ independently. Each cluster $C$ performs a continuous-time SRW with rate $|C|^{-\\alpha}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. Focusing on dimension $d=1$, we show that for $\\alpha>-2$, at time $t$, the cluster size is of order $t^\\frac{1}{\\alpha + 2}$, and for $\\alpha < -2$ we get an infinite cluster in finite time a.s. Additionally, for $\\alpha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.03552","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.03552/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"}