{"paper":{"title":"Phase transition of disordered random networks on quasi-transitive graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Kainan Xiang, Yuelin Liu","submitted_at":"2020-10-04T10:11:53Z","abstract_excerpt":"Given a quasi-transitive infinite graph $G$ with volume growth rate ${\\rm gr}(G),$ a transient biased electric network $(G,\\, c_1)$ with bias $\\lambda_1\\in (0,\\,{\\rm gr}(G))$ and a recurrent biased one $(G,\\, c_2)$ with bias $\\lambda_2\\in ({\\rm gr}(G),\\infty).$ Write $G(p)$ for the Bernoulli-$p$ bond percolation on $G$ defined by the grand coupling. Let $(G,\\, c_1,\\, c_2,\\, p)$ be the following biased disordered random network: Open edges $e$ in $G(p)$ take the conductance $c_1(e)$, and closed edges $g$ in $G(p)$ take the conductance $c_2(g)$. Our main results are as follows: (i) On connected "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.01530","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/2010.01530/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"}