{"paper":{"title":"Extending P\\'olya's random walker beyond probability I. Complex weights","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.HO"],"primary_cat":"math.PR","authors_text":"Martin Klazar, Richard Horsk\\'y","submitted_at":"2025-05-17T23:02:05Z","abstract_excerpt":"Working in combinatorial model $\\mathrm{W_{co}}(d)$, $d=1,2,\\dots$, of P\\'olya's random walker in $\\mathbb{Z}^d$, we prove two theorems on recurrence to a vertex. We obtain an effective version of the first theorem if $d=2$. Using a semi-formal approach to generating functions, we extend both theorems beyond probability to a more general model $\\mathrm{W_{\\mathbb{C}}}$ with complex weights. We relate models $\\mathrm{W_{co}}(d)$ to standard models $\\mathrm{W_{Ma}}(d)$ based on Markov chains. The follow-up article will treat non-Archimedean models $\\mathrm{W_{fo}}(k)$ in which weights are formal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.12170","kind":"arxiv","version":2},"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/2505.12170/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"}