{"paper":{"title":"Yaglom theorem for critical branching random walk on $\\mathbb{Z}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Shen Lin, Xinxin Chen","submitted_at":"2025-12-30T07:44:41Z","abstract_excerpt":"We study the critical branching random walk on $\\mathbb{Z}^d$ started from a distant point $x$ and conditioned to hit some compact set $K$ in $\\mathbb{Z}^d$. We are interested in the occupation time in $K$ and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order $\\|x\\|^{4-d}$ in dimensions $d\\leq 3$, of order $\\log\\|x\\|$ in dimension $d=4$, and of order 1 in dimensions $d\\geq 5$. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (200"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.24047","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/2512.24047/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"}