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The set of vertices and edges visited by at least one of these trajectories is the random interlacement at level u of Sznitman arXiv:0704.2560 . We prove that for any u>0, almost surely, (1) any two vertices in the random interlacement at level u are connected via at most ceiling(d/2) trajectories of the point process, and (2) there are vertices in the random interlacement at level u which can only be connect"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1012.4711","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-12-21T16:01:35Z","cross_cats_sorted":[],"title_canon_sha256":"504a5de0089ff91c11cc028a096e46d094740df520388ad2fd231beeb5ff5bd3","abstract_canon_sha256":"f1771e7f88a853b4d5325762be0157041a8d3e9a9cb221081bb1e7af38dfeb1e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:01.331507Z","signature_b64":"AgCQe3BAiqZINRQgZz60sk5zLPL11NOzNjzqGiJRiSJlF70Mx+HeocGXNRfChvZr1TaSq8VSips6cBu0tvvtCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9e442fc37ff98348de2affeaa6f4c7c2e6e4d0189065454c294ffe55b28e7ff","last_reissued_at":"2026-05-18T04:00:01.331028Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:01.331028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Connectivity properties of random interlacement and intersection of random walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Art\\\"em Sapozhnikov, Bal\\'azs R\\'ath","submitted_at":"2010-12-21T16:01:35Z","abstract_excerpt":"We consider the interlacement Poisson point process on the space of doubly-infinite Z^d-valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times. 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