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Let R = {x \\in Z^d: X_t = x for some t \\ge 0} be the range of X. It is proved that, for almost every realization of the environment, dim_H (R) = dim_P (R) = 2 almost surely, where dim_H and dim_P denote respectively the discrete Hausdorff and packing dimension. Furthermore, given any set A \\subseteq Z^d, a criterion for A to be hit by X_t for arbitrarily larg"},"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":"1110.0581","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-04T05:20:11Z","cross_cats_sorted":[],"title_canon_sha256":"b220791ac38a41a74680eb119b7593281c33c5d38cb8af2737c85723509974f4","abstract_canon_sha256":"17d176765fff4e3f23f67cdb536d27fd70bb2f1fd9424968d4cdbce601fd70fe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:26.020673Z","signature_b64":"XHThRxXHQY8Y1FechUIrMW7/fZIRRRQvxbJokz1DTamIaN0C4apagrITDxy8TvSuz4BYaRBYPzfKJkQyvZnQDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"82b15b05b937f267c7e9f6899f039514a36bc301c4a96033828d7e177d57a4f8","last_reissued_at":"2026-05-18T04:01:26.019870Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:26.019870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discrete Fractal Dimensions of the Ranges of Random Walks in $\\Z^d$ Associate with Random Conductances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xinghua Zheng, Yimin Xiao","submitted_at":"2011-10-04T05:20:11Z","abstract_excerpt":"Let X= {X_t, t \\ge 0} be a continuous time random walk in an environment of i.i.d. random conductances {\\mu_e \\in [1, \\infty), e \\in E_d}, where E_d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice Z^d and d\\ge 3. Let R = {x \\in Z^d: X_t = x for some t \\ge 0} be the range of X. It is proved that, for almost every realization of the environment, dim_H (R) = dim_P (R) = 2 almost surely, where dim_H and dim_P denote respectively the discrete Hausdorff and packing dimension. 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