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Moreover assume that a \"hard wall\" is placed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitouni for the free field on $\\mathbb{Z}^d$, $d \\geq 3$, to the "},"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":"1307.5825","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-07-22T19:18:26Z","cross_cats_sorted":["cond-mat.stat-mech","math-ph","math.MP"],"title_canon_sha256":"4c722091724e4d1d0c2f81d8a4b8a88827a78c3000822533f26cc1c36cc589c5","abstract_canon_sha256":"af6d2bb4662840d998ce4e579c5e35fb4d6af2ff848c12084c70c67394ac6677"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:35.680965Z","signature_b64":"AHfKDeYIdc07qxb2Qn2BlSyzjS8bVCI6DvQc/YAmBiXUQkOx5NuGLZx77GObxN/5iZG84mxLRNIEOWqBoD3CBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7767e0e84035eba41a181ecbfdd41360b8ab8ddc3dd2f56cb87ee18a6a3c5c85","last_reissued_at":"2026-05-18T01:30:35.680266Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:35.680266Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Baris Evren Ugurcan, Joe P. 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