{"paper":{"title":"Fast decay of covariances under $\\delta-$pinning in the critical and supercritical membrane model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alessandra Cipriani, Erwin Bolthausen, Noemi Kurt","submitted_at":"2016-01-07T12:32:21Z","abstract_excerpt":"We consider the membrane model, that is the centered Gaussian field on $\\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a $\\delta-$pinning condition, giving a reward of strength $\\varepsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\\geq 4$ covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in $d\\geq 5$ and logarithmic in $d=4.$ The proof is based on estimates for certain discrete Sobolev norms, an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.01513","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}