{"paper":{"title":"A two cities theorem for the parabolic Anderson model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hubert Lacoin, Nadia Sidorova, Peter M\\\"orters, Wolfgang K\\\"onig","submitted_at":"2011-02-24T08:07:19Z","abstract_excerpt":"The parabolic Anderson problem is the Cauchy problem for the heat equation $\\partial_tu(t,z)=\\Delta u(t,z)+\\xi(z)u(t,z)$ on $(0,\\infty)\\times {\\mathbb{Z}}^d$ with random potential $(\\xi(z):z\\in{\\mathbb{Z}}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $\\xi(z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,{z})={\\mathbh1}_0({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4921","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"}