{"paper":{"title":"Weak and Strong disorder for the stochastic heat equation and the continuous directed polymer in $d\\geq 3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Alexander Shamov, Chiranjib Mukherjee, Ofer Zeitouni","submitted_at":"2016-01-07T20:00:49Z","abstract_excerpt":"We consider the smoothed multiplicative noise stochastic heat equation $$d u_{\\eps,t}= \\frac 12 \\Delta u_{\\eps,t} d t+ \\beta \\eps^{\\frac{d-2}{2}}\\, \\, u_{\\eps, t} \\, d B_{\\eps,t} , \\;\\;u_{\\eps,0}=1,$$ in dimension $d\\geq 3$, where $B_{\\eps,t}$ is a spatially smoothed (at scale $\\eps$) space-time white noise, and $\\beta>0$ is a parameter. We show the existence of a $\\bar\\beta\\in (0,\\infty)$ so that the solution exhibits weak disorder when $\\beta<\\bar\\beta$ and strong disorder when $\\beta > \\bar\\beta$. The proof techniques use elements of the theory of the Gaussian multiplicative chaos."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.01652","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"}