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Our main result is that, under a small rank constraint, the Hamiltonians are automatically frustration-free and they are gapped with a positive probability. This extends previous results on 1D spin chains to all dimensions. The argument additionally controls the local gap. As an application, we obtain a 2D area law for a cut-dependent ground state via recent AGSP methods of Anshu-Arad-Gosset.<\/jats:p>","DOI":"10.22331\/q-2022-09-01-790","type":"journal-article","created":{"date-parts":[[2022,9,1]],"date-time":"2022-09-01T14:16:56Z","timestamp":1662041816000},"page":"790","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Random translation-invariant Hamiltonians and their spectral gaps"],"prefix":"10.22331","volume":"6","author":[{"given":"Ian","family":"Jauslin","sequence":"first","affiliation":[{"name":"Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marius","family":"Lemm","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of T\u00fcbingen, 72076 T\u00fcbingen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2022,9,1]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Itai Arad, Zeph Landau, Umesh Vazirani, and Thomas Vidick. ``Rigorous RG algorithms and area laws for low energy eigenstates in 1D''. 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