{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"b169fea77d9f47966f28d7376b71bb52e4a7b7bfded4621c76cf1e235ce7f2c4","created":"2026-10-03T07:17:58Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"d95bafd39a901cb4a1a59a1ebeb8f4a6372733c33848792364886ef86120b29e","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"cm.anderson-transitions.upper-critical-dimension","kind":"well-posed","literature_status":"contested","n":"1","parents":[],"plain":"For many phase transitions there is a number of spatial dimensions above which the exponents become simple and stop changing. For the localization transition of electrons in a random potential it is unknown whether such a dimension exists or is infinite.","posed_since":"","precise":"For the orthogonal-class Anderson model on a d-dimensional hypercubic lattice, determine whether a finite upper critical dimension $d_u$ exists above which $\\nu$ and the multifractal spectrum are d-independent (proposals include $d_u = 6$ and $d_u = \\infty$), and how the $d \\to \\infty$ limit connects to the Anderson transition on the Bethe lattice and random regular graphs, whose critical scaling differs from ordinary finite-size scaling. An answer is $d_u$, or a demonstration that $d_u = \\infty$ together with the large-d asymptotics of $\\nu(d)$.","problem_ref":null,"references":"","settled_by":"A controlled expansion around the infinite-dimensional (Bethe-lattice) limit, or numerics in $d = 4\\text{ to }7$ with controlled corrections to scaling, showing whether exponents become d-independent at finite d.","status_note":"Numerics in $d = 3\\ \\text{to}\\ 6$ were read as $d_{u} = \\infty$ (Tarquini, Biroli and Tarzia, Physical Review B 95, 094204, 2017); random-matrix descriptions of the infinite-dimensional limit continue (Chen, Giraud, Gong and Lemarie, Physical Review B 110, 014210, 2024).","title":"Upper critical dimension of the Anderson 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