{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"33ce0928a21c0298187e0fbff69f987e98ca71abd9201bba0e91789a59e62d28","created":"2026-10-03T07:17:58Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"ae4132fd66ddd3ff639c2d9dd4d55194df02cb3e2986453b6976eacdcd830f0b","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"cm.chaos-scrambling.lanczos-one-dimension","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"The growth of an operator can be tracked by a sequence of numbers called Lanczos coefficients. In one dimension their exact large-order growth law is not established.","posed_since":"2019","precise":"For a local operator in a generic nonintegrable spin chain (e.g. the mixed-field Ising chain) at infinite temperature, Parker et al. (2019) argued $b_n\\sim A n/\\ln n$ as $n\\to\\infty$ in $d=1$, compared with $b_n\\sim\\alpha n$ in $d\\ge 2$. Growth faster than $n/\\ln n$ is excluded in $d=1$; determine whether generic chaotic chains saturate $b_n\\sim A n/\\ln n$, compute $A$, and relate it to the OTOC front and to the spectral function's high-frequency tail. Answer: the functional form with justification.","problem_ref":null,"references":"","settled_by":"Exact computation of $b_n$ to orders large enough to separate $n/\\ln n$ from competing forms, with an analytic argument linking it to the high-frequency tail.","status_note":"","title":"Asymptotic growth of Lanczos coefficients in one-dimensional chaotic 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