Asymptotic growth of Lanczos coefficients in one-dimensional chaotic chains
In plain words
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.
Precise statement
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.
What would settle it
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.