# Universal Operator Growth Hypothesis The universal operator growth hypothesis is a hypothesis that controls the spectral density at large real frequencies in chaotic systems:$C(\omega) \sim \exp \left(-\frac{\beta_0|\omega|}{2}\right), \quad \omega \rightarrow \pm \infty,$where $\beta_0$ is not necessarily the same as the inverse temperature $\beta$, where $C(\omega)$ is the Fourier transform of the Wightman two-point function$C(t)=\frac{\operatorname{Tr}(\mathcal{O}(t) \mathcal{O}(0))}{\operatorname{Tr}(1)}.$ In terms of Lanczos coefficients, it states that for operators in generic, non-integrable systems, the Lanczos coefficients $b_n$ grow linearly in $n$:$b_n\sim \lambda_K n.$ It is conjectured that$\lambda_L\le \lambda_K\le2\pi/\beta,$where $\lambda_L$ is the [[0466 Lyapunov exponent|Lyapunov exponent]], $\lambda_K$ is the [[0564 Krylov complexity|Krylov exponent]], and $\beta$ is the inverse temperature. ## SYK For [[0201 Sachdev-Ye-Kitaev model|SYK]], at infinite $q$, $\lambda_L=\lambda_K$ for at all temperatures. However, at the next order in the large $q$ expansion, there is a correction in the large-$\beta\mathcal{J}$ expansion:$\lambda_L=\frac{2 \pi}{\beta}\left(1-\left(2+\frac{5 \pi^2-12}{9 q}\right) \frac{1}{\beta \mathcal{J}}+\cdots\right),$while$\lambda_K=\frac{2 \pi}{\beta}\left(1-\left(2-\frac{7 \pi^2+12}{9 q}\right) \frac{1}{\beta \mathcal{J}}+\cdots\right).$ ## Refs - original - [[2018#Parker, Cao, Avdoshkin, Scaffidi, Altman]] - important - [[2019#Avdoshkin, Dymarsky]] - [[2021#Dymarsky, Smolkin]] - SYK - [[2024#Chapman, Demulder, Galante, Sheorey, Shoval]]