Scrambling in Double-Scaled SYK and De Sitter Space
Leonard Susskind
TL;DR
This work analyzes the double-scaled version of the SYK model, where $q$ grows with $N$ as a positive power, and shows that scrambling is hyperfast: the scrambling time scales as $\tau_* = \frac{1}{q-1}\log N$ for fixed $q$, but in the double-scaled limit $q,N\to\infty$ it becomes $\tau_*=1$ with $1-P(\tau) \sim e^{-\tau}$. Through an epidemic-growth description, the author derives the exact recursion and continuum solution for the scrambling fraction, connects late-time decay to quasinormal-mode (QNM) behavior, and identifies a QNM-like decay in the large-$q$ limit where $G_R(\tau) \to e^{-2\tau}$. The paper then argues that hyperfast scrambling aligns with a de Sitter holographic dual, where the degrees of freedom reside on the stretched horizon and no logarithmic entropy factor appears in the scrambling time, contrasting with black-hole scrambling. Overall, the results support a de Sitter dual for the double-scaled SYK and clarify hallmark dynamical signatures that distinguish de Sitter from black-hole holography. The findings emphasize a direct link between QNM decay, scrambling dynamics, and holographic interpretations in a de Sitter context.
Abstract
I want to call attention to a simple previously noted fact about the double-scaled version of the SYK model which suggests that it may be holographically dual to de Sitter space.