# Double-scaled SYK The double-scaled SYK is a limit of the [[0201 Sachdev-Ye-Kitaev model|SYK]] model, where certain things are nicer and more solvable. This limit is defined by $p \rightarrow \infty, \quad N \rightarrow \infty, \quad \lambda=2 p^2 / N=\text { fixed},\quad q \equiv e^{-\lambda},$where $p$ is the number of fermions *p*articipating in each interaction term, and $N$ is the total number of fermions. $q$ or $\lambda$ is an important parameter that is held fixed. For a given disk diagram, there is the bulk Hilbert space space interpretation. The idea is to separate it into three parts: one for the ket, one for the bra, and one for the inner product. The separation must follow certain rules so that there is an unambiguous division. ## Further limits - triple scaling limit: $\lambda\ll 1$ and $E/\mathcal{J}\ll1$: - the theory is governed by the Schwarzian action with coupling $C=N /\left(4 q^2 \mathcal{J}\right)$ - triple scaling & $C/\beta\ll1$ - large quantum fluctuations of the Schwarzian mode ## Refs - bulk Hilbert space - [[2022#Lin]] - symmetry algebra - [[2023#Lin, Stanford]] - high-temperature expansion - [[2023#Okuyama (Apr)]] - one-loop correlators - [[2023#Okuyama, Suzuki]] - [[0204 Quantum complexity|complexity]] - [[2023#Rabinovici, Sanchez-Garrido, Shir, Sonner]] - end-of-the-world brane - [[2023#Okuyama (May)]] - susy SYK - [[2023#Boruch, Lin, Yan]] - discrete Weil-Petersson volume - [[2023#Okuyama (Jun)]] - relation to large-$p$ SYK - [[2023#Mukhametzhanov]] - 2-point function of matter operators - [[2023#Okuyama (Dec)]] - quantum disk (a non-commutative bulk motivated from DSSYK) - [[2024#Almheiri, Popov]] - 3d de Sitter gravity - [[2024#Verlinde]] - [[2024#Verlinde, Zhang]] - coarse-grained chord diagrams and phase transitions - [[2024#Berkooz, Brukner, Jia, Mamroud (Short)]] - [[0415 Von Neumann algebra|vN algebra]] - [[2024#Xu]] - [[2024#Tang]] - high-temperature limit (as a theory on its own) - [[2024#Almheiri, Goel, Hu]] - sine dilaton gravity as a holographic dual - [[2024#Blommaert, Mertens, Papalini]] - [[0051 Baby universes|baby universe]] - [[2024#Okuyama]] - connection to dS via [[0179 Pole skipping|pole skipping]] - [[2024#Yuan, Ge, Kim]] - spread complexity and non-perturbative completion - [[2024#Balasubramanian, Magan, Nandi, Wu]] - quantum cosmology bulk dual - [[2025#Blommaert, Levine]]