arXiv subjects
Publications and source records attributed to Kazumi Okuyama.
We show that the moment of the transfer matrix of the double scaled SYK model is written as an overlap between the stationary state of ASEP (asymmetric simple exclusion process) and a product state. We argue that this overlap is an analogue of the strange correlator appearing in the correspondence between the Levin-Wen string-net model and the Turaev-Viro state sum.
For general one-matrix models in the large $N$ limit, we introduce the cap amplitude $ψ(b)$ as the expansion coefficient of the 1-form $ydx$ on the spectral curve. We find that the dilaton equation for the discrete volume $N_{g,n}$ of the moduli space of genus-$g$ Riemann surfaces with $n$ boundaries is interpreted as gluing the cap amplitude along one of the boundaries. In this process, one of the boundaries is capped and the number of boundaries decreases by one. In a similar manner, the genus-$g$ free energy $F_g$ is obtained by gluing the cap amplitude to $N_{g,1}$.
Average of exponential ${\rm Tr}_R e^X$, i.e. of a group rather than an algebra character, in Gaussian matrix model is known to be an amusing generalization of Schur polynomial, where time variables are substituted by traces of products of non-commuting matrices ${\rm Tr} \left(\prod_i A_{k_i}\right)$ and are thus labeled by weak compositions. The entries of matrices $A_k$ are made from extended Laguerre polynomials, what introduces additional difficulties. We describe the generic sum rules, which express arbitrary traces through convolutions of a single Laguerre polynomial $L_{N-1}^1(z_{k_i})$, what is a considerable simplification.
We extend the notion of chord number in the strict large $N$ double-scaled Sachdev-Ye-Kitaev (DSSYK) model to the corresponding finite $N$ ETH matrix model. The chord number in the strict large $N$ DSSYK model is known to correspond to the discrete length of the Einstein-Rosen bridge in the gravity dual, which reduces to the renormalized geodesic length in JT gravity at weak coupling. At finite $N$, these chord number states form an over-complete basis of the non-perturbative Hilbert space, as the structure of the inner product gets significantly modified due to the Cayley-Hamilton theorem: There are infinitely many null states. In this paper, by considering ``EFT for gravitational observables'' or a version of ``non-isometric code'', we construct a family of chord number operators at finite $N$. While the constructed chord number operator depends on the reference chord number state, it realizes approximate $q$-deformed oscillator algebra and reproduces semiclassical bulk geometry around the reference state. As a special case, we will show that when the reference is chosen to be the chord number zero state, the chord number operator precisely matches with the Krylov state complexity, leading to the ``ramp-slope-plateau'' behavior at late times, implying the formation of ``grey hole''.
It is known that the double-scaled SYK model (DSSYK) reduces to JT gravity with a negative cosmological constant by zooming in on the lower edge $E=-E_0$ of the spectrum. We find that the de Sitter JT gravity (i.e. JT gravity with a positive cosmological constant) is reproduced from DSSYK by taking a scaling limit around the upper edge $E=E_0$ of the spectrum. We also argue that the appearance of de Sitter JT gravity is consistent with the behavior of the classical solution of the sine dilaton gravity.
We construct a holographic tensor network for the double-scaled SYK model (DSSYK). The moment of the transfer matrix of DSSYK can be mapped to the matrix product state (MPS) of a spin chain. By adding the height direction as a holographic direction, we recast the MPS for DSSYK into the holographic tensor network whose building block is a 4-index tensor with the bond dimension three.
It is well-known that the partition function of the Jackiw-Teitelboim (JT) gravity is obtained by an integral transformation of volumes of moduli spaces for Riemann surfaces, also known as the Weil-Petersson volumes. This fact enables us to compute the perturbative genus expansion of the partition function by solving a KdV-type non-linear partial differential equation. In this work, we find that this KdV equation also admits transseries solutions. We give a systematic algorithm to explicitly construct a one-parameter transseries solution to the KdV equation. Our approach is based on general two-dimensional topological gravity, and the results for the JT gravity are easily obtained as a special case. The results in the leading non-perturbative sector perfectly agree with another independent calculation from topological recursions in random matrices.
We study the disk partition function of double-scaled SYK model (DSSYK) in the small $λ$ limit, where $λ=-\log q$ is the coupling of DSSYK. We find that the partition function receives non-perturbative corrections in $λ$, which can be resummed by the cubic power of the Dedekind eta function in a certain low temperature limit. We also discuss a possible bulk interpretation of our findings.
We consider the baby universe operator $\mathcal{B}_a$ in the double-scaled SYK (DSSYK) model, which creates a baby universe of size $a$. We find that $\mathcal{B}_a$ is written in terms of the transfer matrix $T$, and vice versa. In particular, the identity operator on the chord Hilbert space is expanded as a linear combination of $\mathcal{B}_a$, which implies that the disk partition function of DSSYK is written as a linear combination of trumpets. We also find that the thermofield double state of DSSYK is generated by a pair of baby universe operators, which corresponds to a double trumpet. This can be thought of as a concrete realization of the idea of ER=EPR.
We continue our study of the doubled Hilbert space formalism of double-scaled SYK model initiated in [arXiv:2401.07403]. We show that the 1-particle Hilbert space introduced by Lin and Stanford in [arXiv:2307.15725] is related to the doubled Hilbert space $\mathcal{H}_0\otimes \mathcal{H}_0$ of the 0-particle Hilbert space $\mathcal{H}_0$ by some linear isomorphism $\mathcal{C}$. It turns out that the entangler and disentangler appeared in our previous study are given by the dual (or adjoint) of $\mathcal{C}^{-1}$ and $\mathcal{C}$.
We compute the two-point function of matter operators in the double-scaled SYK (DSSYK) model, where the two matter operators are inserted at each end of the cylindrical wormhole. We find that the wormhole amplitude in DSSYK is written as a trace over the chord Hilbert space. We also show that the length of the wormhole is stabilized in the semi-classical limit, by the same mechanism worked for the JT gravity case.
We consider matter correlators in the double-scaled SYK (DSSYK) model. It turns out that matter correlators have a simple expression in terms of the doubled Hilbert space $\mathcal{H}\otimes\mathcal{H}$, where $\mathcal{H}$ is the Fock space of $q$-deformed oscillator (also known as the chord Hilbert space). In this formalism, we find that the operator which counts the intersection of chords should be conjugated by certain ``entangler'' and ``disentangler''. We explicitly demonstrate this structure for the two- and four-point functions of matter operators in DSSYK.
We study the refined Rényi negativity in the matrix model of Jackiw-Teitelboim (JT) gravity. We first consider the JT gravity with dynamical branes, which serves as a toy model of the evaporating black hole. By including the backreaction of branes, we find that the refined Rényi negativity monotonically decreases at late time of the evaporation. Next we define a novel quantity, which we call ``capacity of negativity,'' as a derivative of the refined Rényi negativity with respect to the replica number. We find that the capacity of negativity has two peaks as a function of time, which comes from the exchange of dominance of the different types of replica wormholes.
We study the two-matrix model for double-scaled SYK model, called ETH matrix model introduced by Jafferis et al [arXiv:2209.02131]. If we set the parameters $q_A,q_B$ of this model to zero, the potential of this two-matrix model is given by the Gaussian terms and the $q$-commutator squared interaction. We find that this model is solvable in the large $N$ limit and we explicitly construct the planar one- and two-point function of resolvents in terms of elliptic functions.
We show that the connected correlators of partition functions in double scaled SYK model can be decomposed into ``trumpet'' and the discrete analogue of the Weil-Petersson volume, which was defined by Norbury and Scott. We explicitly compute this discrete volume for the first few orders in the genus expansion and confirm that the discrete volume reduces to the Weil-Petersson volume in a certain semi-classical limit.
We study the end of the world (EOW) brane in double scaled SYK (DSSYK) model. We find that the boundary state of EOW brane is a coherent state of the $q$-deformed oscillators and the associated orthogonal polynomial is the continuous big $q$-Hermite polynomial. In a certain scaling limit, the big $q$-Hermite polynomial reduces to the Whittaker function, which reproduces the wavefunction of JT gravity with an EOW brane. We also compute the half-wormhole amplitude in DSSYK and show that the amplitude is decomposed into the trumpet and the factor coming from the EOW brane.
We study the late time behavior of $n$-point spectral form factors (SFFs) in two-dimensional Witten-Kontsevich topological gravity, which includes both Airy and JT gravities as special cases. This is conducted in the small $\hbar$ expansion, where $\hbar \sim e^{- {1}/{G_N}}$ is the genus counting parameter and nonperturbative in Newton's constant $G_N$. For one-point SFF, we study its absolute square at two different late times. We show that it decays by power law at $t \sim \hbar^{-2/3}$ while it decays exponentially at $t \sim \hbar^{-1}$ due to the higher order corrections in $\hbar$. We also study general $n (\ge 2)$-point SFFs at $t \sim \hbar^{-1}$ in the leading order of the $\hbar$ expansion. We find that they are characterized by a single function, which is essentially the connected two-point SFF and is determined by the classical eigenvalue density $ρ_0(E)$ of the dual matrix integral. These studies suggest that qualitative behaviors of $n$-point SFFs are similar in both Airy and JT gravities, where our analysis in the former case is based on exact results.
We study the spectral form factor (SFF) of general topological gravity in the limit of large time and fixed temperature. It has been observed recently that in this limit, called the tau-scaling limit, the genus expansion of the SFF can be summed up and the late-time behavior of the SFF such as the ramp-plateau transition can be studied analytically. In this paper we develop a technique for the systematic computation of the higher order corrections to the SFF in the strict tau-scaling limit. We obtain the first five corrections in a closed form for the general background of topological gravity. As concrete examples, we present the results for the Airy case and Jackiw-Teitelboim gravity. We find that the above higher order corrections are the Fourier transforms of the corrections to the sine-kernel approximation of the Christoffel-Darboux kernel in the dual double-scaled matrix integral, which naturally explains their structure. Along the way we also develop a technique for the systematic computation of the corrections to the sine-kernel formula, which have not been fully explored in the literature before.