AI 中文总结
该研究针对随机矩阵理论的热场双态，构建了以最大纠缠态为初始矢量的Krylov基半经典希尔伯特空间，发现其在低能连续极限下可约化为JT引力的Liouville哈密顿量，暗示该空间是对偶引力的体希尔伯特空间
AI 中文摘要
我们研究随机矩阵理论中热场双态（TFD）的单侧时间演化，其中哈密顿量取自幺正不变的厄米矩阵系综中的一个D×D随机矩阵。我们认为，以最大纠缠态作为初始矢量的Krylov基，在D→∞极限下、任意O(1)温度和时间下，为随机矩阵理论中的这些TFD态提供了半经典希尔伯特空间描述，这与双标度SYK（DSSYK）模型中的“弦希尔伯特空间”构造非常相似。我们针对D→∞极限下谱密度为偶函数、紧支于区间且具有平方根边缘的系综，详细研究了这种半经典描述。在谱密度解析结构的几个额外条件下，我们观察到，半经典哈密顿量在大Krylov深度处的渐近行为与DSSYK的渐近行为相同，对应的参数𝔮=e⁻λ与谱割附近最近的谱密度零点的位置相关。此外，在对应于DSSYK谱密度紫外变形的一大类模型中，即修改了紫外区的谱同时保留近边缘行为不变的模型中，我们证明，在低能连续极限下，Krylov基中的半经典有效哈密顿量可约化为JT引力的Liouville哈密顿量。这表明我们的半经典希尔伯特空间应被解释为对偶引力描述的体希尔伯特空间。
英文摘要
We study the one-sided time evolution of thermofield double (TFD) states in random matrix theory, where the Hamiltonian is taken to be a $D\times D$ random matrix drawn from a unitarily invariant emsemble of Hermitian matrices. We argue that the Krylov basis with the maximally entangled state taken as the initial vector gives a semiclassical Hilbert space description of these TFD states in random matrix theory at any $O(1)$ temperature and time in the $D\to \infty$ limit, very analogous to the ``chord Hilbert space'' construction in the double-scaled SYK (DSSYK) model. We study this semiclassical description in detail for ensembles where the spectral density in the $D\to \infty$ limit is even, compactly supported on an interval and has square root edges. With a few more conditions on the analytic structure of the spectral density, we observe that the semiclassical Hamiltonian has the same asymptotic behavior at large Krylov depth as that of DSSYK, with the corresponding parameter $\mathfrak{q}=e^{-λ}$ being related to the location of the nearest zero of the spectral density away from the spectral cut. Furthermore, in a large class of models corresponding to ultraviolet deformations of the DSSYK spectral density, i.e., where the spectrum in the UV is modified while leaving the near-edge behavior unchanged, we show that the semiclassical effective Hamiltonian in the Krylov basis reduces to the Liouville Hamiltonian of JT gravity in a low-energy, continuum limit. This suggests that our semiclassical Hilbert space should be interpreted as the bulk Hilbert space of a dual gravity description.
