Cardy随机矩阵理论
Cardy random matrix theory
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中文总结 AI 辅助
该研究构建了以模不变性为唯一输入的Cardy随机矩阵模型，通过粗粒化正则化得到势能，使态密度符合Cardy分布，并揭示多尺度行为，为计算机求解3D量子引力对偶模型奠定基础。
中文摘要 AI 辅助
我们构建并研究了一个“Cardy随机矩阵模型”：一种随机矩阵理论，其在最大无知原则下，唯一的输入是$\mathbb{S}$-模不变性的约束。我们通过对固定宽度的能量窗口进行粗粒化来正则化这一约束，并得到单个随机矩阵的势，其本征值扮演2d CFT中初级算符标度维数的角色。该势将粗粒化的态密度驱动到黑洞阈值以上能隙之上的Cardy分布。我们进行蒙特卡洛模拟以确认这一行为，发现粗粒化态密度符合良好，同时揭示邻近本征值的GUE统计。我们确定了这些现象出现的标度极限。在此极限下，模型实现了尺度的层级：在由窗口大小给定的宏观尺度上，占据数被冻结，自举约束严格地将谱固定为Cardy分布，而在介观尺度（窗口内部），Vandermonde排斥像在随机矩阵理论中那样重新排列本征值。只有在平均能级间距的微观尺度上，单个能级才被分辨。因此，我们的随机矩阵模型展示了一个有效的Thouless时间，由粗粒化窗口的逆能量宽度设定。我们的模型为在计算机上求解与3D量子引力对偶的矩阵/张量模型迈出了第一步。
英文摘要
We construct and study a ``Cardy random matrix model'': a random matrix theory whose only input, in a maximal-ignorance prescription, is the constraint of $\mathbb{S}$-modular invariance. We regularize this constraint by coarse-graining over energy windows of fixed width, and obtain a potential for a single random matrix whose eigenvalues play the role of the scaling dimensions of primaries in a 2d CFT. The potential drives the coarse-grained density of states to the Cardy profile above a gap at the black hole threshold. We perform Monte Carlo simulations to confirm this behavior and find good agreement for the coarse-grained density of states, while at the same time revealing GUE statistics for nearby eigenvalues. We identify the scaling limit in which these phenomena emerge. In this limit, the model realizes a hierarchy of scales: at the macroscopic scale given by the window size, the occupations are frozen and the bootstrap constraint rigidly fixes the spectrum to the Cardy profile, while at the mesoscopic scale (inside a window) the Vandermonde repulsion rearranges the eigenvalues as in a random matrix theory. It is only at the microscopic scale of the mean level spacing that the individual levels are resolved. Accordingly, our random matrix model displays an effective Thouless time, set by the inverse energy width of the coarse-graining window. Our model provides a first step towards solving the matrix/tensor model dual to 3D quantum gravity on a computer.
发表机构
- University of Geneva（日内瓦大学）
- Leinweber Institute for Theoretical Physics and Department of Physics, University of California, Berkeley（加州大学伯克利分校莱因韦伯理论物理研究所及物理系）
- Geneva Quantum Center, University of Geneva（日内瓦大学日内瓦量子中心）
- Dipartimento di Fisica, Università di Milano - Bicocca（米兰比可卡大学物理系）
- INFN, sezione di Milano-Bicocca（意大利国家核物理研究所米兰-比可卡分部）
- Institute for Theoretical Physics, University of Amsterdam（阿姆斯特丹大学理论物理研究所）
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