标题： 随机矩阵叠加谱中的高阶间距与间距比的比较及其在复杂系统中的应用 显示英文标题

标题： Higher-order spacings in the superposed spectra of random matrices with comparison to spacing ratios and application to complex systems

摘要： 随机矩阵与复杂量子系统在适当极限下的谱波动之间的联系可以通过随机矩阵理论的设置来解释。 在圆随机矩阵的$m$叠加谱中，高阶间距统计特性被数值研究。 我们列出了对于给定的$m$、$k$和$\beta$的修改后的 Dyson 指数$\beta'$，其最近邻间距分布与对应于$\beta$和$m$的$k$阶间距分布相同。 这里，我们猜想，对于给定的$m(k)$和$\beta$，得到的$\beta'$序列作为$k(m)$的函数是唯一的。 这个结果可以作为表征系统和在不进行光谱去对称化的情况下确定系统对称结构的工具。 我们通过量子踢转模型验证了 COE 的$m=2$情况下不同希尔伯特空间维度对应的结论。 通过对 COE 和 GOE 的$m=1$和$m=2$情况下的高阶间距和比值进行比较研究，通过改变维度并保持实现次数不变，反之亦然，我们发现 COE 和 GOE 在给定的高阶统计量方面具有相同的渐近行为。 But, we found from our numerical study that within a given ensemble of COE or GOE, the results of spacings and ratios agree with each other only up to some lower $k$, and beyond that, they start deviating from each other. It is observed that for the $k=1$ case, the convergence towards the Poisson distribution is faster in the case of ratios than the corresponding spacings as we increase $m$ for a given $\beta$. Further, the spectral fluctuations of the intermediate map of various dimensions are studied. There, we find that the effect of random numbers used to generate the matrix corresponding to the map is reflected in the higher-order statistics. 显示英文摘要

摘要： The connection between random matrices and the spectral fluctuations of complex quantum systems in a suitable limit can be explained by using the setup of random matrix theory. Higher-order spacing statistics in the $m$ superposed spectra of circular random matrices are studied numerically. We tabulated the modified Dyson index $\beta'$ for a given $m$, $k$, and $\beta$, for which the nearest neighbor spacing distribution is the same as that of the $k$-th order spacing distribution corresponding to the $\beta$ and $m$. Here, we conjecture that for given $m(k)$ and $\beta$, the obtained sequence of $\beta'$ as a function of $k(m)$ is unique. This result can be used as a tool for the characterization of the system and to determine the symmetry structure of the system without desymmetrization of the spectra. We verify the results of the $m=2$ case of COE with the quantum kicked top model corresponding to various Hilbert space dimensions. From the comparative study of the higher-order spacings and ratios in both $m=1$ and $m=2$ cases of COE and GOE by varying dimension, keeping the number of realizations constant and vice-versa, we find that both COE and GOE have the same asymptotic behavior in terms of a given higher-order statistics. But, we found from our numerical study that within a given ensemble of COE or GOE, the results of spacings and ratios agree with each other only up to some lower $k$, and beyond that, they start deviating from each other. It is observed that for the $k=1$ case, the convergence towards the Poisson distribution is faster in the case of ratios than the corresponding spacings as we increase $m$ for a given $\beta$. Further, the spectral fluctuations of the intermediate map of various dimensions are studied. There, we find that the effect of random numbers used to generate the matrix corresponding to the map is reflected in the higher-order statistics.