We study the reduced energy spectrum \(\{E_{i}^{(n)}\}\), which is constructed by picking one level from every n levels of the original spectrum \( \{E_{i}\}\), in a Gaussian ensemble of random matrix with Dyson index \( \beta \in \left( 0,\infty \right) \). It is shown the joint probability distribution of \(\{E_{i}^{(n)}\}\) bears the same form as \(\{E_{i}\}\) with a rescaled parameter \(\gamma =\frac{n(n+1)}{2}\beta +n-1\). Notably, the nth-order level spacing and gap ratio in \(\{E_{i}\}\) become the lowest order ones in \(\{E_{i}^{(n)}\}\), which explains their distributions found separately by recent studies in a consistent way. Our results also establish the higher-order spacing distributions in random matrix ensembles beyond GOE, GUE and GSE and reveal a hierarchy of structures hidden in the energy spectrum.

我们研究了简化的能量谱\（\ {E_ {i} ^ {（n）} \} \），该能量谱是通过从原始频谱的每n个水平中选择一个水平来构造的（\ {E_ {i} \}} \）在戴森索引为\（\ beta \ in \ left（0，\ infty \ right）\）的随机矩阵的高斯整体中。它被示出的联合概率分布\（\ {E_ {I} ^ {（N）} \} \）带有相同的形式\（\ {E_ {I} \} \）与重新缩放的参数\（\ gamma = \ frac {n（n + 1）} {2} \ beta + n-1 \）。显然，\（\ {E_ {i} \} \）中的n阶能级间距和间隙比成为\（\ {E_ {i _ ^^ {（n）} \} \）中的最低阶，以一致的方式解释了最近研究分别发现的分布。我们的结果还建立了GOE，GUE和GSE以外的随机矩阵集合中的高阶间距分布，并揭示了隐藏在能谱中的结构层次。