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Scaling of the reduced energy spectrum of random matrix ensemble

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Abstract

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.

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Data Availibility Statement

This manuscript has associated data in a data repository [Authors comment: The data that support the figures within this paper are available from the corresponding author upon reasonable request.]

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Acknowledgements

This work is supported by the National Natural Science Foundation of China through Grant No. 11904069 and No. 11847005 and No. 11804070.

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Authors and Affiliations

  1. School of Science, Hangzhou Dianzi University, Hangzhou, 310027, China

    Wen-Jia Rao & M. N. Chen

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  1. Wen-Jia Rao
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  2. M. N. Chen
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Correspondence to Wen-Jia Rao.

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Rao, WJ., Chen, M.N. Scaling of the reduced energy spectrum of random matrix ensemble. Eur. Phys. J. Plus 136, 81 (2021). https://doi.org/10.1140/epjp/s13360-020-01067-3

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