Hostname: page-component-857557d7f7-gtc7z Total loading time: 0.001 Render date: 2025-12-01T22:11:52.565Z Has data issue: false hasContentIssue false
  • English
  • Français

Equipartition principle for Wigner matrices

Part of: Probability theory on algebraic and topological structures Equilibrium statistical mechanics

Published online by Cambridge University Press:  27 May 2021

Zhigang Bao ,
László Erdős  and
Kevin Schnelli
Zhigang Bao
Affiliation:
Hong Kong University of Science and Technology, Hong Kong, China; E-mail: mazgbao@ust.hk
László Erdős
Affiliation:
IST Austria, Klosterneuburg, Austria; E-mail: lerdos@ist.ac.at
Kevin Schnelli
Affiliation:
KTH Royal Institute of Technology, Stockholm, Sweden; E-mail: schnelli@kth.se

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.

MSC classification

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see )
Secondary: 82B10: Quantum equilibrium statistical mechanics (general)

Information

Type
Mathematical Physics
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e44
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Benigni, L., ‘Fermionic eigenvector moment flow’, Probab. Theory Related Fields 179(3) (2021), 733–775.CrossRefGoogle Scholar
Benigni, L. and Lopatto, P., ‘Optimal delocalization for generalized Wigner matrices’, Preprint, 2020, arXiv:2007.09585.Google Scholar
Benigni, L. and Lopatto, P., ‘Fluctuations in local quantum unique ergodicity for generalized Wigner matrices’, Preprint, 2021, arXiv:2103.12013.CrossRefGoogle Scholar
Bialas, P., Spiechowicz, J. and Luczka, J., ‘Quantum analogue of energy equipartition theorem’, J. Phys. A 52(15) (2019), 15LT01.CrossRefGoogle Scholar
Bourgade, P. and Yau, H.-T., ‘The eigenvector moment flow and local quantum unique ergodicity’, Comm. Math. Phys. 350(1) (2017), 231–278.CrossRefGoogle Scholar
Cipolloni, G., Erdős, L. and Schröder, D., ‘Eigenstate thermalization hypothesis for Wigner matrices’, Preprint, 2020, arXiv:2012.13215.Google Scholar
Cipolloni, G., Erdős, L. and Schröder, D., ‘Normal fluctuation in quantum ergodicity for Wigner matrices’, Preprint, 2021, arXiv:2103.0673.Google Scholar
Erdős, L., Knowles, A., Yau, H.-T. and Yin, J., ‘The local semicircle law for a general class of random matrices’, Electron. J. Probab. 18(59) (2013), 1–58.CrossRefGoogle Scholar
Erdős, L. and Yau, H.-T., ‘A dynamical approach to random matrix theory’, Courant Lecture Notes in Mathematics, 28, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, (2017).Google Scholar
Erdős, L., Yau, H.-T. and Yin, J., ‘Rigidity of eigenvalues of generalized Wigner matrices’, Adv. Math. 229(3) (2012), 1435–1515.CrossRefGoogle Scholar
He, Y. and Knowles, A., ‘Mesoscopic eigenvalue statistics of Wigner matrices’, Ann. Appl. Probab. 27(3) (2017), 1510–1550.CrossRefGoogle Scholar
He, Y., Knowles, A. and Rosenthal, R., ‘Isotropic self-consistent equations for mean-field random matrices’, Probab. Theory Related Fields 171(1-2) (2018), 203–249.CrossRefGoogle Scholar
Khorunzhy, A., Khoruzhenko, B. and Pastur, L., ‘Asymptotic properties of large random matrices with independent entries’, J. Math. Phys. 37(10) (1996), 5033–5060.CrossRefGoogle Scholar
Knowles, A. and Yin, J., ‘Eigenvector distribution of Wigner matrices’, Probab. Theory Related Fields 155(3-4) (2013), 543–582.CrossRefGoogle Scholar
Lytova, A. and Pastur, L., ‘Central limit theorem for linear eigenvalue statistics of random matrices with independent entries’, Ann. Probab. 37 (2009), 1778–1840.CrossRefGoogle Scholar
Marcinek, J. and Yau, H.-T., ‘High dimensional normality of noisy eigenvectors’, Preprint, 2020, arXiv:2005.08425.Google Scholar
Tao, T. and Vu, V., ‘Random matrices: Universal properties of eigenvectors’, Random Matrices Theory Appl. 1(1) (2012), 1150001.CrossRefGoogle Scholar