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Discrete derivative asymptotics of the β-Hermite eigenvalues

Part of: Probability theory on algebraic and topological structures Stochastic processes

Published online by Cambridge University Press:  17 April 2019

Gopal Goel  and
Andrew Ahn
Gopal Goel*
Affiliation:
Massachusetts Institute of Technology
Andrew Ahn
Affiliation:
Massachusetts Institute of Technology
*
*Corresponding author. Email: [email protected]
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Abstract

We consider the asymptotics of the difference between the empirical measures of the β-Hermite tridiagonal matrix and its minor. We prove that this difference has a deterministic limit and Gaussian fluctuations. Through a correspondence between measures and continual Young diagrams, this deterministic limit is identified with the Vershik–Kerov–Logan–Shepp curve. Moreover, the Gaussian fluctuations are identified with a sectional derivative of the Gaussian free field.

MSC classification

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see )
Secondary: 60G15: Gaussian processes
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 5 , September 2019 , pp. 657 - 674
Copyright
© Cambridge University Press 2019 

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References

Anderson, G. W., Guionnet, A., and Zeitouni, O.(2009) An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics, Cambridge University Press.CrossRefGoogle Scholar
Borodin, A. (2014) CLT for spectra of submatrices of Wigner random matrices. Moscow Math. J. 14 29–38.CrossRefGoogle Scholar
Borodin, A., and Gorin, V. (2014) General β-Jacobi corners process and the Gaussian free field. Comm. Pure Appl. Math. 68 1774–1844.CrossRefGoogle Scholar
Bufetov, A. (2013) Kerov’s interlacing sequences and random matrices. J. Math. Phys. 54 113302.Google Scholar
Dumitriu, I. and Edelman, A. (2002) Matrix models for beta ensembles. J. Math. Phys. 43 5830–5847.CrossRefGoogle Scholar
Dumitriu, I. and Paquette, E. (2018) Spectra of overlapping Wishart matrices and the Gaussian free field. Random Matrices Theory Appl. 7 1850003.CrossRefGoogle Scholar
Edelman, A. (2010) The random matrix technique of ghosts and shadows. In Markov Processes and Related Fields. 16 783–790.Google Scholar
Erdős, L., and Schröder, D. (2016) Fluctuations of functions of Wigner matrices. Electron. Comm. Probab. 21 86.CrossRefGoogle Scholar
Gorin, V. and Zhang, L. (2018) Interlacing adjacent levels of β: Jacobi corners processes. Probability Theory and Related Fields 172 915–981.CrossRefGoogle Scholar
Sodin, S. (2017) Fluctuations of interlacing sequences. J. Math. Phys. Anal. Geom. 13 364–401.Google Scholar
Vershik, A. M. and Kerov, S. V. (1977) Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux. Soviet Math. Doklady 18 527–531.Google Scholar