Skip to main content Accessibility help

Discrete derivative asymptotics of the β-Hermite eigenvalues

  • Gopal Goel (a1) and Andrew Ahn (a1)


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.


Corresponding author

*Corresponding author. Email:


Hide All
[1]Anderson, G. W., Guionnet, A., and Zeitouni, O.(2009) An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics, Cambridge University Press.
[2]Borodin, A. (2014) CLT for spectra of submatrices of Wigner random matrices. Moscow Math. J. 14 2938.
[3]Borodin, A., and Gorin, V. (2014) General β-Jacobi corners process and the Gaussian free field. Comm. Pure Appl. Math. 68 17741844.
[4]Bufetov, A. (2013) Kerov’s interlacing sequences and random matrices. J. Math. Phys. 54 113302.
[5]Dumitriu, I. and Edelman, A. (2002) Matrix models for beta ensembles. J. Math. Phys. 43 58305847.
[6]Dumitriu, I. and Paquette, E. (2018) Spectra of overlapping Wishart matrices and the Gaussian free field. Random Matrices Theory Appl. 7 1850003.
[7]Edelman, A. (2010) The random matrix technique of ghosts and shadows. In Markov Processes and Related Fields. 16 783790.
[8]Erdős, L., and Schröder, D. (2016) Fluctuations of functions of Wigner matrices. Electron. Comm. Probab. 21 86.
[9]Gorin, V. and Zhang, L. (2018) Interlacing adjacent levels of β: Jacobi corners processes. Probability Theory and Related Fields 172 915981.
[10]Sodin, S. (2017) Fluctuations of interlacing sequences. J. Math. Phys. Anal. Geom. 13 364401.
[11]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 527531.

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed