## Electronic Communications in Probability

### Fluctuations of functions of Wigner matrices

#### Abstract

We show that matrix elements of functions of $N\times N$ Wigner matrices fluctuate on a scale of order $N^{-1/2}$ and we identify the limiting fluctuation. Our result holds for any function $f$ of the matrix that has bounded variation thus considerably relaxing the regularity requirement imposed in [7, 11].

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 86, 15 pp.

Dates
Accepted: 13 December 2016
First available in Project Euclid: 2 January 2017

https://projecteuclid.org/euclid.ecp/1483347665

Digital Object Identifier
doi:10.1214/16-ECP38

Mathematical Reviews number (MathSciNet)
MR3600514

Zentralblatt MATH identifier
1355.60011

#### Citation

Erdős, László; Schröder, Dominik. Fluctuations of functions of Wigner matrices. Electron. Commun. Probab. 21 (2016), paper no. 86, 15 pp. doi:10.1214/16-ECP38. https://projecteuclid.org/euclid.ecp/1483347665

#### References

• [1] W. Greg Anderson and Ofer Zeitouni, A CLT for a band matrix model, Probab. Theory Related Fields 134 (2006), no. 2, 283–338.
• [2] Zhigang Bao, Guangming Pan and Wang Zhou, Central limit theorem for partial linear eigenvalue statistics of Wigner matrices, J. Stat. Phys. 150 (2013), 88–129.
• [3] László Erdős and Dominik Schröder, Fluctuations of rectangular young diagrams of interlacing Wigner eigenvalues, ArXiv e-print arXiv:1608.05163 (2016).
• [4] László Erdős, Antti Knowles, Horng-Tzer Yau and Jun Yin, The local semicircle law for a general class of random matrices, Electron. J. Probab 18 (2013), no. 59, 1–58.
• [5] László Erdős, Horng-Tzer Yau, and Jun Yin, Rigidity of eigenvalues of generalized Wigner matrices, Adv. Math. 229 (2012), no. 3, 1435–1515.
• [6] Indrajit Jana, Koushik Saha and Alexander Soshnikov, Fluctuations of linear eigenvalue statistics of random band matrices, Teor. Veroyatnost. i Primenen. 60 (2015), no. 3, 553–592.
• [7] Anna Lytova, On non-Gaussian limiting laws for certain statistics of Wigner matrices, Zh. Mat. Fiz. Anal. Geom. 9 (2013), no. 4, 536–581.
• [8] Anna Lytova and Leonid Pastur, Central limit theorem for linear eigenvalue statistics of random matrices with independent entries, Ann. Probab. 37 (2009), no. 5, 1778–1840.
• [9] Anna Lytova and Leonid Pastur, Fluctuations of matrix elements of regular functions of Gaussian random matrices, J. Stat. Phys. 134 (2009), no. 1, 147–159.
• [10] Anna Lytova and Leonid Pastur, Non-Gaussian limiting laws for the entries of regular functions of the Wigner matrices, ArXiv e-print arXiv:1103.2345 (2011).
• [11] Sean O’Rourke, David Renfrew, and Alexander Soshnikov, On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries, J. Theoret. Probab. 26 (2013), no. 3, 750–780.
• [12] Alessandro Pizzo, David Renfrew, and Alexander Soshnikov, Fluctuations of matrix entries of regular functions of Wigner matrices, J. Stat. Phys. 146 (2012), no. 3, 550–591.
• [13] Åke Pleijel, On a theorem by P. Malliavin, Israel Journal of Mathematics 1 (1963), no. 3, 166–168.
• [14] Mariya Shcherbina, Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices, Zh. Mat. Fiz. Anal. Geom. 7 (2011), no. 2, 176–192, 197, 199.
• [15] Mariya Shcherbina, Fluctuations of linear eigenvalue statistics of $\beta$ matrix models in the multi-cut regime, J. Stat. Phys. 151 (2013), no. 6, 1004–1034.
• [16] Sasha Sodin, Fluctuations of interlacing sequences, ArXiv e-print arXiv:1610.02690 (2016).
• [17] Philippe Sosoe and Percy Wong, Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices, Adv. in Math. 249 (2013), 37–87.
• [18] Nikolai G. Ushakov, Selected topics in characteristic functions, Walter de Gruyter, 1999.
• [19] Eugene P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. 62 (1955), no. 3, pp. 548–564 (English).