The spectral edge of some random band matrices

We study the asymptotic distribution of the eigenvalues of random Hermitian periodic band matrices, focusing on the spectral edges. The eigenvalues close to the edges converge in distribution to the Airy point process if (and only if) the band is sufficiently wide (W >> N^{5/6}.) Otherwise, a different limiting distribution appears.

[1]  Jeffrey Schenker,et al.  Eigenvector Localization for Random Band Matrices with Power Law Band Width , 2008, 0809.4405.

[2]  Martin Bender,et al.  Edge scaling limits for a family of non-Hermitian random matrix ensembles , 2008, 0808.2608.

[3]  L. Pastur,et al.  Limiting eigenvalue distribution for band random matrices , 1992 .

[4]  One more proof of the Erd\H{o}s--Tur\'an inequality, and an error estimate in Wigner's law , 2009, 0901.1620.

[5]  M. Disertori,et al.  Density of States for Random Band Matrices , 2002 .

[6]  SUMMING GRAPHS FOR RANDOM BAND MATRICES , 1996, cond-mat/9610064.

[7]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[8]  E. Wigner.  On the Distribution of the Roots of Certain Symmetric Matrices , 1958 .

[9]  K. Johansson.  From Gumbel to Tracy-Widom , 2005, math/0510181.

[10]  S. Sodin.  Random matrices, nonbacktracking walks, and orthogonal polynomials , 2007, math-ph/0703043.

[11]  M. Pryce.  On a Uniqueness Theorem , 1935, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  N. Alon,et al.  Non-backtracking random walks mix faster , 2006, math/0610550.

[13]  On Asymptotic Expansions and Scales of Spectral Universality in Band Random Matrix Ensembles , 2000, math/0003106.

[14]  I. Ibragimov,et al.  Independent and stationary sequences of random variables , 1971 .

[15]  C. Tracy,et al.  Mathematical Physics © Springer-Verlag 1996 On Orthogonal and Symplectic Matrix Ensembles , 1995 .

[16]  Antti Knowles,et al.  Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model , 2010, 1002.1695.

[17]  Uzy Smilansky,et al.  Quantum chaos on discrete graphs , 2007, 0704.3525.

[18]  Y. Fyodorov,et al.  STATISTICAL PROPERTIES OF EIGENFUNCTIONS OF RANDOM QUASI 1D ONE-PARTICLE HAMILTONIANS , 1994 .

[19]  A. Soshnikov.  Universality at the Edge of the Spectrum¶in Wigner Random Matrices , 1999, math-ph/9907013.

[20]  S. Péché.  Universality results for largest eigenvalues of some sample covariance matrix ensembles , 2007, 0705.1701.

[21]  S. P'ech'e,et al.  The largest eigenvalues of sample covariance matrices for a spiked population: Diagonal case , 2008, 0812.2320.

[22]  D. Thouless,et al.  Maximum metallic resistance in thin wires , 1977 .

[23]  Helmut Bender,et al.  On the uniqueness theorem , 1970 .

[24]  Ohad N. Feldheim,et al.  A Universality Result for the Smallest Eigenvalues of Certain Sample Covariance Matrices , 2008, 0812.1961.

[25]  D. Thouless.  A relation between the density of states and range of localization for one dimensional random systems , 1972 .

[26]  Estimates for moments of random matrices with Gaussian elements , 2005, math-ph/0507060.

[27]  Z. Bai,et al.  Limit of the smallest eigenvalue of a large dimensional sample covariance matrix , 1993 .

[28]  Antti Knowles,et al.  Quantum Diffusion and Delocalization for Band Matrices with General Distribution , 2010, 1005.1838.

[29]  L. Pastur,et al.  On the level density of random band matrices , 1991 .

[30]  A. Soshnikov.  A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices , 2001, math/0104113.

[31]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .