Spectral statistics for the discrete Anderson model in the localized regime (Spectra of Random Operators and Related Topics)

We report on recent results on the spectral statistics of the discrete Anderson model in the localized phase. Our results show, in particular, that, for the discrete Anderson Hamiltonian with smoothly distributed random potential at sufficiently large coupling, the limit of the level spacing distribution is that of i.i.d. random variables distributed according to the density of states of the random Hamiltonian. This text is a contribution to the proceedings of the conference "Spectra of Random Operators and Related Topics" held at Kyoto University, 02-04/12/09 organized by N. Minami and N. Ueki.

[1]  Fr'ed'eric Klopp,et al.  Spectral statistics for random Schr\"odinger operators in the localized regime , 2010, 1011.1832.

[2]  J. Combes,et al.  Poisson Statistics for Eigenvalues of Continuum Random Schr , 2008, 0807.0455.

[3]  J. Combes,et al.  Generalized Eigenvalue-Counting Estimates for the Anderson Model , 2008, 0804.3202.

[4]  Werner Kirsch,et al.  An Invitation to Random Schr¨ odinger operators , 2007 .

[5]  R. Killip,et al.  Eigenfunction Statistics in the Localized Anderson Model , 2007 .

[6]  F. Nakano.  DISTRIBUTION OF LOCALIZATION CENTERS IN SOME DISCRETE RANDOM SYSTEMS , 2007, math-ph/0701042.

[7]  F. Nakano.  The Repulsion Between Localization Centers in the Anderson Model , 2006, math-ph/0603029.

[8]  A. Klein,et al.  New Characterizations of the Region of Complete Localization for Random Schrödinger Operators , 2005, math-ph/0503017.

[9]  Manning Marable,et al.  A discussion , 2000 .

[10]  A. Mirlin.  Statistics of energy levels and eigenfunctions in disordered systems , 1999, cond-mat/9907126.

[11]  M. Janssen.  STATISTICS AND SCALING IN DISORDERED MESOSCOPIC ELECTRON SYSTEMS , 1997, cond-mat/9703196.

[12]  A. Klein,et al.  Smoothness of the density of states in the Anderson model at high disorder , 1988 .

[13]  D. Thouless,et al.  Electrons in disordered systems and the theory of localization , 1974 .

[14]  L. Weiss.  The Stochastic Convergence of a Function of Sample Successive Differences , 1955 .

[15]  M. Aizenman,et al.  Communications in Mathematical Physics Finite-Volume Fractional-Moment Criteria for Anderson Localization , 2001 .

[16]  Nariyuki Minami,et al.  Local fluctuation of the spectrum of a multidimensional Anderson tight binding model , 1996 .