Extremal Theory for Spectrum of Random Discrete Schrodinger Operator. II. Distributions with Heavy Tails

作者：Astrauskas A^*

来源：Journal of Statistical Physics, 2012, 146(1): 98-117.

DOI：10.1007/s10955-011-0402-9

• Anderson Hamiltonian
• Random potential
• Largest eigenvalues
• Extreme value theory
• Heavy tails
• Rank one perturbation

We study the asymptotic structure of the first K largest eigenvalues lambda (k,V) and the corresponding eigenfunctions psi(therefore lambda(k,V)) of a finite-volume Anderson model (discrete Schrodinger operator) H-V =kappa Delta V + xi(.) on the multidimensional lattice torus V increasing to the whole of lattice Z(v), provided the distribution function F(.) of i.i.d. potential xi(.) satisfies condition -log(1 - F(t)) = o(t(3)) and some additional regularity conditions as t ->infinity. For z is an element of V, denote by lambda(0)(z) the principal eigenvalue of the "single-peak" Hamiltonian. kappa Delta V + xi(z)delta(z) in l(2)(V), and let lambda(0)(k), V be the kth largest value of the sample lambda(0)(.) in V. We first show that the eigenvalues lambda(k, V) are asymptotically close to lambda(0)(k, V). V. We then prove extremal type limit theorems (i. e., Poisson statistics) for the normalized eigenvalues (lambda(k, V) - B-V)a(V), where the normalizing constants a(V) > 0 and B-V are chosen the same as in the corresponding limit theorems for lambda(0)(k, V). The eigenfunction psi(therefore lambda(k, V)) is shown to be asymptotically completely localized (as V up arrow Z) at the sites z(k, V) is an element of V defined by lambda(0)(z(k, V)) = lambda(0)(k, V). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrdinger operator when potential peaks are sparse.

• 出版日期2012-1