Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder. (English) Zbl 1035.82023

Wang, Wei-Min

Invent. Math. 146, No. 2, 365-398 (2001).

The author considers the model described by a discrete random Schrödinger operator of the following type \(H = \Delta + \gamma V\). It is defined on \(\ell^2 (\mathbb{Z}^d)\). Here \(\gamma>0\) is a parameter and the matrix elements (in the canonical base) of \(\Delta\) and \(V\) are as \(\Delta_{ij} = \delta_{| i - j| ,1}\), \(V_{ij} = \delta_{ij} v_i\), where \(| i-j| \) is the \(\ell^1\)-distance in \(\mathbb{Z}^d\) and \(\{v_i \}_{i \in \mathbb{Z}^d}\) is a family of independent identically distributed random variables with distribution \(g\). It is well-known that the Laplacian \(\Delta\) has an absolutely continuous spectrum \(\sigma (\Delta)= [-2d , 2d]\). It is also known that \[ \sigma(H) = [-2d , 2d] + \gamma \cdot \text{supp\,}g, \quad \text{a.s.} \] For \(\gamma \gg 1\) and \(d\), \(\sigma (H)\) is an almost surely pure point (Anderson localization). It is believed that for \(d\geq 3\) in the region of small \(\gamma\) there is a transition from the point to continuous spectrum (Anderson transition). This is the reason why multidimensional Anderson models with weak disorder attract much attention in the recent years. In the present article it is proven that under certain conditions on \(g\) there exists \(\gamma_0>0\) such that for all \(\gamma \in (0, \gamma_0)\), with probability 1 \(\sigma (H) \cap I(\gamma) = \sigma_{\text{pp}} (H)\). Here \(I(\gamma)\) is a nonempty sub-interval of \([-2d , 2d]\) depending on \(\gamma \) and \(g\). It is also proven that the eigenfunctions \(\psi_E\) corresponding to \(E \in I(\gamma)\) decay at least exponentially with the mean-field rate \(\sqrt{\gamma}\). The statistics of the eigenvalue spacing is proven to be asymptotically Poissonian. The proofs rely on supersymmetric arguments.

Reviewer: Yuri Kozitsky (Lublin)

MSC:

82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
35P20	Asymptotic distributions of eigenvalues in context of PDEs
60K35	Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

Anderson localization; pure point spectrum; Lyapunov exponent; supersymmetry approach