Positivity of Lyapunov Exponents for a Continuous Matrix-Valued Anderson Model

Abstract

We study a continuous matrix-valued Anderson-type model. Both leading Lyapunov exponents of this model are proved to be positive and distinct for all energies in (2, +∞) except those in a discrete set, which leads to absence of absolutely continuous spectrum in (2, +∞). This result is an improvement of a previous result with Stolz. The methods, based upon a result by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a criterion by Goldsheid and Margulis, allow for singular Bernoulli distributions.

Localization for a Matrix-valued Anderson Model

“…In a previous article of the author, [2], we proved separability of the Lyapunov exponents of H 1 (ω) for large energies, but only for N = 2. It was done by proving p-contractivity and L p -strong irreducibility of the Fürstenberg group for energies E > 2 (for N = 2, λ max = 2) and away from a discrete set of R. Point (ii) of Theorem 2 is based upon this result.…”

“…This model corresponds to the case N = 2 and ℓ = 1 of H ℓ (ω). The main difference between [2] and the proof we have just given is that in [2] we could not let ℓ get small and then just control E, to ensure that ℓX…”

“…Then, for E / ∈ S, the expression of these logarithms being not simple, we could not use an algebraic result like Lemma 1 to prove that the Lie algebra generated by the logarithms is sp N (R) . That is why we had to restrict ourselves to N = 2 in [2].…”

Hölder Continuity of the Integrated Density of States for Matrix-Valued Anderson Models

“…random variables with common law ν such that {0, 1} ⊂ supp ν. This model has already been studied by the author in [3] as an improvement of a result by Stolz and the author in [5]. We proved in [5] absence of absolutely continuous spectrum and pointed out that the improvement made in [3] was necessary to be able to prove local Hölder continuity of the IDS.…”

Absence de spectre absolument continu pour un opérateur d'Anderson à potentiel d'interaction générique

“…the group generated by the support of the common law of the transfer matrices, defined at (4)) is equal to the whole symplectic group Sp N (R) , for all energies in I(N, ℓ), except those in a finite set. According to (4), it is sufficient to prove that the group generated by 2 N transfer matrices, [2] and [3], we use a denseness criterion for finitely generated subgroups of real semisimple connected Lie groups, due to Breuillard and Gelander and stated at theorem 3.2. This criterion gives us the plan of the proof of proposition 3.1 which implies theorem 2.1.…”