Dependence of the Density of States on the Probability Distribution for Discrete Random Schrödinger Operators

Abstract

We prove that the the density of states measure (DOSm) for random Schrödinger operators on Z d is weak- * Hölder-continuous in the probability measure. The framework we develop is general enough to extend to a wide range of discrete, random operators, including the Anderson model on the Bethe lattice, as well as random Schrödinger operators on the strip. An immediate application of our main result provides quantitive continuity estimates for the disorder dependence of the DOSm and the integrated density of states (IDS) in the weak disorder regime. These results hold for a general compactly supported single-site probability measure, without any further assumptions. The few previously available results for the disorder dependence of the IDS valid for dimensions d 2 assumed absolute continuity of the single-site measure and thus excluded the Bernoulli-Anderson model. As a further application of our main result, we establish quantitative continuity results for the Lyapunov exponent of random Schrödinger operators for d = 1 in the probability measure with respect to the weak- * topology.

Dependence of the Density of States on the Probability Distribution. Part II: Schrödinger Operators on $$\pmb {\mathbb {R}}^d$$ and Non-compactly Supported Probability Measures

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We extend our results in [15] on the quantitative continuity properties, with respect to the single-site probability measure, of the density of states measure and the integrated density of states for random Schrödinger operators. For lattice models on Z d , with d 1, we treat the case of non-compactly supported probability measures with finite first moments.

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“…For lattice models on Z d , with d 1, we treat the case of non-compactly supported probability measures with finite first moments. For random Schrödinger operators on R d , with d 1, we prove results analogous to those in [15] for compactly supported probability measures. The method of proof makes use of the Combes-Thomas estimate and the Helffer-Sjöstrand formula.…”

“…This is the second of a pair of papers, the first being [15], in which we study the dependency of almost sure quantities such as the density of states measure and the integrated density of states of random Schrödinger operators on the single-site probability measure.…”

“…Models for N > 1 arise for instance in the study of multi-dimensional random polymers [9,18,7]. More generally, as in part one [15] of this two-part series (see section 6 in [15]), our treatment allows us to consider more general finite-difference operators in more general settings, such as graphs. In particular, our methods apply to the Anderson model on the Bethe lattice.…”

“…In our recent paper [15], we showed that for single-site probability measures supported on a fixed compact set [−C, C], the map ν → n (∞) ν is Hölder continuous (1.3) in the weak- * topology associated with the dual of C([−C, C]). The purpose of this article is twofold.…”