[Paper Review] Discrete alloy-type models: Regularity of distributions and recent results

Martin Tautenhahn, Ivan Veselić|arXiv (Cornell University)|Mar 28, 2014

Spectral Theory in Mathematical Physics22 references3 citations

TL;DR

This paper investigates spectral regularity and localization in discrete alloy-type random Schrödinger operators on $\ell^2(\mathbb{Z}^d)$, where the potential is a linear combination of i.i.d. random variables with possibly sign-changing coefficients. It establishes that the model is not uniformly $\tau$-Hölder continuous, proves fractional moment bounds via a reverse Hölder inequality, and derives Minami estimates and Poisson eigenvalue statistics, enabling rigorous localization results under large disorder.

ABSTRACT

We consider discrete random Schrödinger operators on $\ell^2 (\mathbb{Z}^d)$ with a potential of discrete alloy-type structure. That is, the potential at lattice site $x \in \mathbb{Z}^d$ is given by a linear combination of independent identically distributed random variables, possibly with sign-changing coefficients. In a first part we show that the discrete alloy-type model is not uniformly $τ$-Hölder continuous, a frequently used condition in the literature of Anderson-type models with general random potentials. In a second part we review recent results on regularity properties of spectral data and localization properties for the discrete alloy-type model.

Motivation & Objective

• To analyze the regularity of spectral distributions in discrete alloy-type models, which are correlated random potentials arising in disordered quantum systems.
• To address the failure of uniform $\tau$-Hölder continuity in these models, a condition often assumed in prior Anderson localization theory.
• To establish rigorous fractional moment bounds and spectral localization via a reverse Hölder inequality, extending methods from the fractional moment method.
• To clarify measurability and conditional distribution issues in the context of alloy-type potentials, correcting common misapplications in the literature.
• To generalize Minami estimates and Poisson eigenvalue statistics to a broader class of alloy-type models beyond the standard Anderson model.

Proposed method

• Uses the fractional moment method with a reverse Hölder inequality as the central analytical tool to derive uniform bounds on Green's function moments.
• Applies Cramér's rule to express Green's functions as rational functions of single-site random variables, enabling moment estimates via polynomial bounds.
• Employs conditional expectation techniques to handle measurability of supremum over uncountable sets in concentration functions.
• Derives a subharmonicity inequality for large disorder, adapted from [ESS14], to establish exponential decay of fractional moments.
• Utilizes Lemma 6.1 and Proposition 6.2 to bound inverse moments of the potential, crucial for the reverse Hölder argument.
• Combines Wegner-type estimates, fractional moment bounds, and Minami estimates to prove spectral localization and Poisson statistics.

Experimental results

Research questions

• RQ1 Is the discrete alloy-type model uniformly $\tau$-Hölder continuous, and does this condition fail in general?
• RQ2 Can the fractional moment method be adapted to alloy-type potentials with sign-changing coefficients and bounded random variables?
• RQ3 What role does the reverse Hölder inequality play in deriving uniform bounds on fractional moments of Green's functions?
• RQ4 How do conditional distributions of alloy-type potentials behave, and what measurability issues arise in their analysis?
• RQ5 Can Minami estimates and Poisson eigenvalue statistics be extended beyond the standard Anderson model to general alloy-type potentials?

Key findings

• The discrete alloy-type model is not uniformly $\tau$-Hölder continuous, invalidating a common assumption in prior localization proofs.
• A reverse Hölder inequality serves as the pivotal estimate in the proof of fractional moment bounds, as shown in [ESS14] and adapted here.
• For large disorder, the fractional moment bound (19) implies exponential decay of Green's function moments, leading to localization.
• Minami estimates are generalized to a class of discrete alloy-type models, confirming Poisson statistics of eigenvalues in the localized regime.
• The paper provides a corrected treatment of conditional distributions and measurability, including a counterexample showing pathological behavior even in simple alloy models.
• Uniform bounds on fractional moments are established under Assumptions (J), (K), and (L), with explicit dependence on disorder strength $\lambda$ and model parameters.