Anderson Localization in Discrete Random Displacements Models

Abstract

We prove spectral and strong dynamical localization in a random displacements model on the lattice, with singular (including Bernoulli) probability distribution of the displacements and an infinite-range site potential which can be asymmetric and sign-indefinite. Our techniques also apply to the models in continuous configuration spaces.

Appendix A: Estimates of Characteristic Functions of Sums of Weakly Dependent Random Variables

Lemma 7

(Cf. [10, Proposition 4.1]) Assume that random variables ${\text {X}}_k({\omega }) = {\text {Y}}_k({\omega }) + {\text {y}}_k({\omega }) = a_k\left( \xi _k + {\text {y}}_k({\omega })\right)$, $1\le k \le n$ satisfy the following conditions:

• (1) The family ${\left\{ \,\xi _k\,, \; k\in \llbracket 1, n \rrbracket \,\right\} }$ is independent, and for all $k\in \llbracket 1,n \rrbracket$ one has $\Vert \xi _k\Vert _{\infty }\le 1$, $\mathbb {E}\left[ \, \xi _k\, \right] = 0$, $\mathbb {E}\left[ \, \xi _k^2\, \right] \ge c >0.{I^{I^I}}$

• (2) $\max \limits _{1 \le k \le n} \Vert {\text {y}}_k\Vert _{\infty } \le n^{-\gamma }$ for some $\gamma >1$.

Denote $\sigma _k^2 = \mathbb {E}\left[ \, {\text {Y}}_k^2\, \right]$, $S_n({\omega }) = \sum _{k=1}^n {\text {X}}_k({\omega })$, and let $\Psi _n(t) = \mathbb {E}\left[ \, \text {e}^{ \text {i}t S_n({\omega })}\, \right]$, $t\in \mathbb {R}$. There exists $C_*>0$ such that, for any $\theta \in (0, \, \frac{1}{2})$, one has:

$$\begin{aligned} {{\,\mathrm{\forall }\,}}t \in \left[ c_* T^*_\theta , T^*_\theta \right] \quad |\Psi _n(t)| \le C \, \text {e}^{- \left( 1 + \text {o}\left( 1\right) \right) \frac{t^2}{2}\sum _{k=1}^n \sigma _k^2} \,, \quad T^*_\theta := C_* a n^{-\theta }. \end{aligned}$$

(A.1)

Furthermore, for any $c_*\in (0,1)$ and $c_* T^*_\theta \le |t| \le T^*_\theta$, it holds that

$$\begin{aligned} |\Psi _n(t)| \le C \, \text {e}^{ - |t|^{\kappa } }\,, \quad \kappa = \frac{ \beta (1 - 2\theta ) }{ 1 - \theta \beta } >0 \,. \end{aligned}$$

(A.2)

The next result is a well-known smoothing lemma (cf. [4, 12, 16, 17]).

Lemma 8

Let $\mu$ be a probability measure on $\mathbb {R}$ with Fourier transform $\hat{\mu }$. Then for any interval $I_\epsilon \subset \mathbb {R}$ of length $\epsilon >0$, one has

$$\begin{aligned} \mu ( I_\epsilon ) \le \frac{8}{\pi } \int _{|t| \le \epsilon ^{-1} } | \hat{\mu }(t)| \, dt \,. \end{aligned}$$

(A.3)

Proof

For any $T>0$, the function $g_{T}: t \mapsto \left( 1 - T^{-1}|t| \right) {{\,\mathrm{{{\textbf {1}}}}\,}}_{[-T,T]}(t)$ is the Fourier transform of the probability measure with density $p_{T}(x) = T \, \frac{1 - \cos (T x)}{\pi (T x)^2}$ (cf., e.g., [9, Section XVI.3]). Further, for any $B>0$ and $T = \epsilon ^{-1}$, $p_{\epsilon ^{-1}}(B\epsilon ) = \epsilon ^{-1} \, \frac{1 - \cos B }{\pi B ^2}$ . In particular, one finds by a numerical calculation that for $B=\pi /3$

$$\begin{aligned} {{\,\mathrm{\forall }\,}}x\in [-B\epsilon , B\epsilon ] \quad \epsilon p_{\epsilon ^{-1}}(x) \ge \epsilon \cdot \epsilon ^{-1} \, \frac{1 - \cos (\pi /3)}{\pi ^3/9} > \frac{1}{8}. \end{aligned}$$

Since $\pi /3>1$, we have ${{\,\mathrm{{{\textbf {1}}}}\,}}_{[-\epsilon , \epsilon ]}(x) \le {{\,\mathrm{{{\textbf {1}}}}\,}}_{[-B\epsilon , B \epsilon ]}(x) \le 8 \epsilon p_{\epsilon ^{-1}}(x)$, and so the claim follows by Parseval’ identity.