Random centers of localization for random operators

Raphael Ducatez

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Distributed under a Creative Commons Attribution 4.0 International License
Random centers of localization for random operators

Raphael Ducatez

June 12, 2024

Abstract

We propose a new random process to construct the eigenvectors of some random operators which make a short and clean connection with the resolvent. In this process the center of localization has to be chosen randomly.

# 1 Introduction

We consider a matrix $H \in \mathbb{R}^{|\Lambda| \times |\Lambda|}$, $\Lambda$ a finite set, defined as

$$H = T + V$$

where $T$ is a fixed symmetric matrix and $V = \text{diag}((v_x)_{x \in \Lambda})$ is a diagonal matrix whose entries $(v_x)$ are random, independent and with law $\rho_x$ and we are interested in the following question.

**Question 1.** Assuming $\lambda$ is an eigenvalue of $H$ what does the eigenvector $\phi_\lambda$ look like ?

Such a question were first asked by P. Anderson [And58] who predicted that in the tight binding model with high disorder the eigenvectors should be localized and since then the subject has been very active field of research in physics. A rigorous proof of the now called Anderson localization has been first given in [KS80] for the random Schrodinger operator in one dimension and then in [FS83, AM93] in any dimension with high disorder. See also the books [CL12] and [AW15] for an introduction of the topics and the many related questions (dynamical localization, local statistic of the eigenvalues, delocalization or trees,...).

The present paper is motivated by the following remarks : In the literature the mechanics of the proofs heavily relied on the resolvent $(H-z)^{-1}$, $z \in \mathbb{C}$. The reason is that one can use the resolvent formula, which allows a lot of algebraic computations to obtain very good estimates on $(H-z)^{-1}$. On the others hand the results on the eigenvectors are mostly qualitative. The difficulty here is that eigenvalue $\lambda$ is random itself and then the associated eigenvector $\phi_\lambda$ is not
as easy to manipulate as the resolvent. In these notes, our main result is to give a new random process to construct the eigenvector which make a clear and direct connection with the resolvent. The key input is to choose the center of localization randomly. Such a process already appeared in [RV18] and [Duc19] but in a less general statement.

## 2 Two random constructions of the eigenvectors

We assume that $T$ and $V$ are such that

1. The law of the $(v_x)$ are absolutely continuous,

2. $\text{Spec}(H) \subset [0, 1]$ a.s,

3. The eigenvalues are non degenerate a.s.

For the second condition one should just think $T$ and $V$ as bounded matrices that has been re-scaled for convenience. The third point actually follows from Minami estimate [Min96]. We add it just to make sure that the definition of $\phi_\lambda$ the eigenvector associated to an eigenvalue $\lambda$ is not ambiguous (up to a phase). Then the first condition is by far the most critical one.

We propose two random processes on $\Omega = \mathbb{R}^\Lambda \times [0, 1] \times \mathbb{S}^{\Lambda-1} \times \Lambda$ that we will denote $\mu_1$ and $\mu_2$ and where we denote $\mathbb{S}^{\Lambda-1} = \{u \in \mathbb{R}^\Lambda : \|u\|_{L^2} = 1\}$.

### Construction of $\mu_1$:

The law $\mu_1$ should be seen as the standard way to construct an eigenvector and it is the process we are interested in.

1. Draw the random diagonal $V$ (with law $\otimes_{x \in \Lambda} \rho_x$)

2. Diagonalize $H = T + V$, choose a random eigenvalue $\lambda \in \text{Spec}(H)$ uniformly and denote $\phi_\lambda$ its corresponding eigenvector.

3. Choose a random point $x^* \in \Lambda$ with the conditional law $\mathbb{P}(x^* = x | V, \lambda, \phi_\lambda) = |\phi_\lambda(x)|^2$. We will call $x^*$ a random center of $\phi_\lambda$.

We have then obtained $(V, \lambda, \phi_\lambda, x^*) \in \Omega$ and $\mu_1$ is the law of this construction. Remark that if $\phi_\lambda$ is localized, the point $x^*$ should be a good guess of its domain of localization, which is the reason we also refer $x^*$ as the center of localization of $\phi_\lambda$.

We now construct the process $\mu_2$, but first we will need the following very standard Lemma (see for example [AW15, Theorem 5.3]). For $x \in \Lambda$, we denote $H_{v_x=0}$ such that $H = H_{v_x=0} + v_x 1_x 1_x^*$.

**Lemma 2.** For any $\lambda \in \mathbb{R}$, $x \in \Lambda$ and $(v_y) \in \mathbb{R}^{\Lambda \setminus \{x\}}$ there exists at most a unique $v_x \in \mathbb{R}$ such that $\lambda$ is an eigenvalue of $T+V$ with $\phi_\lambda(x) \neq 0$. Moreover

$$v_x = - \frac{1}{(H_{v_x=0} - \lambda)_{xx}^{-1}} \quad \text{and} \quad \phi_\lambda = \frac{(H_{v_x=0} - \lambda)^{-1} 1_x}{\|(H_{v_x=0} - \lambda)^{-1} 1_x\|}.$$
Proof of Lemma 2. If such a $v_x$ exists, we have $(H_{v_x=0} - \lambda + v_x 1_x 1_x^*)\phi_\lambda = 0$ and then

$$ (H_{v_x=0} - \lambda)\phi_\lambda = -v_x\phi_\lambda(x)1_x $$

so that we have

$$ \phi_\lambda = -v_x\phi_\lambda(x)(H_{v_x=0} - \lambda)^{-1}1_x. $$

Here $v_x\phi_\lambda(x) \in \mathbb{R}$ and can be obtained using the normalization $\|\phi_\lambda\| = 1$. Moreover, $\phi_\lambda(x) = -v_x\phi_\lambda(x)(H_{v_x=0} - \lambda)_{xx}^{-1}$ which finishes the proof of the Lemma. $\square$

### Construction of $\mu_2$:

The second process $\mu_2$ is also natural and has a clear connection with the resolvent. It will be a very useful tool for the study of $\mu_1$.

1. Draw $\lambda \in [0, 1]$ randomly with uniform law.

2. Choose a random point $x^* \in \Lambda$ with uniform law.

3. Draw the random diagonal entries of $V$ except for $v_{x^*}$ (with law $\otimes_{y \in \Lambda \setminus \{x^*\}} \rho_y$).

4. Construct $v_{x^*}$ and $\phi_\lambda$ as in Lemma 2 such that $\lambda$ is an eigenvalue of $H = T + V$.

We then obtain $((v_y)_{y \neq x^*}, v_{x^*})$, $\lambda, \phi_\lambda, x^*) \in \Omega$ and denote $\mu_2$ the law of this construction. We can now state the main result of this paper.

**Theorem 3.** We have the following Radon-Nikodym derivative

$$ \frac{d\mu_1}{d\mu_2}(V, \lambda, \phi_\lambda, x^*) = \frac{d\rho_{x^*}}{dv_{x^*}}(v_{x^*}) $$

Proof of Theorem 3. Let $f: \Omega \to \mathbb{R}$ a test function and denoting $\omega = (V, \lambda, \phi_\lambda, x^*)$ we have

$$ \begin{aligned} \mathbb{E}_{\mu_1}(f(\omega)) &= \frac{1}{|\Lambda|} \int_{\mathbb{R}^\Lambda} \sum_{\lambda \in \text{Spec}(H)} \sum_{x^* \in \Lambda} f(\omega) |\phi_\lambda(x^*)|^2 \prod_{y \in \Lambda} d\rho_y(v_y) \\ &= \frac{1}{|\Lambda|} \sum_{x^* \in \Lambda} \sum_{i=1}^{|\Lambda|} \int_{\mathbb{R}^{\Lambda \setminus \{x^*\}}} \int_{\mathbb{R}} f(\omega) |\phi_{\lambda_i}(x^*)|^2 d\rho_{x^*}(v_{x^*}) \prod_{y \in \Lambda \setminus \{x^*\}} d\rho_y(v_y) \end{aligned} $$

where we denote $\lambda_1 \le \dots \le \lambda_{|\Lambda|}$ the eigenvalues of $H$. We also denote $\lambda_1^{(x^*)} \le \dots \le \lambda_{|\Lambda|-1}^{(x^*)}$ the eigenvalues of the restricted matrix $H|_{\Lambda \setminus \{x^*\}}$ and by interlacing we have $\lambda_i \in [\lambda_{i-1}^{(x^*)}, \lambda_i^{(x^*)}]$ for all $1 \le i \le |\Lambda|$ where $\lambda_0^{(x^*)} = 0$ and $\lambda_{|\Lambda|}^{(x^*)} = 1$. With $(v_y)_{y \neq x^*}$ fixed, the eigenvalues can be seen as functions of $v_{x^*}: \lambda_i = $
$\lambda_i(v_{x^*})$ and we make the following change of variable $v_{x^*} \to \lambda_i$. Because $\frac{\partial \lambda_i}{\partial v_{x^*}} = |\phi_{\lambda_i}(x^*)|^2$ we obtain

$$
\begin{align*}
\mathbb{E}_{\mu_1}(f(\omega)) &= \frac{1}{|\Lambda|} \sum_{x^* \in \Lambda} \sum_{i=1}^{|\Lambda|} \int_{\mathbb{R}^{\Lambda \setminus \{x^*\}}} \left( \int_{\lambda_{i-1}^{(x^*)}}^{\lambda_i^{(x^*)}} f(\omega) \frac{d\rho_{x^*}}{dv_{x^*}} d\lambda \right) \prod_{y \in \Lambda \setminus \{x^*\}} d\rho_y(v_y) \\
&= \frac{1}{|\Lambda|} \sum_{x^* \in \Lambda} \int_{\mathbb{R}^{\Lambda \setminus \{x^*\}}} \int_{[0,1]} f(\omega) \frac{d\rho_{x^*}}{dv_{x^*}} d\lambda \prod_{y \in \Lambda \setminus \{x^*\}} d\rho_y(v_y) \\
&= \mathbb{E}_{\mu_2} \left( f(\omega) \frac{d\rho_{x^*}}{dv_{x^*}} \right).
\end{align*}
$$

□

Remark 4. The hypothesis the entries $(v_x)$ are independent can be removed if we replace $\frac{d\rho_{x^*}}{dv_{x^*}}$ by the law of $v_{x^*}$ conditionally on $(v_y)_{y \neq x^*}$. The proof is exactly the same in that case.

Theorem 3 propose an interesting answer to Question 1. Indeed, conditionally on $\lambda$, steps 2-4 in the construction of $\mu_2$ gives a recipe to construct $\phi_\lambda$. The answer here is that we have a direct link between the resolvent and the eigenvector but with one particular feature : in order to be able to use Lemma 2 the starting point $x^* \in \Lambda$ has to be chosen randomly.

Remark 5. We denote $\|\rho\|_{\infty} = \sup_{x \in \Lambda} \|\frac{d\rho_x}{dv_x}\|_{\infty}$. Then for any $A \subset \Omega$

$$
\mu_1(A) \le \|\rho\|_{\infty} \mu_2(A).
$$

In particular, if $\|\rho\|_{\infty} < \infty$, any event $A$ that occurs with high probability for $\mu_2$ (ie $\mu_2(\Omega \setminus A) \ll 1$) occurs with high probability for $\mu_1$.

3 Applications

We now state a few consequences of Theorem 3.

3.1 Law of the eigenvalues and eigenvectors

We denote $\mathbb{E}_{\neq x}$ the mean on all the entries of $V$ but $x$ and similarly $\mathbb{P}_{\neq x}$ the probability

$$
\mathbb{E}_{\neq x}(\cdots) = \int_{\mathbb{R}^{\Lambda \setminus \{x\}}} \cdots \prod_{y \neq x} d\rho(v_y) \quad \text{and} \quad \mathbb{P}_{\neq x}(A) = \mathbb{E}_{\neq x}(1_A) \text{ for all } A.
$$

The first point of the following Corollary is a standard result which is usually proved considering the limit $\lim_{\epsilon \to 0^+} \mathbb{E} \left( \frac{1}{|\Lambda|} \mathfrak{Tr}((H - \lambda + i\epsilon)^{-1}) \right)$ (see for example [AW15, Chapter 4]). We claim that it is just a particular case of Theorem 3 which is much more general because it also gives the law for the eigenvector.
**Corollary 6.** 1. The density of state $\nu$ is given by

$$ \frac{d\nu}{d\lambda}(\lambda) = \frac{1}{|\Lambda|} \sum_{x \in \Lambda} \mathbb{E}_{\neq x} \left[ \frac{d\rho_x}{dv_x} \left( \frac{1}{(\lambda - H_{v_x=0})_{xx}^{-1}} \right) \right] $$

In particular $\|\frac{d\nu}{d\lambda}\|_{\infty} \le \|\rho\|_{\infty}$ (Wegner Estimate).

2. For any subset $U \subset S^{\Lambda-1}$

$$ \mathbb{P}(\phi_{\lambda} \in U | \lambda \in \text{Spec}(H)) = \frac{1}{|\Lambda| \frac{d\nu}{d\lambda}(\lambda)} \sum_{x \in \Lambda} \mathbb{E}_{\neq x} \left[ 1_U \left( \frac{(H_{v_x=0} - \lambda)^{-1} 1_x}{\| (H_{v_x=0} - \lambda)^{-1} 1_x \|} \right) \frac{d\rho_x}{dv_x} \left( \frac{1}{(\lambda - H_{v_x=0})_{xx}^{-1}} \right) \right] $$

*Proof of Corollary 6.* With a test function that only depends on $\lambda$, $f(\omega) = f(\lambda)$ we have

$$ \begin{aligned} \mathbb{E}_{\mu_1}(f(\lambda)) &= \mathbb{E}_{\mu_2} \left( f(\lambda) \frac{d\rho_{x^*}}{dv_{x^*}} \right) \\ &= \int_0^1 f(\lambda) \left( \frac{1}{|\Lambda|} \sum_{x \in \Lambda} \mathbb{E}_{\neq x} \left[ \frac{d\rho_x}{dv_x} \left( \frac{1}{(\lambda - H_{v_x=0})_{xx}^{-1}} \right) \right] \right) d\lambda \end{aligned} $$

and we deduce the first point of the Corollary. The second point follows similarly

$$ \begin{aligned} \mathbb{E}_{\mu_1}(f(\lambda)1_{\phi_{\lambda} \in U}) &= \mathbb{E}_{\mu_2} \left( f(\lambda)1_{\phi_{\lambda} \in U} \frac{d\rho_{x^*}}{dv_{x^*}} \right) \\ &= \int_0^1 f(\lambda) \left( \frac{1}{|\Lambda|} \sum_{x \in \Lambda} \mathbb{E}_{\neq x} \left[ 1_U \left( \frac{(H_{v_x=0} - \lambda)^{-1} 1_x}{\| (H_{v_x=0} - \lambda)^{-1} 1_x \|} \right) \frac{d\rho_x}{dv_x} \left( \frac{1}{(\lambda - H_{v_x=0})_{xx}^{-1}} \right) \right] \right) d\lambda \\ &= \int_0^1 f(\lambda) \mathbb{E}_{\mu_1} (1_{\phi_{\lambda} \in U} | \lambda) \frac{d\nu}{d\lambda}(\lambda) d\lambda \end{aligned} $$

□

## 3.2 From the localization of the resolvent to the localization of the eigenvector

The resolvent is by far the main tool to study the Anderson model and most of the computation and estimate is done on this object. Because Theorem 3 gives a direct connection between the law of the eigenvector and the resolvent one have a very short proof to deduce “eigenvector localization” from the localization of the resolvent. We introduce the following localization event

$$ U_{\text{loc}}^{\eta} = \{ \phi \in S^{\Lambda-1} : \exists x \in \Lambda, \forall y \in \Lambda | \phi(y) \le \eta(x, y) \} $$

for some function $\eta: \Lambda^2 \to \mathbb{R}$. Such a function describes a localization phenomenon if $\eta(x, y) \to 0$ when $y$ is far away from $x$. For example, in the case $\Lambda \subset \mathbb{Z}^d$, one usually chooses $\eta(x, y) \sim \exp(-c|x - y|)$ or $\eta(x, y) \sim \frac{1}{(1 + |x - y|)^k}$.
**Corollary 7.** For $\lambda \in [0, 1]$,

$$ \mathbb{P}(\phi_{\lambda} \notin U_{\text{loc}}^{\eta} | \lambda \in \text{Spec}(H)) \leq \left( \frac{d\nu}{d\lambda}(\lambda) \right)^{-1} \|\rho\|_{\infty} \alpha^{\eta} $$

where

$$ \alpha^{\eta} = \sup_{x \in \Lambda} \mathbb{P}_{\neq x} \left[ \exists y \in \Lambda : \left| (H_{v_x=0} - \lambda)_{xy}^{-1} \right| > \eta(x, y) \left\| (H_{v_x=0} - \lambda)^{-1} 1_x \right\| \right] $$

*Proof of Corollary 7.* From Corollary 6 we have

$$ \mathbb{P}(\phi_{\lambda} \notin U_{\text{loc}}^{\eta} | \lambda \in \text{Spec}(H)) \leq \frac{\|\rho\|_{\infty}}{\frac{d\nu}{d\lambda}(\lambda)} \sup_{x \in \Lambda} \mathbb{P}_{\neq x} \left[ \frac{(H_{v_x=0} - \lambda)^{-1} 1_x}{\| (H|_{v_x=0} - \lambda)^{-1} 1_x \|} \notin U_{\text{loc}}^{\eta} \right] $$

and the result follows from

$$ \left\{ \frac{(H_{v_x=0} - \lambda)^{-1} 1_x}{\|(H_{v_x=0} - \lambda)^{-1} 1_x\|} \notin U_{\text{loc}}^{\eta} \right\} \subset \left\{ \exists y \in \Lambda : \frac{|(H_{v_x=0} - \lambda)_{xy}^{-1}|}{\|(H_{v_x=0} - \lambda)^{-1} 1_x\|} > \eta(x, y) \right\}. \quad \square $$

To prove localization of the eigenvector it then enough prove that $\alpha^{\eta}$ is small. In the case of random Schrodinger operator in dimension one usually use the product of 2×2 random matrices of the form $\begin{pmatrix} \lambda - v_x & -1 \\ 1 & 0 \end{pmatrix}$ as in [CL12]. Notice that in that case $\lambda \in \mathbb{R}$ is fixed. In the case of random Schrodinger operator in any dimension with large disorder, it can be done using the so called multi-scaled analysis [FS83] or more directly the Fractional Moment Method [AM93]

$$ \mathbb{E}(|(H_{v_x=0} - \lambda)_{xy}^{-1}|^s) \leq C e^{-c|x-y|} $$

for some $s, c, C > 0$ and a Markov estimate.

## 3.3 The 1-dimensional random Schrodinger operator

In [RV18] Rifkind and Virag consider the scaling limit of the eigenvector of the critical 1-dimensional random Schrodinger operator

$$ H = -\Delta + \frac{1}{\sqrt{n}} V $$

defined on $\ell^2([1, n])$ where $\Delta$ is the discrete Laplacian on this set. The authors proved the following very nice Theorem.

**Theorem 8.** [RV18] As $n \to \infty$ the form of the eigenvector behaves as

$$ \phi_{\lambda}(|nt|)^2 \sim \frac{1}{C} \exp \left( -\tau(\lambda) \left( \frac{|t-U|}{4} + \frac{B_{t-U}}{\sqrt{2}} \right) \right) \quad \text{for } t \in [0, 1] $$

with U uniform on $[0, 1]$, B an independent Brownian motion, $\tau$ an explicit function and C a normalizing constant.
We claim that one could read this Theorem and compare it with our Theorem 3 as follows

• $U$ is the scaling limit of the center of localization $x^*$ chosen uniformly on $[0, n]$

• $\exp\left(-\tau(\lambda)\left(\frac{|t-U|}{4} + \frac{B_{t-U}}{\sqrt{2}}\right)\right)$ is the scaling limit of $f(y) = \left|(H_{v_x=0} - \lambda)_{x^*y}^{-1}\right|^2$ for $y \in [0, n]$

A similar generalization in the finite 1D-discrete model were also proposed in [Duc19].

## References

[AM93] Michael AIZENMAN et Stanislav MOLCHANOV : Localization at large disorder and at extreme energies: An elementary derivations. Communications in Mathematical Physics, 157:245–278, 1993.

[And58] Philip W ANDERSON : Absence of diffusion in certain random lattices. Physical review, 109(5):1492, 1958.

[AW15] Michael AIZENMAN et Simone WARZEL : Random operators, volume 168. American Mathematical Soc., 2015.

[CL12] René CARMONA et Jean LACROIX : Spectral theory of random Schrödinger operators. Springer Science & Business Media, 2012.

[Duc19] Raphael DUCATEZ : A forward-backward random process for the spectrum of 1d anderson operators. 2019.

[FS83] Jürg FRÖHLICH et Thomas SPENCER : Absence of diffusion in the anderson tight binding model for large disorder or low energy. Communications in Mathematical Physics, 88(2):151–184, 1983.

[KS80] Hervé KUNZ et Bernard SOUILLARD : Sur le spectre des opérateurs aux différences finies aléatoires. Communications in Mathematical Physics, 78:201–246, 1980.

[Min96] Nariyuki MINAMI : Local fluctuation of the spectrum of a multidimensional anderson tight binding model. Communications in mathematical physics, 177:709–725, 1996.

[RV18] Ben RIFKIND et Balint VIRAG : Eigenvectors of the 1-dimensional critical random schrödinger operator. Geometric and Functional Analysis, 28:1394–1419, 2018.