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\copyrightinfo{2005}{Evgenij Kritchevski}


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\begin{document}

\title{Hierarchical Anderson Model.}
\author{Evgenij Kritchevski}
\address{Department of Mathematics and Statistics.
McGill University, 805 Sherbrooke Street West Montreal, QC, H3A
2K6 Canada} \curraddr{}

\email{ekritc@math.mcgill.ca}
\thanks{This work was supported in part
by an FQRNT grant.}



\subjclass[2000]{Primary 47B80, 47A55, 93A13.}
% details
%93A13 Hierarchical systems
%47A55 Perturbation theory
%47B80 Random operators

\dedicatory{Dedicated to Stanislav Molchanov on the occasion of his 65'th
birthday.}
\date{Friday, April 21, 2006}

%\begin{abstract}
%\end{abstract}

\maketitle

\section{introduction}

In this article, we will review the spectral localization problem in the
hierarchical Anderson model. We will present the original result
of Molchanov (see Theorem \ref{Stas} below) on localization at
arbitrary disorder in any spectral dimension, when the random
perturbations are i.i.d. with a Cauchy distribution, as well as generalization to the case of mixed Cauchy distributions. We will also prove
the new result (see Theorem \ref{maintheorem} below), that in spectral dimension $\rm{d}\leq 4$,
localization holds for very general distributions of the
perturbation. This is an improvement of the main theorem of
\cite{K1}. Our notation will be consistent with \cite{K1} and this
article should be a complementary reading to \cite{K1} and
\cite{M2}. For additional information about the hierarchical
structures and the hierarchical Anderson model we refer the reader
to \cite{D, BS, Bo, K1,K2,M1, M2}.




We first recall the
definition of the model and its basic properties. Let $X$ be an infinite countable set.  Throughout the paper,
$\delta_x$ will denote the Kronecker delta function at $x\in X$. A
partition $\cP$ of $X$ is a collection of its disjoint subsets
whose union is equal to $X$. Let $\mathbf{n}=(n_r)_{r\geq 0}$ be a
sequence of positive integers and $\mathbf{P}=(\cP_r)_{r\geq 0}$ a
sequence of partitions of $X$. The elements of $\cP_r$ are called
"clusters" of rank $r$.  We say that $(X, \mathbf{P},\mathbf{n})$
is a hierarchical structure if the following hold:
\begin{enumerate}
    \item $n_0=1$ and every $Q\in\cP_0$ has exactly one element.
    \item For $r\geq 1$, every $Q\in \cP_r$ is a  disjoint union of
    $n_r$ clusters in $\cP_{r-1}$.
    \item Given $x,y\in X$, there is a cluster $Q$ of some rank
    containing both $x$ and $y$.

\end{enumerate}
Let us state some immediate consequences of this definition. Every
cluster of rank $r\geq 0$ has size $N_r:=\prod_{s=0}^{r}n_s$.
Given $x\in X$ and $r\geq 0$, there is a unique cluster of rank
$r$ containing $x$. We denote this cluster  by $Q_r(x)$.  The map
 \[ d(x,y):=\min\set{r:y\in
Q_r(x)},\] is a metric on $X$ and  $Q_r(x)=\set{y:d(x,y)\leq r}$.
Note that $Q_r(x)=Q_r(y)$ whenever $d(x,y)\leq r$. Given an
integer $n\geq 2$, a hierarchical structure is called
\emph{homogeneous of degree} $n$ if $n_r=n$ for all $r\geq 1$.
Figure \ref{fig} illustrates the first three levels of hierarchy in a
hierarchical structure with $n_1=3,n_2=2,n_3=3$.

\begin{figure}
$$\overbrace{\overbrace{\overbrace{\underbrace{x_1}_{Q_0(x_1)}, x_2,x_3}^{Q_1(x_3)},\overbrace{x_4, x_5,x_6}}^{Q_2(x_3)=Q_2(x_5)},\overbrace{\overbrace{x_7, x_8,x_9},\overbrace{x_{10}, x_{11},x_{12}}^{Q_1(x_8)}}^{Q_2(x_9)}}^{Q_3(x_1)},\overbrace{x_{13}, x_{14},x_{15}},\dots$$
here $d(x_1,x_1)=0,d(x_1,x_3)=1,d(x_1,x_6)=2,d(x_1,x_8)=3$, etc
\caption{A hierarchical structure} \label{fig}
\end{figure}


The free Laplacian on the hierarchical structure $(X,
\mathbf{P},\mathbf{n})$ is defined as follows.
For each $r\geq 0$, let $E_r:l^2(X)\To l^2(X)$ be
the averaging operator
$$(E_r\psi)(x):=\frac{1}{N_r}\sum_{d(x,y)\leq r}\psi(y).$$  Let
 $\mathbf{p}=(p_r)_{r\geq 1}$ be a sequence of positive numbers such that
$\sum_{r=1}^{\infty}p_r=1$.  In the sequel we set $p_0:=0$ and
\[
\lambda_{r}:=\sum_{s=0}^r p_s,\qquad r=0, 1, \cdots, \infty.
\]
The hierarchical Laplacian $\Delta$ on
$l^2(X)$ is defined by
$$\Delta:=\sum_{r=0}^{\infty}p_rE_r.$$
Clearly, $\Delta$ is a bounded self-adjoint
operator and  $0\leq\Delta\leq 1$.

A hierarchical model is a hierarchical structure
$(X,\mathbf{P},\mathbf{n})$ together with the hierarchical
Laplacian $\Delta$. The basic spectral theorem for $\Delta$ is the following.

\begin{thm}\label{spectral_theorem}
{\rm (1)} The spectrum of $\Delta$  is given by

    $$\sigma(\Delta)=\{\lambda_r\,:\, r=0, \cdots, \infty\}.$$
Each $\lambda_r$, $r<\infty$, is an eigenvalue of $\Delta$ of infinite multiplicity.
The point $\la_\infty=1$ is not
an eigenvalue. \newline
{\rm (2)}  $E_r-E_{r+1}$ is the orthogonal projection onto the
    eigenspace of $\la_r$ and
$$\Delta=\sum_{r=0}^{\infty}\la_r(E_r-E_{r+1}).$$
{\rm (3)} For every $x\in X$, the spectral measure
    for $\delta_x$ and $\Delta$
is given by
$$\mu=\sum_{r=0}^{\infty}\of{\frac{1}{N_r}-\frac{1}{N_{r+1}}}\delta({\la_r}),$$
where $\delta({\la_r})$ stands for the Dirac unit mass at
$\la_r$. Note that $\mu$ does not depend on $x$.
\end{thm}
A proof can be found in \cite{K1} or \cite{M2}.

We now define the hierarchical Anderson model associated to $(X,
{\bf P}, {\bf n})$ and the hierarchical Laplacian $\Delta$.
Consider the probability space $(\Omega, {\mathcal F}, {\mathbb
P})$ where  $\Omega:=\Real^X$,  ${\mathcal F}$ is the  usual Borel
$\sigma$-algebra in $\Omega$, and  ${\mathbb P}$ is a given
probability measure on $(\Omega, {\mathcal F})$. For $\omega \in
\Omega$, we set
$$V_{\omega}:=\sum_{x\in X}\omega(x)\scalp{\delta_x}{\cdot}\delta_x. $$
$V_\omega$ is a self-adjoint (possibly unbounded) multiplication operator on
$l^2(X)$. Let
$$H_{\omega}:=\Delta + V_\omega, \qquad \omega\in\Omega.$$
The family of self-adjoint operators $\{H_\omega\}_{\omega \in
\Omega}$ indexed by the events of the probability space $(\Omega,
{\mathcal F}, {\mathbb P})$ is called the \emph{hierarchical
Anderson model}. The main problem is to understand the spectral properties of $H_\omega$. The first natural question is to describe the spectrum $\sigma(H_\omega)$ of $H_\omega$ as a set. Note that, unlike the spectrum of the standard discrete Laplacian on $\mathbb{Z}^{\rm d}$, the
spectrum $\sigma(\Delta)$ of the hierarchical Laplacian is a disconnected set.
\begin{lem}\label{nonrandomspectrum} Suppose that the random variables $\set{\omega(x):x\in
X}$ are i.i.d. with distribution $\nu$, i.e. $\mathbb P=\otimes_{x\in X}\nu$. Let
$S$ be the support of $\nu$ and $ch(S)$ the convex hull of $S$. Then, for $\mathbb P$-a.e. $\omega\in\Omega$,
$$\sigma(\Delta)+S\subset\sigma(H_\omega)\subset (\sigma(\Delta)+ch(S))\cap ([0,1]+S).$$
In particular, if $S$ is connected, then $\sigma(H_\omega)=\sigma(\Delta)+S$, for
$\mathbb P$-a.e. $\omega\in\Omega$.
\end{lem}

The proof of Lemma \ref{nonrandomspectrum} (given in the appendix) is a straight forward adaptation of the proof of the corresponding well known result for the standard Anderson model on $\mathbb Z^{\rm d}$, where the spectrum is almost surely equal to $[0,2\rm d]+\rm{supp} (\nu)$ (cf for example Proposition V.3.5. in \cite{CL}).


We denote by $\sigma_{\rm ac}(H_\omega)$ the
absolutely continuous part of the spectrum of $H_\omega$ and by
$\sigma_{\rm cont}(H_\omega)$ the continuous part. In \cite{M2}, Molchanov proved the following localization result:
\begin{thm}\label{Stas} Suppose that
\begin{equation}\label{Molchanov_condition}
\sum_{r=1}^{\infty}p_r r^{1+\eps}<\infty,
\end{equation}
for some $\eps>0$. If the random variables $\set{\omega(x):x\in
X}$ are i.i.d. with a Cauchy distribution,
$$k_{a,h}(e)de:=\frac{1}{\pi}\frac{h}{(e-a)^2+h^2}de,$$ for some
parameters $a\in\Real$, $h>0$, then $\sigma_{\rm
cont}(H_\omega)=\emptyset$ for ${\mathbb P}$-a.e. $\omega$.
\end{thm}
A proof will be given in section 3. A generalization of Molchanov's theorem will be stated and proven in section 4.

Molchanov's theorem is remarkable for several reasons. First of all, it is a result valid  for any disorder. Given a coupling constant $c>0$, the random operator $\Delta + c V_\omega$ can be rewritten as $\Delta + \sum_{x\in X}c\omega(x)\scalp{\delta_x}{\cdot}\delta_x.$ If $\omega(x)$ has a Cauchy distribution $k_{a,h}$, then $c \omega(x)$ has a Cauchy distribution $k_{c a, c h}$. Hence, under the hypotheses of the theorem, $\Delta + c V_\omega$ has pure point spectrum for ${\mathbb P}$-a.e. $\omega$. The next observation is that the condition
\eqref{Molchanov_condition} is very weak, it particular it does not at all depend  on $n_r$. To appreciate the generality of \eqref{Molchanov_condition}, one has to look at the density of states and at the spectral dimension of $\Delta$. Let $x_0\in X$ be given and consider the increasing sequence of
clusters $Q_r(x_0)$, $r\geq 0$. Let  $P_r$ be the orthogonal
projection onto the $N_r$-dimensional subspace
$$l^2(Q_r(x_0)):=\set{\psi\in l^2(X): \psi(x)=0 \textrm{ for }
x\notin Q_r(x_0)}.$$ Let $e_1^{(r)}\leq e_2^{(r)}\leq\dots\leq
e_{N_r}^{(r)}$ be the eigenvalues of the restricted Laplacian $P_r
\Delta P_r$ acting on $l^2(Q_r(x_0))$ and
$$\nu_r:=\frac{1}{N_r}\sum_{s=1}^{r}\delta(e_{s}^{(r)}),$$
the corresponding counting measure.

\begin{prop}\label{densityofstates} The weak-* limit $\lim_{r \rightarrow \infty}\nu_r$ exists and
is equal to $\mu$.
\end{prop}  A detailed proof can be found in \cite{K1}. In the original paper \cite{M2}, an analogous result is proven, but with a different definition of $\nu_r$. The content of the proposition is that the spectral measure $\mu$ can be naturally interpreted as the integrated density
of states of the operator  $\Delta$.

If
\[\lim_{t\downarrow 0}\frac{\log \mu([1-t,1])} {\log t} ={\rm d}/2,\]
then the number ${\rm d}$ is called the \emph{spectral dimension}
of $\Delta$. This definition is motivated by analogy with the
edge asymptotics of the density of states of the standard
discrete Laplacian on $\mathbb{Z}^{\rm d}$,  for which the spectral
and spatial dimensions coincide.

The relation $\sum_{y\in
X}\scalp{\delta_x}{\Delta\delta_y}=1$ yields that $\Delta$ generates a
random walk on $X$.
 We recall that the random walk on ${\mathbb Z}^{\rm d}$ generated by the  standard discrete
 Laplacian is recurrent if ${\rm d}=1, 2$ and transient if ${\rm d}> 2$. The corresponding
 result for the hierarchical Laplacian is:
\begin{prop}\label{dimension} Consider a homogeneous
hierarchical structure of degree $n\geq 2$. Suppose that there
exist constants $C_1>0,C_2>0$ and $\rho>1$ such that
\[C_1\rho^{-r}\leq
p_r\leq C_2 \rho^{-r},\]
 for $r$ big enough. Then: \newline {\rm
(1)} The spectral dimension of this model is
   $${\rm d}(n,\rho)=2\frac{\log n}{\log \rho}.$$
    Hence $0<{\rm d}(n,\rho)\leq 2$ iff $ n\leq\rho$. \newline
{\rm (2)} The random walk generated by $\Delta$ is recurrent if
    $0<{\rm d}(n,\rho)\leq 2$ and transient if  ${\rm d}(n,\rho)>2$.
\end{prop}
See \cite{K1}, \cite{M2} for a proof. We can now see that Theorem \ref{Stas} holds in arbitrary spectral dimension. Indeed, let $n\geq 2$ be given and let $\rm{d}$ be a positive real number. Then there is a unique $\rho>1$ with
$\rm{d}=\rm{d}(n,\rho)$. Set $p_r=C\rho^{-r}$, where $C$ is a normalization constant making $\sum_{r=1}^{\infty}p_r=1$. Then $p_r$ satisfies \eqref{Molchanov_condition}. Therefore, we can construct homogeneous hierarchical Anderson models of arbitrary spectral dimension for which Theorem \ref{Stas} holds.

Since Cauchy random variables play a very special role in the
theory of random Schr\"{o}dinger operators, it is natural to ask
whether one can extend Theorem \ref{Stas} to distributions other
than Cauchy. A partial answer to this question is that one can
prove localization for very general distributions of $\omega(x)$,
at any disorder, provided that one imposes stronger decay
conditions on $p_r$ than \eqref{Molchanov_condition}. We will
prove the following new result:
\begin{thm}\label{maintheorem} Assume that there exists a sequence
$u_r>0$ with  $\sum_{r=1}^{\infty}u_r^{-1}<\infty$ and
\begin{equation}\label{main_hypothesis}
\sum_{r=1}^{\infty}p_r u_r\sqrt{N_r}<\infty.
\end{equation}
Then: \newline {\rm (1)} For all $\omega \in \Omega$, $\sigma_{\rm
ac}(H_\omega)=\emptyset$. \newline {\rm (2)} If $\mathbb{P}$ is
conditionally a.c. then $\sigma_{\rm cont}(H_\omega)=\emptyset$
for ${\mathbb P}$-a.e. $\omega$.
\end{thm}
A proof will be given in section 3.

\noindent{\bf Remark 1.} The condition \eqref{main_hypothesis} is
more demanding than \eqref{Molchanov_condition}. The required
decay of $p_r$ imposes an upper bound on the spectral dimension of
$\Delta$. Theorem \ref{maintheorem} and Proposition
\ref{dimension} allow us to construct hierarchical models with
spectral dimension ${\rm d}\leq 4$ that exhibit localization at arbitrary disorder. If $(X,
\mathbf{P},\mathbf{n})$ is a homogeneous hierarchical structure of
degree $n\geq 2$ and $p_r=C \rho^{-r}$ with $\rho>\sqrt{n}$, then
the hypothesis \eqref{main_hypothesis} is fulfilled for
$u_r=r^{1+\eps}$. Given $0<{\rm d}<4$ one can adjust
$\rho>\sqrt{n}$ to make $d(n,\rho)={\rm d}$. If
$p_r=Cr^{-2-\eps}n^{-r/2}$, then the model has spectral dimension
${\rm d}=4$ and \eqref{main_hypothesis} is verified for
$u_r=r^{1+\eps/2}$. One can also construct trivial models with
${\rm d}=0$ by taking $p_r$ to decrease faster than $\rho^{-r}$ for
any $\rho$.  We emphasize that homogeneity of the hierarchical
structure is not required for Theorem \ref{maintheorem}.

\noindent{\bf Remark 2.} Theorem \ref{maintheorem} is an
improvement of the main localization theorem of \cite{K1}. In
\cite{K1}, instead of \eqref{main_hypothesis}, it is assumed that
$\sum_{r=1}^{\infty}p_r N_{r-1}u_{r-1}u_r<\infty$, which imposes
the spectral dimension to be $\leq 2$.

\section{Localization criterion} In this section we will formulate and prove a sufficient condition for $H_\omega$ to have $\sigma_{\rm
ac}(H_\omega)=\emptyset$ for all $\omega \in \Omega$,  and a sufficient condition for $H_\omega$ to  have $\sigma_{\rm cont}(H_\omega)=\emptyset$
for ${\mathbb P}$-a.e. $\omega$. Both theorems \ref{Stas} and \ref{maintheorem} will follow from this localization criterion and will be proven in the next section.

Consider the truncated hierarchical Laplacian
$$\Delta_r:=\sum_{s=0}^{r}p_sE_s, \qquad r \geq 0,$$
and set
$$H_{\omega,r}:=V_{\omega} +\Delta_r, \qquad r \geq 0.$$
Fix $\omega\in\Omega$. Note that for any $Q_r\in\cP_r$, the subspace
$l^2(Q_r)$ is invariant for $H_{\omega,r}$. This is the main reason for working with the truncated hierarchical Laplacian. Let
$\sigma(\omega,Q_r)$ be the set of the eigenvalues of the
restricted operator $H_{\omega,r}\upharpoonright l^2(Q_r)$ and
$\sigma_{\omega}:=\bigcup \sigma(\omega,Q_r)$ where the union is
over all clusters of all ranks. Clearly,  $\sigma_{\omega}$ is a
countable subset of $\Real$, and hence of zero Lebesgue measure. For
$z\in\Comp\backslash\sigma_{\omega}$, $r\geq 0$, and $x,y\in X$,
we set
$$G_{\omega,r}(x,y;z):=\scalp{\delta_x}{(H_{\omega,r}-z)^{-1}\delta_y}.$$
For $z\in\Comp\backslash\sigma_{\omega}$, $r\geq 0$ and $t\in X$,
let $g_{\omega,r}(t;z)$ be the average of
$G_{\omega,r}(\cdot,t;z)$ over the cluster $Q_r(t)$, i.e.
$$g_{\omega,r}(t;z):=\frac{1}{N_r}\sum_{d(t',t)\leq r}G_{\omega,r}(t',t;z).$$
Since the joint spectral measure for $\delta_t$, $\delta_{t'}$ and
$H_{\omega,r}$ is real,
$G_{\omega,r}(t',t;z)=G_{\omega,r}(t,t';z)$ and
\begin{equation}\label{symmetry_of_G}
g_{\omega,r}(t;z)=\frac{1}{N_r}\sum_{d(t',t)\leq
r}G_{\omega,r}(t,t';z)=\frac{1}{N_r}\scalp{\delta_t}{(H_{\omega,r}-z)^{-1}\mathbf{1}_{Q_r(t)}}.
\end{equation}
For $Q_r\in\cP_r$, let
$$\gamma_{\omega}(Q_r;z):=\frac{1}{N_r}\scalp{\mathbf{1}_{Q_r}}{(H_{\omega,r}-z)^{-1}\mathbf{1}_{Q_r}}.$$

We denote by $m$ the Lebesgue measure on $\Real$. For $x\in X$
and $\omega\in\Omega$, we denote by $\mu_x^{\omega}$ the spectral
measure for $\Delta+V_\omega$ and $\delta_x$, by
$\mu_{x,\mathrm{cont}}^{\omega}$ the continuous part of
$\mu_x^{\omega}$ and by $\mu_{x,\mathrm{ac}}^{\omega}$ the a.c.
part.
The main spectral localization criterion for the hierarchical Anderson model is the following
\begin{thm}\label{loc_criterion} Assume that there exists a sequence
$u_r>0$ with  $\sum_{r=1}^{\infty}u_r^{-1}<\infty$ and
\begin{equation}\label{weakest_hypothesis}
\sum_{r=1}^{\infty}p_r u_r<\infty.
\end{equation}
Let $x\in X$ and a Borel set $B\subset\Real$ be given. Then: \newline {\rm (1)} If for a fixed $\omega\in\Omega$,
\begin{equation}\label{localization}
\sum_{s=1}^{\infty} p_s\abs{\gamma_{\omega}(Q_s(x);e)}<\infty,
\end{equation}
for $m$-a.e. $e\in B$, then $\mu_{x,\mathrm{ac}}^{\omega}(B)=0$.
\newline {\rm (2)} If $\mathbb{P}$ has a conditional density along the $x$'th coordinate, and \eqref{localization} holds for $\mathbb{P}\otimes m$-a.e.
$(\omega,e)\in\Omega\times B$, then $\mu_{x,\mathrm{cont}}^{\omega}(B)=0$
for ${\mathbb P}$-a.e. $\omega\in\Omega$. \newline {\rm (3)} Assume that for $\mathbb{P}\otimes m$-a.e.
$(\omega,e)\in \Omega\times B$, there is a finite constant $C_{\omega,e,x}$ such that
\begin{equation}\label{improvement}
\abs{\gamma_{\omega}(Q_r(x);e)}\leq C_{\omega,e,x} u_r,
\end{equation}
for all $r\geq 0$. If $\mathbb{P}$ has a conditional density along the $x$'th coordinate, then $\mu_{x,\mathrm{cont}}^{\omega}(B)=0$
for ${\mathbb P}$-a.e. $\omega\in\Omega$.
\end{thm}
To facilitate the exposition, we will first prove three simple propositions (\ref{expansion_of_resolvent},  \ref{universal_bound} and \ref{bound_for_G}). Then we will review the Simon-Wolff localiztion criterion (Theorem \ref{SWthm}) and give the proof of Theorem \ref{loc_criterion}.
\begin{prop}\label{expansion_of_resolvent}
Let $\omega\in\Omega$, $x,y\in X$,
$z\in\Comp\backslash\sigma_{\omega}$ and $r\geq 0$ be given. Then
\begin{equation}\label{expansion_formula}
G_{\omega,r}(x,y;z)=G_{\omega,0}(x,y;z)-\sum_{s=d(x,y)}^{r}p_sN_{s-1}g_{\omega,s-1}(x;z)g_{\omega,s}(y;z),
\end{equation}
and
\begin{equation}\label{prod_expansion_formula}
g_{\omega,r}(x;z)=\frac{1}{N_r}
G_{\omega,0}(x,x;z)\prod_{s=1}^{r}(1-p_s\gamma_{\omega}(Q_s(x);z)).
\end{equation}
\end{prop}
\begin{proof} The formula \eqref{expansion_formula} holds for $r=0$ since $p_0=0$.  For $s\geq 1$, the resolvent identity yields
$$(H_{\omega,s}-z)^{-1}\delta_y-(H_{\omega,s-1}-z)^{-1}\delta_y=-(H_{\omega,
s-1}-z)^{-1}p_s E_s(H_{\omega,s}-z)^{-1}\delta_y.$$ Observe that
$E_s(H_{\omega,s}-z)^{-1}\delta_y
=g_{\omega,s}(y;z)\mathbf{1}_{Q_s(y)}$. Taking
$\scalp{\delta_x}{\cdot}$ in the above equation yields
\begin{equation}\label{iteration}
G_{\omega,s}(x,y;z)-G_{\omega,s-1}(x,y;z)=-p_s
g_{\omega,s}(y;z)\scalp{\delta_x}{(H_{\omega,s-1}-z)^{-1}\mathbf{1}_{Q_s(y)}}.
\end{equation}
Note that by \eqref{symmetry_of_G}, $$\scalp{\delta_x}{(H_{\omega,s-1}-z)^{-1}\mathbf{1}_{Q_s(y)}}=\left\{%
\begin{array}{ll}
    N_{s-1} g_{\omega,s-1}(x;z), & \hbox{if } d(x,y)\leq s, \\
    0, & \hbox{if } d(x,y)>s. \\
\end{array}
\right.$$ The formula \eqref{expansion_formula} follows after
adding \eqref{iteration} for $s=1,2,\cdots,r$.

The proof of \eqref{prod_expansion_formula} is similar. The
resolvent identity yields
$$\scalp{\delta_x}{(H_{\omega,r}-z)^{-1}\mathbf{1}_{Q_r(x)}}=(1-p_r\gamma_{\omega}(Q_r(x);z))\scalp{\delta_x}{(H_{\omega,r-1}-z)^{-1}\mathbf{1}_{Q_{r-1}(x)}},$$
and \eqref{prod_expansion_formula} follows after iterating the
above formula.
\end{proof}


\begin{prop}\label{universal_bound} Let
$u_r>0$ be a sequence with  $\sum_{r=1}^{\infty}u_r^{-1}<\infty$. Let $\omega\in\Omega$ and $x\in X$ be fixed. Then for $m$-a.e. $e\in\Real$, there exists a finite constant $C_e$ such that

\begin{equation}\label{bound_for_gry}
\of{\sum_{d(x,y)\leq s}\abs{g_{\omega,s}(y;e)}^2}^{1/2}\leq
C_e u_s,
\end{equation}
for all $s\geq 0$.
\end{prop}

\begin{proof} Since $l^2(Q_r(x))$ is an $N_r$-dimensional invariant subspace for
$H_{\omega,r}$ and since
$\norm{\mathbf{1}_{Q_r(x)}}_2=\sqrt{N_r}$, we have by Lemma
\ref{Matrix_lemma}(see appendix) that for $M_r>0$,

$$m\of{\set{e\in\Real\backslash\sigma_{\omega}:\norm{(H_{\omega,r}-e)^{-1}\mathbf{1}_{Q_r(x)}}_2^2\geq M_r}}\leq \frac{4N_r}{\sqrt{M_r}}.$$
Let  $M_r:=(u_r N_r)^2$. Then
$\sum_{r=1}^{\infty}N_rM_r^{-1/2}<\infty$. By the Borel-Cantelli
lemma, for $m$-a.e. $e\in\Real\backslash\sigma_{\omega}$, there
exists a finite constant $C_e$ such that
\begin{equation}\label{bound_from_BC}
\norm{(H_{\omega,r}-e)^{-1}\mathbf{1}_{Q_r(x)}}_2<C_e \sqrt{M_r},
\end{equation}
for all $r\geq 0$.
Observe that
\begin{equation*}
\begin{split}
\of{\sum_{d(x,y)\leq s}\abs{g_{\omega,s}(y;e)}^2}^{1/2} &=
\of{\sum_{d(x,y)\leq
s}\abs{\frac{1}{N_s}\scalp{\delta_y}{(H_{\omega,s}-e)^{-1}\mathbf{1}_{Q_s(y)}}}^2}^{1/2}
\\
&=\frac{1}{N_s}\of{\sum_{d(x,y)\leq
s}\abs{\scalp{\delta_y}{(H_{\omega,s}-e)^{-1}\mathbf{1}_{Q_s(x)}}}^2}^{1/2}
\\
&= \frac{1}{N_s}\norm{(H_{\omega,s}-e)^{-1}\mathbf{1}_{Q_s(x)}}_2.
\end{split}
\end{equation*}
Inequality \eqref{bound_from_BC} yields \eqref{bound_for_gry}.
\end{proof}




The key step
in proving Theorem \ref{loc_criterion} is:

\begin{prop}\label{bound_for_G}
Assume that there exists a sequence
$u_r>0$ with  $\sum_{r=1}^{\infty}u_r^{-1}<\infty$ and
$\sum_{r=1}^{\infty}p_r u_r<\infty$.
Let $B\subset\Real$ be a Borel set and let $\omega\in\Omega$, $x\in X$ be fixed.
Suppose that
\begin{equation}\label{hyp_gamma}
\sum_{s=1}^{\infty} p_s\abs{\gamma_{\omega}(Q_s(x);e)}<\infty,
\end{equation}
for $m$-a.e. $e\in B$. Then, for $m$-a.e. $e\in B$,
\begin{equation}\label{eq_bound_for_G}
\sup_{r\geq 0}\sum_{y\in X}\abs{G_{\omega,r}(x,y;e)}^2<\infty.
\end{equation}
\end{prop}

\begin{proof} For all $e\in\Comp\backslash\sigma_{\omega}$, the representation formula \eqref{expansion_formula} and Cauchy-Schwarz inequality yield
\begin{equation*}
\begin{split}
\of{\sum_{y\in X}\abs{G_{\omega,r}(x,y;e)}^2}^{1/2}& \leq \abs{G_{\omega,0}(x,x;e)}\\
&+\sum_{s=1}^{r}p_sN_{s-1}\abs{g_{\omega,s-1}(x;e)}\of{\sum_{d(x,y)\leq
s}\abs{g_{\omega,s}(y;e)}^2}^{1/2}.\\
\end{split}
\end{equation*}
Hence, by Proposition \ref{universal_bound}, for $m$-a.e. $e\in \Real$,

\begin{equation}\label{cauchy_schwarz}
\of{\sum_{y\in X}\abs{G_{\omega,r}(x,y;e)}^2}^{1/2} \leq \abs{G_{\omega,0}(x,x;e)}+C_e\sum_{s=1}^{r}p_s u_s N_{s-1}\abs{g_{\omega,s-1}(x;e)}.
\end{equation}
Hypothesis \eqref{hyp_gamma} implies that for $m$-a.e. $e\in B$,
the product
$\prod_{s=1}^{\infty}(1-p_s\gamma_{\omega}(Q_s(x);e))$ converges. For such $e$, it follows from \eqref{prod_expansion_formula} that there is a finite constant $C^{'}_{e}$ such that
\begin{equation}\label{bound_for_grx}
N_{s}\abs{g_{\omega,s}(x;e)}\leq C^{'}_{e},
\end{equation}
for all $s\geq 0$.

Combination of \eqref{cauchy_schwarz} with \eqref{bound_for_grx}
yields the estimate
$$\of{\sum_{y\in X}\abs{G_{\omega,r}(x,y;e)}^2}^{1/2}\leq \abs{G_{\omega,0}(x,x;e)} + C_e C^{'}_{e}\sum_{s=1}^{r}p_s u_s,$$
and the result follows.
\end{proof}

Before proving Theorem \ref{loc_criterion},
let us recall the Simon-Wolff localization criterion. Define the function $G_{\omega,x}:\Real\To [0,+\infty]$ by
$$G_{\omega,x}(e):=\int_{\Real}\frac{d \mu_x^{\omega}(\lambda)}{(e-\la)^2}
=\lim_{\epsilon\downarrow
0}\norm{(\Delta+V_\omega-e-i\epsilon)^{-1}\delta_x}^2.$$ By the
Theorem of de la Vall\'{e} Poussin,
$$d\mu_{x,\mathrm{ac}}^{\omega}(e)=\pi^{-1}\of{\lim_{\epsilon\downarrow
0}\eps\norm{(\Delta+V_\omega-e-i\epsilon)^{-1}\delta_x}^2}de.$$
Hence, if for a fixed $\omega\in\Omega$ we have that
$G_{\omega,x}(e)<\infty$ for $m$-a.e. $e\in\Real$, then
$\mu_{x,\mathrm{ac}}^{\omega}=0$.

The Simon-Wolff localization criterion is summarized in:
\begin{thm}\label{SWthm}  Assume
that $\mathbb{P}$ has a conditional density along the $x$'th
coordinate. Let $B\subset\Real$ be a Borel set such that
$G_{\omega,x}(e)<\infty$ for $\mathbb{P}\otimes m$-a.e.
$(\omega,e)\in\Omega\times B$. Then
$\mu_{x,\mathrm{cont}}^{\omega}(B)=0$ for $\mathbb{P}$-a.e.
$\omega\in\Omega$.
\end{thm}
Theorem \ref{SWthm} is a well known consequence of the rank-1
Simon-Wolff theorem \cite{SW} and the conditional Fubini's theorem.

Now we can prove the main localization criterion.
\newline\emph{Proof of Theorem \ref{loc_criterion}.} Fix $\omega\in\Omega$
and fix $e\in\Real\backslash\sigma_{\omega}$ for which \eqref{weakest_hypothesis} holds. Proposition \ref{bound_for_G} yields that then we also have the bound \eqref{eq_bound_for_G}. By monotone convergence
$$ \int_{\Real}\frac{d \mu_x^{\omega}(\lambda)}{(e-\la)^2}=
\lim_{\eps\downarrow 0} \int_{\Real}\frac{d
\mu_x^{\omega}(\lambda)}{(e-\la)^2+\eps^2}=
\sup_{\eps>0}\int_{\Real}\frac{d
\mu_x^{\omega}(\lambda)}{(e-\la)^2+\eps^2}.$$ Since for any
$z\in\Comp\backslash\Real$,
$$\lim_{r\To\infty}\norm{(H_{\omega,r}-z)^{-1}-
(H_{\omega}-z)^{-1}}=0,$$ we have that the weak-* limit
$\lim_{r\To\infty}\mu_{x,r}^{\omega}$ equals $\mu_x^{\omega}$, where
$\mu_{x,r}^{\omega}$ is the spectral measure for $H_{\omega,r}$
and $\delta_x$. Therefore
\begin{equation*}
\begin{split}\int_{\Real}\frac{d \mu_x^{\omega}(\lambda)}{(e-\la)^2}=
\sup_{\eps>0}\lim_{r\To\infty}\int_{\Real}\frac{d
\mu_x^{\omega,r}(\lambda)}{(e-\la)^2+\eps^2} &\leq \sup_{\eps>0,r\geq1}\int_{\Real}\frac{d
\mu_x^{\omega,r}(\lambda)}{(e-\la)^2+\eps^2}\\
&=\sup_{r\geq1}\int_{\Real}\frac{d
\mu_x^{\omega,r}(\lambda)}{(e-\la)^2}\\
&=\sup_{r\geq 1}\norm{(H_{\omega,r}-e)^{-1}\delta_x}^2\\
&=\sup_{r\geq 1}\sum_{y\in X}\abs{G_{\omega,r}(x,y)}^2<\infty.
\end{split}
\end{equation*}
In the final equality we used the fact that $\set{\delta_y:y\in
X}$ is an orthonormal basis for $l^2(X)$. Since
$m\of{\sigma_\omega}=0$ and since the bound \eqref{eq_bound_for_G}
holds for $m$-a.e. $e\in\Real\backslash\sigma_\omega$, we have
that for every fixed $\omega\in\Omega$, $G_{\omega,x}(e)<\infty$ for $m$-a.e. $e\in\Real$. This proves part (1). Part (2) follows from the fact that $G_{\omega,x}(e)<\infty$ for $\mathbb{P}\otimes m$-a.e.
$(\omega,e)\in\Omega\times \Real$ and the Simon-Wolff criterion.  Part (3) is an immediate consequence of part (2) and assumption \eqref{weakest_hypothesis}. $\Box$


\section{Proofs of localization theorems} In this section we show how to apply the localization criterion (Theorem \ref{loc_criterion}) to prove theorems
\ref{maintheorem} and \ref{Stas}.

\emph{Proof of Theorem \ref{maintheorem}.} Since $\gamma_{\omega}(Q_s(x);e)=\sum_{d(x,y)\leq s}g_{\omega,s}(y;e)$, Cauchy-Schwarz inequality and Proposition \ref{universal_bound} yield that for $m$-a.e. $e\in\Real$,
\begin{equation}\label{easyCSbound}
\abs{\gamma_{\omega}(Q_s(x);e)}\leq C_eu_s \sqrt{N_s}.
\end{equation}
Hence $\sum_{s\geq 0}p_s\abs{\gamma_{\omega}(Q_s(x);e)}<\infty$
and the result follows from parts (1) and (2) of Theorem \ref{loc_criterion}.
$\Box$.


The proof of Molchanov's theorem \ref{Stas} requires more work.  Let us introduce some notation. It is convenient to form the complex number $v=a+ih\in\Comp_+$ and to denote  by $k_v$ the Cauchy distribution $k_{a,h}$. If $\nu$ is a Borel probability measure on the Riemann sphere $\rs:=\Comp\cup\set{\infty}$, and $\tau:\rs\To\rs$ is a Borel measurable map, then $\tau\mu$ will denote the induced Borel probability measure on $\rs$:
$$(\tau\nu)(B):=\nu(\tau^{-1}(B)),$$
for Borel sets $B\subset\rs$. If $n\geq 2$ is a integer, and $\nu$ is a Borel probability measure on $\rs$, we let
$$A_n\nu:=\tau(\underbrace{\nu\star\dots\star\nu}_{n \textrm{ times}}),$$
where $\tau(z)=z/n$, and $\star$ is the usual convolution of
measures. Hence, if $Y_1,\dots Y_n$ are i.i.d. random variables on
$\rs$ with distribution $\nu$, then $A_n\nu$ is the distribution
of $(Y_1+\dots+Y_n)/n$. The following proposition, whose proof is
left an an easy exercise for the reader, summarizes some of the
well known classical facts about Cauchy distributions.
\begin{prop}\label{Cauchy} Cauchy distributions have the following properties:
\begin{enumerate}
\item If $v_1,v_1\in\Comp_+$, then $$k_{v_1}\star k_{v_2}=k_{v_1+v_2}.$$
\item If $\tau(z)=-\overline{z}$ or if $\tau(z)=(az+b)/(cz+d)$, where $a,b,c,d\in\Real$, $ad-bc>0$, then
$$\tau k_v=k_{\tau v},$$
for all $v\in\Comp_+$.
\item For all $v\in\Comp_+$ and integers $n\geq 2$, $$A_nk_v=k_v.$$

\end{enumerate}
\end{prop}

We will use fractional linear transformations of the special form
$$\tau_p(z)=\frac{z}{1+pz},$$
where $p\geq 0$.  Note that $\tau_{p+p'}=\tau_p\circ \tau_{p'}$. The transformations $\tau_p$ are important in the hierarchical Anderson model because of the following recursive relation, proven by Molchanov in \cite{M2}.
\begin{prop}\label{recursion} Let $\omega\in\Omega$ and $z\in\Comp\backslash \sigma_\omega$ be given. Then for all $x\in X$,
\begin{equation}\label{mobius1}
\gamma_{\omega}(Q_0(x);z)=\frac{1}{\omega(x)-z}.
\end{equation}
 Moreover, for all $r\geq 1$ and $Q_r\in\cP_r$, we have
\begin{equation}\label{mobius_recursion}
\gamma_{\omega}(Q_r;z)=\tau_{p_r}\of{\frac{1}{n_r}\sum_{Q_{r-1}^j\subset{Q_r}}\gamma_{\omega}(Q_{r-1}^j;z)},
\end{equation}
where the sum contains $n_r$ terms, one for each rank $r-1$ subcluster $Q_{r-1}^{j}$ of $Q_r$.
\end{prop}
\begin{proof} The formula \eqref{mobius1} is clear since
$G_{\omega,0}(x,y;z):=(\omega(x)-z)^{1}\scalp{\delta_x}{\delta_y}$.
 For $r\geq 1$, the resolvent identity yields
$$(H_{\omega,r}-z)^{-1}\mathbf{1}_{Q_r}-(H_{\omega,r-1}-z)^{-1}\mathbf{1}_{Q_r}=-p_r\gamma_{\omega}(Q_r;z)(H_{\omega,
r-1}-z)^{-1}\mathbf{1}_{Q_r}.$$ Taking
$\scalp{\mathbf{1}_{Q_r}}{\cdot}$ in the above equation yields
$$N_r\gamma_{\omega}(Q_r;z)-\sum_{Q_{r-1}^j\subset{Q_r}}N_{r-1}\gamma_{\omega}(Q_{r-1}^j;z)=-p_r\gamma_{\omega}(Q_r;z)\sum_{Q_{r-1}^j\subset{Q_r}}N_{r-1}\gamma_{\omega}(Q_{r-1}^j;z).$$ The formula \eqref{mobius_recursion} follows after dividing by $N_r$ in the above equation and then solving for $\gamma_{\omega}(Q_r;z)$.
\end{proof}

\emph{Proof of Theorem \ref{Stas}.}

Since for every fixed $\omega$, $m(\sigma_\omega)=0$, we have that for $\mathbb{P}\otimes m$-a.e.
$(\omega,e)\in\Omega\times \Real$, $e\notin\sigma_w$. Hence, there is a Borel mesurable set $E\subset\Real$ with the properties
\begin{enumerate}
\item $m(\Real\backslash E)=0$, and \item for every fixed $e\in
E$, there is a set $\Omega_e\in\mathcal{F}$ with
$\mathbb{P}(\Omega_e)=1$ and such that $e\notin\sigma_w$ for all
$\omega\in\Omega_e$.
\end{enumerate}
Let us fix $e\in E$. Then for all $\omega\in\Omega_e$, and hence
for $\mathbb{P}$-a.e. $\omega\in\Omega$, all the representation
formulas in propositions \ref{expansion_of_resolvent} and
\ref{recursion} are valid. It follows from \eqref{mobius1} and
from Proposition \ref{Cauchy} that
$\set{\gamma_{\omega}(Q_0(x);e): x\in X}$ are i.i.d. Cauchy random
variables with distribution $k_{v_0}$, where $v_0(e)=-(e-a+ih)^{-1}$. Moreover,
\eqref{mobius_recursion} and Proposition \ref{Cauchy} yield that
for $r\geq 1$, $\set{\gamma_{\omega}(Q_r;e): Q_r\in P_r}$ are
i.i.d. Cauchy random variables with distribution $k_{v_r}$, where
$v_r(e)=\tau_{\lambda_r}v_0(e)$. Since
$\lambda_r\To\lambda_\infty=1$ as $r\To\infty$, the closure of the
orbit
$$V(e):=\bigcup_{r\geq 0}\set{v_r(e)},$$
is equal to $V(e)\cup \set{\tau_1v_0(e)}$, a compact set in $\Comp_+$. Since
$\sup_{r\geq 0}\abs{v_r(e)}<\infty$, there is a constant $K(e)<\infty$, such that
$$\mathbb{P}\of{\abs{\gamma_{\omega}(Q_r;e)}>u}\leq \frac{K(e)}{u},$$
for all real $u>0$, integer $r\geq 0$ and $Q_r\in P_r$. Let us now fix $x\in X$ and
take $u_r=r^{1+\eps}$. By the Borel-Cantelli lemma, for
$\mathbb{P}$-a.e. $\omega\in\Omega_e$, there exists a finite
constant $L_\omega(e)$ such that
\begin{equation}\label{improvement}
\abs{\gamma_{\omega}(Q_r(x);e)}\leq L_\omega(e) u_r,
\end{equation}
for all $r\geq 0$. Part (3) of Theorem \ref{loc_criterion} yields that
$\mu_{x,\mathrm{cont}}^{\omega}=0$
for ${\mathbb P}$-a.e. $\omega\in\Omega$. Since $x\in X$ is arbitrary, the result follows.
$\Box$.

Note that, in the proof given above, \eqref{improvement}  is a significant improvement of \eqref{easyCSbound} since the factor $\sqrt{N_r}$ is no longer present. 

\section{A generalization of Molchanov's theorem}

In this section we generalize Theorem \ref{Stas} to convex combinations of Cauchy distributions.
Let $\cM$ denote the set of Borel probability measures on $\Comp_+$. Given $\nu\in\cM$ and a Borel set $B\subset\Real$,
we set
$$k_\nu(B):=\int k_v(B)d\nu(v).$$
Then $k_\nu$ is a Borel probability measure on $\Real$ and for every bounded Borel function $f:\Real\To\Comp$,
$$\int_\Real f d k_\nu=\int\of {\int_\Real f d k_v}d\nu(v).$$
We call $k_\nu$ a \emph{mixed Cauchy distribution}. Note that the usual Cauchy distribution $k_{a+ih}$ is a special case of the above general defintion: $k_{a+ih}=k_\nu$ where $\nu=\delta(a+ih)$.
Let $S$ be the set of mixed Cauchy distributions $k_\nu$, such that the distance from $\Real$ to the support of $\nu$ is strictly positive.


\begin{thm}\label{generalStas}Assume that there exists a sequence
$u_r>0$ with  $\sum_{r=1}^{\infty}u_r^{-1}<\infty$ and
\begin{equation*}
\sum_{r=1}^{\infty}p_r u_r<\infty.
\end{equation*}
If the random variables $\set{\omega(x):x\in
X}$ are i.i.d. with a mixed Cauchy distrubution $k_\nu\in S$, then $\sigma_{\rm
cont}(H_\omega)=\emptyset$ for ${\mathbb P}$-a.e. $\omega$.
\end{thm}
\noindent{\bf Remark 1.} Theorem \ref{generalStas} is valid at arbitrary disorder. Indeed, if $\omega(x)$ has a distribution $k_\nu\in S$, and $c>0$, then $c\omega$ has again a distribution of the form $k_{\nu'}\in S$. Precisely, $\nu'=\tau\nu$, where $\tau(z)=cz$.

\noindent{\bf Remark 2.} 
If $f$ is any compactly supported continuous probability density on $\Real$, and $\eps>0$, then there is a mixed Cauchy distribution $k_\nu\in S$ s.t. $$\sup_{e\in\Real}\abs{f(e)-dk_{\nu}/de}<\eps.$$ To see this, recall that the harmonic function $F(v)=\int_\Real f(e) dk_v(e)$, $v\in\Comp_+$, converges uniformly to $f$ as $\Im v\downarrow 0$, and note that the integral can be approximated by finite Riemann sums.

The proof of Theorem \ref{generalStas} parallels that of Molchanov's theorem \ref{Stas}. First, we need a generalization of Proposition \ref{Cauchy} to the case of mixed Cauchy distributions.
\begin{prop}\label{mixedCauchy} Mixed Cauchy distributions have the following properties:
\begin{enumerate}
\item If $\nu_1,\nu_1\in\cM$, then $$k_{\nu_1}\star k_{\nu_2}=k_{\nu_1\star\nu_2}.$$
\item If $\tau(z)=-\overline{z}$ or if $\tau(z)=(az+b)/(cz+d)$, where $a,b,c,d\in\Real$, $ad-bc>0$, then
$$\tau k_\nu=k_{\tau \nu},$$
for all $\nu\in\cM $.
\item For all $\nu \in\cM$ and integers $n\geq 2$, $$A_nk_\nu=k_{A_n\nu}.$$
\end{enumerate}
\end{prop}
The proof is a straight forward consequence of the definitions together with Propostion \ref{Cauchy} and Fubini's theorem.

\emph{Proof of Theorem \ref{Stas}.}
 Let $E$ be the set of full Lebesgue measure as in the beginning of the proof of Theorem \ref{Stas}, and let $e\in E$ be fixed. 
It follows from \eqref{mobius1} and
from Proposition \ref{mixedCauchy} that
$\set{\gamma_{\omega}(Q_0(x);e): x\in X}$ are i.i.d. mixed Cauchy random
variables with distribution $k_{\nu_0}$, where 
$\nu_0=\tau_0\nu$, $\tau_0(z)=-(e-\overline{z})^{-1}$. Since the distance from the support of $\nu$ to $\Real$ is strictly positive, there is a closed disk $D_0\subset \Comp_+$ such that $\rm{supp}(\nu_0)\subset D_0$.
The relation
\eqref{mobius_recursion} and Proposition \ref{mixedCauchy} yield that
for $r\geq 1$, $\set{\gamma_{\omega}(Q_r;e): Q_r\in P_r}$ are
i.i.d. mixed Cauchy random variables with distribution $k_{\nu_r}$, where
$$\nu_r=\tau_{p_r} A_{n_r}\nu_{r-1}.$$

For $r\geq 0$, let $D_r:=\tau_{\la_r}D_0$. Since $\tau_{\la_r}$ is a fractional linear transformation taking $\Comp_+$ onto itself, $D_r$ is again a closed disk in $\Comp_+$.
We claim that for all $r\geq 0$, $$(*)\qquad \rm{supp}(\nu_r)\subset D_r.$$
The proof is by induction. By construction of $D_0$, $(*)$ holds for $r=0$. Let $r\geq 1$ and suppose $(*)$ holds for $r-1$. Since $\rm{supp}(\nu_{r-1})\subset D_{r-1}$, and $D_{r-1}$ is convex, we have that
$\rm{supp}(A_{n_r}\nu_{r-1})\subset D_{r-1}$. 
Hence $support(\tau_{p_r}A_{n_r}\nu_{r-1})\subset \tau_{p_r}D_{r-1}=D_r$, and $(*)$ holds for $r$, proving the claim.

Let $D$ be the closure of $\cup_{r\geq 0}D_r$. $D$ is a compact subset of $\Comp_+$ and for all $r\geq 0$, $\rm{supp}(\nu_r)\subset D$. It follows that there is a constant 
$K(e)<\infty$, such that
$$\mathbb{P}\of{\abs{\gamma_{\omega}(Q_r;e)}>u}\leq \frac{K(e)}{u},$$
for all real $u>0$, integer $r\geq 0$ and $Q_r\in P_r$. The rest of the proof is the same as in Theorem \ref{Stas}, with the more general sequence $u_r$ instead of $r^{1+\eps}$.
$\Box$


\section{Appendix}
\emph{Proof of Lemma \ref{nonrandomspectrum}.}
Let $e\in S$. Given a cluster $Q_r$ or rank $r$ and given $\eps>0$, consider the event
$$B(Q_r,\eps):=\set{\omega: \max_{x\in Q_r}\abs{\omega(x)-e}<\eps}.$$
Since $\nu((e-\eps,e+\eps))>0$ and the random variables $\set{\omega(x):x\in Q_r}$ are independent, we have that
$$\mathbb{P}\of{B(Q_r,\eps)}=\of{\nu((e-\eps,e+\eps))}^{N_r}>0.$$
According to the hierarchical structure, $X$ is a countable union of disjoint clusters of rank $r$: $X=\bigcup_{j=1}^{\infty}Q_{r,j}$. Let $B_{r,\eps}:=\limsup_j B(Q_{r,j},\eps)$.
Since the events $\set{B(Q_{r,j},\eps),j=1,\dots,\infty}$ are independent and $\sum_{j=1}^{\infty}\mathbb{P}\of{B(Q_{r,j},\eps)}=\infty$, the second Borel-Cantelli Lemma yields that $\mathbb{P}(B_{r,\eps})=1$. Letting $B:=\cap_{r\geq 0}B_{r,1/(r+1)}$, we also have
$\mathbb{P}(B)=1$. Hence, for $\omega\in B$, there is a sequence $(Q_{r,j_r}:r\geq 0)$ of clusters of increasing rank with the property that
$$(*)\qquad \max_{x\in Q_{r,j_r}}\abs{\omega(x)-e}<\frac{1}{r+1},$$
for all $r\geq 0$. Let $\la\in\sigma(\Delta)$. Then $\la=\la_s$ for some $s\in\set{1,\dots,\infty}$. For $r\geq 0$, we let $q_r=\min(r,s)$. It is easy to see that for each $r\geq 0$, there exists a unit eigenvector $\psi_r$ of $\Delta$ corresponding to the eigenvalue $\la_{q_r}$,
and such that $\psi_r(x)=0$ for $x\notin Q_{r,j_r}$. Indeed, take a unit vector $\psi_r$ constant on every rank $q_r$ subcluster of $Q_{r,j_r}$, zero outside $Q_{r,j_r}$ and such that $E_{q_r+1}\psi_r=0$. Then
$$\lim_{r\To\infty}\norm{\Delta\psi_r-\la_s\psi_r}=0,$$
and, by (*),
$$\lim_{r\To\infty}\norm{H_\omega \psi_r-(\la_s+e)\psi_r}=0,$$
for $\omega\in B$. Hence, $\psi_r$ is a Weil sequence for the operator $H_\omega$ and
$\la_s+e\in \sigma(H_\omega)$. We have shown that for $\mathbb P$-a.e. $\omega\in\Omega$,
$\sigma(\Delta)+S\subset\sigma(H_\omega)$. The inclusion $\sigma(H_\omega)\subset (\sigma(\Delta)+ch(S))\cap ([0,1]+S)$ follows from general facts about the spectrum of a sum of two self-adjoint operators.
$\Box$.
\begin{lem}\label{Matrix_lemma} Let $A$ be a hermitian $N\times N$ matrix and
$v\in\Comp^N$. Then for all $M>0$
$$m\of{\set{e: \norm{(A-e)^{-1}v}_2^2\geq M}}\leq 4\sqrt{\frac{N}{M}}\norm{v}_2$$
\end{lem}
\begin{proof} (Following \cite{M2}) Let $\la_1,\dots,\la_N$ be the eigenvalues of $A$
and $\psi_1,\dots,\psi_N$ the corresponding orthonormal basis of eigenvectors.
In this basis, we have $v=\sum_{i=1}^{N}v_i\psi_i$,
$\norm{v}_2^2=\sum_{i=1}^{N}\abs{v_i}^2$ and
$(A-e)^{-1}v=\sum_{i=1}^{N}(\la_i-e)^{-1}v_i\psi_i$. For each
$i=1,\dots,N$, consider the open interval
$\Delta_i=(\la_i-v_i/\sqrt{M}, \la_i+v_i/\sqrt{M})$. The total
length of these $N$ intervals is
$$m\of{\bigcup_{i=1}^{N}\Delta_i}\leq \sum_{i=1}^{N}m\of{\Delta_i}=\frac{2}{\sqrt{M}}\sum_{i=1}^{N}\abs{v_i}\leq \frac{2}{\sqrt{M}}\sqrt{N}\norm{v}_2.$$

Chebyshev inequality yields
\begin{equation*}
\begin{split}
m\of{\set{e\in\Real\backslash \bigcup_{i=1}^{N}\Delta_i: \norm{(A-e)^{-1}v}_2^2\geq M}}&\leq \frac{1}{M}\int_{\Real\backslash \bigcup_{i=1}^{N}\Delta_i} \norm{(A-e)^{-1}v}_2^2 de\\
&=\frac{1}{M}\int_{\Real\backslash \bigcup_{i=1}^{N}\Delta_i} \sum_{i=1}^{N}\abs{\frac{v_i}{\la_i-e}}^2 de\\
&\leq \frac{1}{M} \sum_{i=1}^{N}\int_{\Real\backslash\Delta_i}\abs{\frac{v_i}{\la_i-e}}^2 de\\
&=\frac{1}{M}\sum_{i=1}^{N}2\int_{\abs{v_i}/\sqrt{M}}^{\infty}\frac{\abs{v_i}^2}{t^2}dt\\
&=\frac{2}{M}\sqrt{M} \sum_{i=1}^{N}\abs{v_i}\\
&\leq \frac{2}{\sqrt{M}}\sqrt{N}\norm{v}_2,
\end{split}
\end{equation*}
and the result follows.
\end{proof}

\textbf{Acknowledgements.} We are grateful to Vojkan
Jaksic for suggesting this research project and for giving us constructive feedback. We would also
like to thank Stanislav Molchanov for very helpful
discussions and enthusiastic encouragements.
\begin{thebibliography}{aaaaa}

\bibitem[Bo]{Bo}  Bovier, A.: The density of states in the Anderson model at weak disorder: a renormalization group analysis of the hierarchical model.
 J. Statist. Phys. {\bf 59} (1990), no. 3-4, 745--779.

\bibitem[BS]{BS} Bleher, P. M., Sinai, Ya. G.: Investigation of the
Critical Point in Models of the Type of Dyson's Hierarchical
Models. Commun. Math. Phys. {\bf 33}, (1973).
\bibitem[CL]{CL} Carmona R., Lacroix J.: Spectral Theory of Random Schrodinger Operators. Birkhauser, 1990.

\bibitem[D]{D} Dyson, F.J.: Existence of a phase-transition in a one dimensional Ising Ferromagnet.
Comm. Math. Phys. {\bf 12}, {\bf 91} ( 1969).

\bibitem[K1]{K1} Kritchevski, E.: Spectral Localization in the Hierarchical Anderson Model. To appear in the Proceedings of AMS.

\bibitem[K2]{K2} Kritchevski, E.: PhD. thesis, McGill University. In preparation.

\bibitem[M1]{M1} Molchanov, S.: Lectures on random media. Lectures on probability
theory (Saint-Flour, 1992), 242--411, Lecture Notes in Math.,
{\bf 1581}, Springer, Berlin, 1994.

\bibitem[M2]{M2} Molchanov S.:
Hierarchical random matrices and operators. Application to
Anderson model. Multidimensional statistical analysis and theory
of random matrices (Bowling Green, OH, 1996), 179--194, VSP,
Utrecht, 1996.

\bibitem[SW]{SW} Simon B., Wolff T.: Singular continuous spectrum under rank one perturbations
and localization for random Hamiltonians. Communications in Pure
and Applied Mathematics {\bf 49} (1986), 75.

\end{thebibliography}

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