# Localization for quasi-one-dimensional Dirac operators

Hakim Boumaza  LAGA, Université Sorbonne Paris Nord, 99 avenue J.B. Clément,
F-93430 Villetaneuse<EMAIL_ADDRESS>and  Sylvain Zalczer  KIT,
Englerstrasse 2, 76131, Karlsruhe, Germany<EMAIL_ADDRESS>
###### Abstract.

We consider a random family of Dirac operators on $N$ parallel real lines,
modelling for example a graphene nanoribbon. We establish a localization
criterion involving properties on the group generated by transfer matrices. In
particular, we consider not only the case where this group is the symplectic
group but also a strict subgroup of it. We establish under quite general
hypotheses that the sum of the Lyapunov exponents and the integrated density
of states are Hölder continuous. Moreover, for a set of concrete cases where
the potentials are on Pauli matrices, we compute the transfer matrices and
prove either localization or delocalization, depending on the potential and on
the parity of $N$.

The research of H.B was partially funded by the ANR Project RAW,
ANR-20-CE40-0012

The research of S.Z was funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – Project-ID 258734477 – SFB 1173.

##  1\. Introduction

Dirac introduced the operator bearing his name in order to describe the motion
of a relativistic electron. While nonrelativistic quantum matter is described
by the Schrödinger operator, which is a second-order differential operator
acting on scalar-valued functions, the Dirac operator has order 1 and acts on
vector-valued functions. A standard reference on the Dirac operator from a
mathematics point of view is the book of Thaller [32].

In the last twenty years, the Dirac operator has been used as an effective
Hamiltonian to describe graphene samples, although no relativistic effects are
considered. Graphene is a 2-dimensional material made of carbon atoms arranged
according to a honeycomb structure. The setting which we are mostly interested
in this paper is the one of graphene nanoribbons, which consist in an infinite
band of graphene, bounded transversely. In [4], a model of Dirac operator on a
waveguide is studied.

The question of localization on graphene nanoribbons has already been
considered by physicists, although they generally consider a discrete tight-
binding model instead of the Dirac operator. In [30], some localization is
obtained for a quasi-periodic perturbation. In [25] and [15], different
Anderson-like models are considered. In [33], several interesting phenomena
are highlighted. First, different types of disorder can lead to different
localization regimes. Second, in some cases there is localization for all
nonzero energy but delocalization for energy zero, for symmetry reasons.
Third, the Lyapunov exponents can be grouped into pairs, here again because of
some symmetry of the system. We will be able to recover these properties in
our results.

From a mathematical point of view, a few papers have already considered random
Dirac operators. In [3], Barbaroux, Cornean and the second author proved
localization at band edges for a gapped Dirac Hamiltonian in any dimension.
The latter continued with [34], where he proved the Lipschitz regularity of
the integrated density of states for the same model, and [35], in which
Anderson localization is proven at any energy for a one-dimensional model.
Discretized versions of the Dirac operator have been studied by Prado and de
Oliveira, either for a Bernoulli random potential [26, 28] or for a potential
with an absolutely continuous distribution [27]. In [29], Sadel and Schulz-
Baldes consider a quasi-one-dimensional random Dirac operator with some
symmetry property. They look at the Lyapunov exponents and establish
delocalization in some cases, but they do not prove localization.

In the present paper, we consider a semi-discretized model: the nanoribbon is
represented by a finite number of parallel infinite lines. This makes it
possible to use techniques which are specific to one-dimensional systems,
based on transfer matrices and Lyapunov exponents. These methods have already
been used to prove localization for Schrödinger operators with random
potentials even with singular distributions, in [23] in the discrete case and
in [8] in the continuous case. A survey article on these methods has recently
been published [10]. The proof of the localization itself relies on the method
called _multiscale analysis_ , in its version developped by Germinet and Klein
in [16].

Our paper is organized as follows. In Section 1, we present the model as well
as the main tools used in the analysis, and we state our results. In Section
2, we prove a general criterion ensuring localization for a quasi-one-
dimensional random Dirac operators. In Section 3, we prove that we can apply
this criterion to some explicit examples. Finally, in Appendix A, we study the
properties of a group appearing in one of the cases considered in Section 3
and in Appendix B, we prove an initial length-scale estimate for a quasi-one-
dimensional random Schrödinger operator, correcting a mistake of [8].

###  1.1. Quasi-one-dimensional operators of Dirac type

Given an integer $N\geq 1$, the free Dirac operator on $N$ parallel straight
lines is

(1.1) $D_{0}^{(N)}:=J\frac{\mathrm{d}}{\mathrm{d}x},\text{ with
}J:=\begin{pmatrix}0&-I_{N}\\\ I_{N}&0\end{pmatrix}\text{ and
}\mathrm{Dom}(D_{0}^{(N)})=H^{1}(\mathbb{R})\otimes\mathbb{C}^{2N}.$

It is easy to see that this operator is self-adjoint.

We add to this free operator a random potential. Let
$(\Omega,\mathcal{A},\mathbb{P})$ a complete probability space and $\ell>0$ be
a disorder parameter: the smaller $\ell$ is, the ”more disordered” the system
is. The random potential $(V_{\omega}^{(n)})_{n\in\mathbb{Z}}$ is a sequence
of independent and identically distributed (i.i.d. for short) random variables
such that, for every $n\in\mathbb{Z}$, the function $x\mapsto
V_{\omega}^{(n)}(x)$ takes values in the Hermitian matrices, is supported in
$[0,\ell]$ and is uniformly bounded in $x$, $n$ and $\omega$.

We consider the random family $\\{D_{\omega}\\}_{\omega\in\Omega}$ of quasi-
one-dimensional Dirac operators defined for every realization
$\omega\in\Omega$ by:

(1.2)
$D_{\omega}:=D_{0}^{(N)}+\sum_{n\in\mathbb{Z}}V_{\omega}^{(n)}(\cdot-\ell n).$

Under such conditions, for each $\omega\in\Omega$, the operator $D_{\omega}$
is self-adjoint on the Sobolev space $H^{1}(\mathbb{R})\otimes\mathbb{C}^{2N}$
and thus, for every $\omega\in\Omega$, the spectrum of $D_{\omega}$, denoted
by $\sigma\left(D_{\omega}\right)$, is included in $\mathbb{R}$.

The random potential is such that $\\{D_{\omega}\\}_{\omega\in\Omega}$ is a
$\ell\mathbb{Z}$-ergodic random family of operators. As a consequence, there
exists $\Sigma\subset\mathbb{R}$ such that, for ${\mathbb{P}}$-almost every
$\omega\in\Omega$, $\Sigma=\sigma(D_{\omega})$. There also exist
$\Sigma_{\mathrm{pp}}$, $\Sigma_{\mathrm{ac}}$ and $\Sigma_{\mathrm{sc}}$,
subsets of $\mathbb{R}$, such that, for $\mathbb{P}$-almost every
$\omega\in\Omega$, $\Sigma_{\mathrm{pp}}=\sigma_{\mathrm{pp}}(D_{\omega})$,
$\Sigma_{\mathrm{ac}}=\sigma_{\mathrm{ac}}(D_{\omega})$ and
$\Sigma_{\mathrm{sc}}=\sigma_{\mathrm{sc}}(D_{\omega})$, respectively the pure
point, absolutely continuous and singular continuous spectrum of $D_{\omega}$
(see for example [21, Theorem 4.3]).

We aim at proving that under conditions involving notions coming from the
theory of dynamical systems, the phenomenon of Anderson localization occurs
for $\\{D_{\omega}\\}_{\omega\in\Omega}$ at all energies, except maybe those
in a discrete set. There are several mathematical definitions to translate the
Anderson localization phenomenon for a general family of random operators. Let
$\\{H_{\omega}\\}_{\omega\in\Omega}$ be a family of self-adjoint random
operator on a Hilbert space, which will be the space
$H^{1}(\mathbb{R})\otimes\mathbb{C}^{2N}$ in our model (1.2).

######  Definition 1.1 (AL).

Let $I$ be an interval of $\mathbb{R}$. We say that the family
$\\{H_{\omega}\\}_{\omega\in\Omega}$ of almost-sure spectrum $\Sigma$ has the
property of _Anderson localization_ in $I$ when:

  1. (1)

$\Sigma\cap I=\Sigma_{\mathrm{pp}}\cap I\neq\emptyset$ and
$\Sigma_{\mathrm{ac}}\cap I=\Sigma_{\mathrm{sc}}\cap I=\emptyset$,

  2. (2)

the eigenfunctions associated with the eigenvalues in $\Sigma\cap I$ decay
exponentially to $0$ at infinity.

Note that if $\\{H_{\omega}\\}_{\omega\in\Omega}$ exhibits Anderson
localization in $I$, ${\mathbb{P}}$-almost surely the point spectrum of
$H_{\omega}$ is dense in $\Sigma\cap I$. The definition of Anderson
localization is a stationary definition, involving only the Hamiltonian
$H_{\omega}$ and not the associated one-parameter group. The following
definition takes into account the dynamics in time of the wave packets.

######  Definition 1.2 (DL).

Let $I$ be an interval of $\mathbb{R}$. We say that the family
$\\{H_{\omega}\\}_{\omega\in\Omega}$ is _dynamically localized_ in $I$, when

  1. (1)

$\Sigma\cap I\neq\emptyset$,

  2. (2)

for every compact interval $I_{0}\subset I$, every $\psi\in
L^{2}(\mathbb{R})\otimes\mathbb{C}^{2N}$ with compact support and every $p\geq
0$,

(1.3)
${\mathbb{E}}\left(\sup_{t\in\mathbb{R}}\left\|(1+|\cdot|^{2})^{\frac{p}{2}}\mathrm{e}^{-\mathrm{i}tH_{\omega}}\mathbf{1}_{I_{0}}(H_{\omega})\psi\right\|_{L^{2}(\mathbb{R})}^{2}\right)<+\infty$

where $\mathbf{1}_{I_{0}}(H_{\omega})$ denotes the spectral projector on
$I_{0}$ associated with $H_{\omega}$.

The definition 1.2 is dynamical in nature and follows the evolution of wave
packets over time. It tells us that the solutions of the Schrödinger equation
are localized in space in the vicinity of their initial position and this,
uniformly over time. This reflects the absence of quantum transport. More
precisely, let $|x|$ be the position operator, _i.e._ , the multiplication
operator by $x\mapsto(1+||x||_{2}^{2})^{\frac{1}{2}}$ on
$L^{2}(\mathbb{R})\otimes\mathbb{C}^{2N}$. For any state $\psi\in
L^{2}(\mathbb{R})\otimes\mathbb{C}^{2N}$, if we denote $\psi_{I}(t)$ as the
evolution of the spectral projection of $\psi$ at time $t$, then the moments
of the position operator are bounded in $t$:

$\exists C_{\psi,I}>0,\ \forall p\geq 0,\ \forall t\in\mathbb{R},\
\left\langle\psi_{I}(t),|x|^{p}\psi_{I}(t)\right\rangle\leq C_{\psi,I}.$

On the contrary, we will say that there is quantum transport in an interval
$I^{\prime}$ when

$\exists\alpha>0,\ \exists p\geq 0,\ \forall t\in\mathbb{R},\
\langle\psi_{I^{\prime}}(t),|x|^{p}\psi_{I^{\prime}}(t)\rangle\geq|t|^{\alpha}.$

Let us point out that the use of multiscale analysis as done in this paper
will imply both Anderson localization and dynamical localization [14]. Note
that dynamical localization implies absence of continuous spectrum but does
not imply in general, the exponential decay of the eigenfunctions as in (AL).
It is also possible to define even stronger notions of localization, all of
them being implied by the use of multiscale analysis. For an exhaustive
presentation of these notions we refer to the third part of [22].

Before stating our main result, a localization criterion for quasi-one-
dimensional operators of Dirac type, we need to introduce the Lyapunov
exponents and the Furstenberg group of such operators.

###  1.2. Transfer matrices, Lyapunov exponents and the Furstenberg group

In order to determine the almost-sure spectrum of
$\\{D_{\omega}\\}_{\omega\in\Omega}$ and to study the asymptotic behaviour of
the corresponding generalized eigenfunctions, one considers the equation for
the generalized eigenvalues, for every $\omega\in\Omega$,

(1.4) $D_{\omega}u=Eu,\quad\mbox{ where }E\in\mathbb{C}\ \mbox{ and }\
u=\left(\begin{smallmatrix}u^{\uparrow}\\\
u_{\downarrow}\end{smallmatrix}\right)\ :\ \mathbb{R}\to\mathbb{C}^{2N}\ .$

The notation $u=\left(\begin{smallmatrix}u^{\uparrow}\\\
u_{\downarrow}\end{smallmatrix}\right)$ refers to the decomposition spin up /
spin down of the solution of the Dirac equation.

Equation (1.4) is a linear differential system of order $1$. We introduce, for
$E\in\mathbb{C}$ and every $x,y\in\mathbb{R}$, the transfer matrix
$T_{x}^{y}(E)$ of $D_{\omega}$ from $x$ to $y$ which maps a solution
$(u^{\uparrow},u_{\downarrow})$ at time $x$ to the same solution at time $y$.
It is defined by the relation

(1.5) $\left(\begin{array}[]{c}u^{\uparrow}(y)\\\
u_{\downarrow}(y)\end{array}\right)=T_{x}^{y}(E)\left(\begin{array}[]{c}u^{\uparrow}(x)\\\
u_{\downarrow}(x)\end{array}\right)$

and in particular, $T_{x}^{x}(E)=I_{\mathrm{2N}}$ for every $x\in\mathbb{R}$.
The transfer matrices are elements of the complex symplectic group

(1.6)
$\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})=\\{M\in\mathcal{M}_{\mathrm{2N}}(\mathbb{C})\
|\ M^{*}JM=J\\}$

with $J=\left(\begin{smallmatrix}0&-I_{N}\\\ I_{N}&0\end{smallmatrix}\right)$.
Indeed, for $E\in\mathbb{C}$ and $x\in\mathbb{R}$ fixed $y\mapsto
T_{x}^{y}(E)$ satisfies $D_{\omega}(T_{x}^{y}(E))=E(T_{x}^{y}(E))$ on
$\mathbb{R}$. It implies
$\left(\frac{d}{dy}T_{x}^{y}(E)\right)^{*}JT_{x}^{y}(E)+(T_{x}^{y}(E))^{*}J\frac{d}{dy}T_{x}^{y}(E)=0$.
Hence, the function $y\mapsto(T_{x}^{y}(E))^{*}JT_{x}^{y}(E)$ is constant on
$\mathbb{R}$. Taking the value at $y=x$ one obtains $J$ and
$(T_{x}^{y}(E))^{*}JT_{x}^{y}(E)=J$ for every $y\in\mathbb{R}$.

######  Remark 1.3.

Note that despite its name, $\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$ is a
real Lie group since it is a $C^{\infty}$ manifold and not a holomorphic
manifold because of the presence of a conjugation in its definition.

For $E\in\mathbb{C}$ fixed and two couples $(x,y)$ and
$(x^{\prime},y^{\prime})$ in $\mathbb{R}^{2}$, the random matrices
$T_{x}^{y}(E)$ and $T_{x^{\prime}}^{y^{\prime}}(E)$ are not necessarily
independent. In order to apply the results of the theory of products of i.i.d.
random matrices, we also introduce, for every $n\in\mathbb{Z}$, the transfer
matrices $T_{\omega^{(n)}}(E)=T_{\ell n}^{\ell(n+1)}(E)$ from $\ell n$ to
$\ell(n+1)$. The transfer matrix $T_{\omega^{(n)}}(E)$ is therefore defined by
the relation

(1.7) $\left(\begin{array}[]{c}u^{\uparrow}(\ell(n+1))\\\
u_{\downarrow}(\ell(n+1))\end{array}\right)=T_{\omega^{(n)}}(E)\left(\begin{array}[]{c}u^{\uparrow}(\ell
n)\\\ u_{\downarrow}(\ell n)\end{array}\right)$

for all $n\in\mathbb{Z}$.

The sequence $(T_{\omega^{(n)}}(E))_{n\in\mathbb{Z}}$ is a sequence of i.i.d.
matrices because of the i.i.d. character of the $V_{\omega}^{(n)}$’s and the
disjointness of their supports for different values of $n$.

By iterating the relation (1.7) we get the asymptotic behaviour of
$\left(\begin{smallmatrix}u^{\uparrow}\\\
u_{\downarrow}\end{smallmatrix}\right)$. More precisely, we introduce, for
$E\in\mathbb{C}$ fixed, the cocycle
$\Phi_{E}:\mathbb{Z}\times\Omega\to\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$
defined by : $\forall n\in\mathbb{Z},\ \forall\omega\in\Omega,$

$\Phi_{E}(n,\omega)\;=\;\left\\{\begin{array}[]{lcl}T_{\omega^{(n-1)}}(E)\cdots
T_{\omega^{(0)}}(E)&\mbox{ if }&n>0\\\ I_{\mathrm{N}}&\mbox{ if }&n=0\\\
(T_{\omega^{(n)}}(E))^{-1}\cdots(T_{\omega^{(-1)}}(E))^{-1}&\mbox{ if
}&n<0\end{array}\right.$

To get the _exponential_ asymptotic behaviour of
$\left(\begin{smallmatrix}u^{\uparrow}\\\
u_{\downarrow}\end{smallmatrix}\right)$, we define the exponential growth (or
decay) exponents of the product of random matrices
$T_{\omega^{(n-1)}}(E)\cdots T_{\omega^{(0)}}(E)$.

######  Definition 1.4.

Let $E\in\mathbb{C}$. The _Lyapunov exponents_
$\gamma_{1}^{\pm}(E),\ldots,\gamma_{2N}^{\pm}(E)$ associated with the sequence
$(T_{\omega^{(n)}}(E))_{n\in\mathbb{Z}}$ are defined inductively by

(1.8)
$\sum_{i=1}^{p}\gamma_{i}^{\pm}(E)=\lim_{n\to\pm\infty}\frac{1}{|n|}\mathbb{E}(\log||\wedge^{p}\Phi_{E}(n,\omega)||)$

for every $p\in\\{1,\ldots,2N\\}$. Here, $\wedge^{p}M$ denotes the $p$-th
exterior power of the matrix $M$, acting on the $p$-th exterior power of
$\mathbb{C}^{2N}$.

Since the transfer matrices all lie in
$\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$, for every $i\in\\{1,\ldots,2N\\}$,
$\gamma_{i}^{+}(E)=\gamma_{i}^{-}(E)$. Indeed, for each
$M\in\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$, $||M||=||M^{-1}||$.

One can link the Lyapunov exponents to the singulars values of the cocycle
$\Phi_{E}(n,\omega)$.

######  Proposition 1.5 ([5],Proposition A.III.5.6).

If $s_{1}(\Phi_{E}(n,\omega))\geq\ldots\geq s_{2N}(\Phi_{E}(n,\omega))>0$ are
the singular values of $\Phi_{E}(n,\omega)$, then, for $\mathsf{P}$-almost
every $\omega\in\Omega$,

$\forall p\in\\{1,\ldots,2N\\},\
\gamma_{p}^{\pm}(E)=\lim_{n\to\pm\infty}\frac{1}{|n|}{\mathbb{E}}(\log
s_{p}(\Phi_{E}(n,\omega))=\lim_{n\to\pm\infty}\frac{1}{|n|}\log
s_{p}(\Phi_{E}(n,\omega)).$

This implies in particular that $\gamma_{1}(E)\geq\ldots\geq\gamma_{2N}(E)$.
Moreover, the symplecticity of the transfer matrices also implies the
following symmetry property (_cf._ [5, Proposition A.IV.3.2])

$\forall i\in\\{1,\ldots,N\\},\ \gamma_{2N-i+1}(E)=-\gamma_{i}(E).$

To study the properties of the Lyapunov exponents, we introduce the group
which contains all the different products of transfer matrices, the so-called
Furstenberg group.

######  Definition 1.6.

For every $E\in\mathbb{C}$, let $\mu_{E}$ be the common distribution of the
random matrices $T_{\omega^{(n)}}(E)$. We define the _Furstenberg group_ of
$\\{D_{\omega}\\}_{\omega\in\Omega}$ at $E$ as the closed group generated by
the support of $\mu_{E}$,

$G(E)=\overline{<\operatorname{supp}\mu_{E}>},$

where the closure is taken for the usual topology in
$\mathcal{M}_{2N}(\mathbb{C})$.

We already remark that for all $E\in\mathbb{C}$, $G(E)$ is a subgroup of
$\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$.

###  1.3. Localization criterion for quasi-one-dimensional operators of Dirac
type

The formalism of transfer matrices, Lyapunov exponents and the Furstenberg
group enables to state a criterion of dynamical localization for quasi-one-
dimensional operators of Dirac type.

Before that, we introduce several definitions in order to fix the framework in
which we are able to obtain such a criterion of dynamical localization.

We introduce two properties concerning the Furstenberg group. Let
$p\in\\{1,\ldots,N\\}$. The first property is called _$p$ -contractivity_.

######  Definition 1.7.

A subset $T$ of $\operatorname{GL}_{\mathrm{2N}}(\mathbb{C})$ is called _p-
contracting_ if there exists a sequence $(M_{n})$ of elements of $T$ such that
$\|\Lambda^{p}M_{n}\|^{-1}\Lambda^{p}M_{n}$ converges to a rank-one matrix.

Let $L\geq 1$ an integer. For $l\in\\{1,\ldots,L\\}$ we denote indifferently
by $b_{l}$ a bilinear form on $\mathbb{C}^{2N}$ or its matrix in the canonical
basis of $\mathbb{C}^{2N}$. We also denote by $b_{0}$, or most simply by $J$,
the symplectic bilinear form on $\mathbb{C}^{2N}$ associated with the matrix
$J=\left(\begin{smallmatrix}0&-\mathrm{I}_{\mathrm{N}}\\\
\mathrm{I}_{\mathrm{N}}&0\end{smallmatrix}\right)$.

For any $p\in\\{1,\ldots,N\\}$, let $(J,b_{1},\ldots,b_{L})$-$L_{p}$ be the
vector subspace of $\Lambda^{p}\mathbb{C}^{2N}$ whose elements are
$p$-decomposable vectors $u_{1}\wedge\cdots\wedge u_{p}$ such that

$\forall i,j\in\\{1,\ldots,p\\},\ \forall l\in\\{0,\ldots,L\\},\
b_{l}(u_{i},u_{j})=0$

_i.e_ , the family $(u_{1},\ldots,u_{p})$ is orthogonal for all the bilinear
forms $b_{l}$ (and in particular each $u_{i}$ is orthogonal to itself for all
$b_{l}$).

The second property is called the $(J,b_{1},\ldots,b_{L})$-$L_{p}$-strong
irreducibility. It generalizes the notion of $L_{p}$-strong irreducibility as
defined in [5] in the setting of the real symplectic group.

######  Definition 1.8.

We say that a subset $T$ of $\operatorname{GL}_{\mathrm{2N}}(\mathbb{C})$ is
_$(J,b_{1},\ldots,b_{L})$ -$L_{p}$-strongly irreducible_ if there does not
exist any $W$, finite union of proper vector subspaces of
$(J,b_{1},\ldots,b_{L})$-$L_{p}$, such that $(\Lambda^{p}M)(W)=W$ for all $M$
in $T$.

We now state two localization criteria for
$\\{D_{\omega}\\}_{\omega\in\Omega}$ in some particular cases of the group
$G$. The first one states for the group
$G=\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$.

######  Theorem 1.9.

We fix a compact interval $I\subset\mathbb{R}$. We assume that there exists an
open interval $\tilde{I}$ containing $I$ and such that for every
$E\in\tilde{I}$:

  1. (1)

the Furstenberg group $G(E)$ is included in
$\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$ ;

  2. (2)

for every $p\in\\{1,\ldots,N\\}$, $G(E)$ is $p$-contracting and
$J$-$L_{p}$-strongly irreducible.

Then $\\{D_{\omega}\\}_{\omega\in\Omega}$ exhibits dynamical localization in
$\Sigma\cap I$.

In order to state a second theorem, we introduce the matrix

(1.9) $S:=\left(\begin{array}[]{cc}K&0\\\
0&K\end{array}\right)\in\mathcal{M}_{2N}(\mathbb{R}),$

where $K$ is the diagonal matrix with $(-1)^{i+1}$ in position $i$. We define
the group

(1.10)
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R}):=\left\\{M\in\mathcal{M}_{2N}(\mathbb{R}),\
^{t}MJM=J,\ ^{t}MSM=S\right\\}.$

The second localization criterion states for the group
$G=\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$.

######  Theorem 1.10.

Assume that $N$ is even. We fix a compact interval $I\subset\mathbb{R}$. We
assume that there exists an open interval $\tilde{I}$ containing $I$ and such
that for every $E\in\tilde{I}$:

  1. (1)

the Furstenberg group $G(E)$ is included in
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ ;

  2. (2)

for every $2p\in\\{1,\ldots,N\\}$, $G(E)$ is $2p$-contracting and
$(J,S)$-$L_{2p}$-strongly irreducible.

Then $\\{D_{\omega}\\}_{\omega\in\Omega}$ exhibits dynamical localization in
$\Sigma\cap I$.

Theorems 1.9 and 1.10 are comparable to [8, Theorem 1] which is a criterion of
localization for quasi-one-dimensional operators of Schrödinger type. This
former result deals only with the case of
$G=\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$.

The proofs of Theorems 1.9 and 1.10 involve several steps as detailed in
Section 2:

  1. (1)

The assumptions of the two theorems lead to an integral formula for the
Lyapunov exponents which implies their Hölder regularity.

  2. (2)

We then deduce the same Hölder regularity for the integrated density of
states, using a Thouless formula.

  3. (3)

From this regularity of the integrated density of states, we get a weak
Wegner’s estimate adapted to Bernoulli randomness.

  4. (4)

Finally we apply a multiscale analysis scheme which involves the proof of an
Initial Length Scale Estimate.

Actually, we will see in Section 2 that most of these steps except the proof
of the Initial Length Scale Estimate in the last one are true with more
general hypothesis. Let us state them now.

######  Assumption ($\mathbf{L^{(N)}}$).

We fix a compact interval $I\subset\mathbb{R}$. We assume that there exists
$L_{N}$ a vector subspace of $\Lambda^{N}\mathbb{C}^{2N}$ and an open interval
$\tilde{I}$ containing $I$ such that, for all $E\in\tilde{I}$:

  * $\mathbf{(L_{1}^{(N)}})$

for all $g\in G(E)$, $(\Lambda^{N}g)(L_{N})\subset L_{N}$;

  * $\mathbf{(L_{2}^{(N)}})$

for all $x\neq 0$ in $L_{N}$,

$\lim_{n\to\infty}\frac{1}{n}{\mathbb{E}}(\log\|\Lambda^{N}\Phi_{E}(n,\cdot)x\|)=\sum_{i=1}^{N}\gamma_{i}(E);$

  * $\mathbf{(L_{3}^{(N)}})$

there exists a unique probability measure $\nu_{N,E}$ on $\mathsf{P}(L_{N})$
which is $\mu_{E}$-invariant, such that

$\gamma_{1}(E)+\dots+\gamma_{N}(E)=\int_{G(E)\times\mathsf{P}(L_{N})}\log\frac{\|(\Lambda^{N}g)x\|}{\|x\|}{\rm
d}\mu_{E}(g){\rm d}\nu_{N,E}(\bar{x});$

  * $\mathbf{(L_{4}^{(N)}})$

$\gamma_{1}(E)+\dots+\gamma_{N}(E)>0$.

The properties of $N$-contractivity and $J$ or $(J,S)$-$L_{N}$-strong
irreducibility imply Assumption $\mathbf{(L^{(N)}})$. It is a consequence of
[5, Proposition A.IV.3.4] in the case of
$\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$ since one has the identification
between $\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$ and
$\mathrm{Sp}_{2\mathrm{N}}(\mathbb{R})$ using the following application which
split the real and imaginery parts of the matrices in
$\mathcal{M}_{\mathrm{2N}}(\mathbb{C})$:

$\pi\ :\
\begin{array}[]{cll}\mathcal{M}_{\mathrm{2N}}(\mathbb{C})&\to&\mathcal{M}_{\mathrm{4N}}(\mathbb{R})\\\\[5.69054pt]
A+iB&\mapsto&\left(\begin{smallmatrix}A&-B\\\
B&A\end{smallmatrix}\right).\end{array}$

In the case of $\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, it is proven in
Proposition 2.5.

In the following Section, we will present some explicit cases of the model
$\\{D_{\omega}\\}_{\omega\in\Omega}$ for which we are able to verify the
assumptions of Theorem 1.9 or Theorem 1.10.

###  1.4. Application of the localization criterion to a class of splitting
potentials

In this Section, we introduce a particular case of quasi-one-dimensional
operators of Dirac type whose potentials split in a sum of two tensorized
Pauli matrices. Recall the usual notations for Pauli matrices :

$\sigma_{0}:=\mathrm{I}_{2},\ \sigma_{1}:=\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right),\
\sigma_{2}:=\left(\begin{smallmatrix}0&-\mathrm{i}\\\
\mathrm{i}&0\end{smallmatrix}\right),\
\sigma_{3}:=\left(\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\right).$

Consider the particular family $\\{D_{\omega,\ell}^{(N)}\\}_{\omega\in\Omega}$
where the potential split into a periodic part and a random part :

(1.11) $D_{\omega,\ell}^{(N)}:=D_{0}^{(N)}+V_{\mathrm{per}}+V_{\omega}.$

The potential $V_{\mathrm{per}}$ is a $\ell$-periodic function, linear
combination of tensorized Pauli matrices of the form

(1.12)
$V_{\mathrm{per}}:=\left(\alpha_{0}\sigma_{0}+\alpha_{1}\sigma_{1}+\alpha_{2}\sigma_{2}+\alpha_{3}\sigma_{3}\right)\otimes\hat{V}_{\mathrm{per}},$

where $\alpha_{0},\ldots,\alpha_{3}$ are real numbers and
$\hat{V}_{\mathrm{per}}$ is a $\ell$-periodic function with value in the space
of the $N$-by-$N$ real symmetric matrices denoted by
$\mathrm{S}_{\mathrm{N}}(\mathbb{R})\,$. Note that :

$\sigma_{0}\otimes V:=\left(\begin{smallmatrix}V&0\\\
0&V\end{smallmatrix}\right)\in\mathcal{M}_{\mathrm{2N}}(\mathbb{C})$

and the same for the other tensor products.

We construct the random potential $V_{\omega}$ in the following way. Let
$(\tilde{\Omega}_{1},\tilde{\mathcal{A}}_{1},\tilde{\mathbb{P}}_{1}),\ldots,(\tilde{\Omega}_{N},\tilde{\mathcal{A}}_{N},\tilde{\mathbb{P}}_{N})$
be $N$ complete probability spaces. We take

$(\Omega,\mathcal{A},\mathbb{P})=\left(\operatorname*{\vartimes}_{n\in\mathbb{Z}}\tilde{\Omega}_{1}\times\cdots\times\tilde{\Omega}_{N},\bigotimes_{n\in\mathbb{Z}}\tilde{\mathcal{A}}_{1}\otimes\cdots\otimes\tilde{\mathcal{A}}_{N},\bigotimes_{n\in\mathbb{Z}}\tilde{\mathbb{P}}_{1}\otimes\cdots\otimes\tilde{\mathbb{P}}_{N}\right).$

For $i\in\\{1,\ldots,N\\}$, the sequences
$(\omega_{i}^{(n)})_{n\in\mathbb{Z}}$ are independent of each other and each
one is a sequence of i.i.d. real-valued random variables on
$(\widetilde{\Omega}_{i},\widetilde{\mathcal{A}}_{i},\widetilde{\mathbb{P}}_{i})$.
Let $\nu_{i}$ be the common law of the $\omega_{i}^{(n)}$’s. We assume that
$\\{0,1\\}\subset\operatorname{supp}\nu_{i}$ and $\operatorname{supp}\nu_{i}$
is bounded. In particular, the $\omega_{i}^{(n)}$’s can be Bernoulli random
variables which is the most difficult case of randomness to deal with since it
will imply the smallest possible Furstenberg group. We also set, for every
$n\in\mathbb{Z}$, $\omega^{(n)}=(\omega_{1}^{(n)},\ldots,\omega_{N}^{(n)})$,
which is a random variable on
$(\widetilde{\Omega}_{1}\times\cdots\times\widetilde{\Omega}_{N},\widetilde{\mathcal{A}}_{1}\otimes\cdots\otimes\widetilde{\mathcal{A}}_{N},\widetilde{\mathbb{P}}_{1}\otimes\cdots\otimes\widetilde{\mathbb{P}}_{N})$
of law $\nu=\nu_{1}\otimes\cdots\otimes\nu_{N}$. For each $n$ in $\mathbb{Z}$,
we introduce the random function

$\hat{V}_{\omega^{(n)}}:=\left(\begin{smallmatrix}\omega_{1}^{(n)}v_{1}(\cdot-\ell
n)&&0\\\ &\ddots&\\\ 0&&\omega_{N}^{(n)}v_{N}(\cdot-\ell n)\\\
\end{smallmatrix}\right),$

where the $v_{i}$ are measurable functions from $[0,\ell)$ to $\mathbb{R}$. We
take

(1.13)
$V_{\omega}=\left(\beta_{0}\sigma_{0}+\beta_{1}\sigma_{1}+\beta_{2}\sigma_{2}+\beta_{3}\sigma_{3}\right)\otimes\sum_{n\in\mathbb{Z}}\hat{V}_{\omega^{(n)}}$

where $\beta_{0},\ldots,\beta_{3}$ are real numbers.

In order to compute the Furstenberg group associated with
$\\{D_{\omega}\\}_{\omega\in\Omega}$, we will express the transfer matrices as
matrix exponentials (_cf._ Section 3.1). This is possible only when the
potentials are constant on each interval $(n\ell,(n+1)\ell)$. For this reason,
we only consider the case where $\hat{V}_{\mathrm{per}}$ is a constant
function and $v_{1}=\dots=v_{N}=\mathbf{1}_{[0,\ell]}$. By a small abuse of
notation, we will denote by $\hat{V}_{\mathrm{per}}$ (resp.
$\hat{V}_{\omega^{(n)}}$) the unique value of the function
$\hat{V}_{\mathrm{per}}$ (resp. $\hat{V}_{\omega^{(n)}}$).

Moreover, we will restrict ourselves to particular combinations of non-
vanishing $\alpha_{i}$’s and $\beta_{j}$’s. For simplicity we will only
consider real-valued potentials $V_{\mathrm{per}}$ and $V_{\omega}$, which
corresponds to the absence of magnetic field and to the following assumption.

######  Assumption.

We assume that $\alpha_{2}=\beta_{2}=0$.

Hence we only consider potentials which are on $\sigma_{0}$, $\sigma_{1}$ and
$\sigma_{3}$ which implies in particular that the corresponding Furstenberg
groups will be included in $\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$ instead of
$\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$. Next, we consider only splitting
potentials with one deterministic term and one random term which allows to
reduce the number of cases in which we should compute the Furstenberg group
from 43 to 9.

######  Assumption.

We assume that one and only one among $\alpha_{0}$, $\alpha_{1}$ and
$\alpha_{3}$ is different from zero and one and only one among $\beta_{0}$,
$\beta_{1}$ and $\beta_{3}$ is different from zero.

Since we have to choose one random potential and one deterministic one, there
are _a priori_ nine possibilities. Nevertheless, it is possible to reduce this
number to five in the following way. If one sets for
$(V_{0},V_{1},V_{3})\in(\mathcal{M}_{\mathrm{N}}(\mathbb{R}))^{3}$,

$D(V_{0},V_{1},V_{3})=D_{0}^{(N)}+\sigma_{0}\otimes V_{0}+\sigma_{1}\otimes
V_{1}+\sigma_{3}\otimes V_{3}$

acting on $H^{1}(\mathbb{R})\otimes\mathbb{R}^{2}$, then

$\forall(V_{0},V_{1},V_{3})\in(\mathcal{M}_{\mathrm{N}}(\mathbb{R}))^{3},\
D(V_{0},V_{1},V_{3})=P(-D(-V_{0},-V_{3},-V_{1}))P^{*}$

with $P$ the unitary matrix defined by

(1.14)
$P=\frac{1}{\sqrt{2}}\left(\begin{matrix}I_{\mathrm{N}}&I_{\mathrm{N}}\\\
I_{\mathrm{N}}&-I_{\mathrm{N}}\end{matrix}\right).$

Hence, for any
$(V_{0},V_{1},V_{3})\in(\mathcal{M}_{\mathrm{N}}(\mathbb{R}))^{3}$, the
operators $D(V_{0},V_{1},V_{3})$ and $-D(-V_{0},-V_{3},-V_{1})$ have the same
spectrum and also the same pure point, absolutely continuous and singular
continuous spectra. These two operators have transfer matrices which are
unitarily equivalent (through $P$ defined at (1.14)), hence their Lyapunov
exponents are equal. They also have Furstenberg groups which are unitarily
equivalent (again through $P$) and there is localization for
$D(V_{0},V_{1},V_{3})$ if and only if there is localization for
$-D(-V_{0},-V_{3},-V_{1})$. As a consequence, there are five cases, as
explained in the following table.

$V_{\omega}$ $\backslash$ $V_{per}$ | $\sigma_{0}$ | $\sigma_{1}$ | $\sigma_{3}$
---|---|---|---
$\sigma_{0}$ | 1 | 5 | 5
$\sigma_{1}$ | 3 | 2 | 4
$\sigma_{3}$ | 3 | 4 | 2
Table 1. The five possible cases

For each of these cases, we prove either localization or delocalization.

######  Theorem 1.11.

Denote by $V$ the unique value of the function $\hat{V}_{\mathrm{per}}$. For
almost-every real symmetric matrix
$V\in\mathrm{S}_{\mathrm{N}}(\mathbb{R})\,$, there exist a finite set
$\mathcal{S}_{V}\subset\mathbb{R}$ and $\ell_{C}:=\ell_{C}(N,V)>0$ such that,
for every $\ell\in(0,\ell_{C})$, there exists a compact interval
$I(N,V,\ell)\subset\mathbb{R}$ such that if $I\subset
I(N,V,\ell)\setminus\mathcal{S}_{V}$ is an open interval with $\Sigma\cap
I\neq\emptyset$, then :

  * (i)

in case $1$, $\Sigma\cap I$ is purely a.c.;

  * (ii)

in cases $2$, $3$ and $4$, $\\{D_{\omega,\ell}^{(N)}\\}_{\omega\in\Omega}$
exhibits Anderson and dynamical localization in $\Sigma\cap I$;

  * (iii)

in case $5$, $\\{D_{\omega,\ell}^{(N)}\\}_{\omega\in\Omega}$ exhibits Anderson
and dynamical localization in $\Sigma\cap I$ if $N$ is even and there is
presence of a.c. spectrum of multiplicity $2$ if $N$ is odd.

Remark that from its construction given at Section 3.1, the interval
$I(N,V,\ell)$ tends to the whole real line $\mathbb{R}$ when $\ell$ tends to
$0$.

The localization results come from applications of our localization criteria
in Theorems 1.9 and 1.10. The presence of a.c. spectrum in case $1$ and in
case $5$ when $N$ is odd is a consequence of the following theorem of Sadel
and Schulz-Baldes.

######  Theorem 1.12 ([29], Theorem 4).

Let $\\{D_{\omega}\\}_{\omega\in\Omega}$ be as in (1.2). Then, for
$k\in\\{1,\dots,N\\}$, the set

$S_{k}:=\\{E\in\mathbb{R},\text{ exactly }2k\text{ Lyapunov exponents vanish
at }E\\}$

is an essential support of the almost-sure absolutely continuous spectrum of
multiplicity $2k$.

This result of Kotani’s theory for quasi-one-dimensional operators of Dirac
type has to be seen as a _delocalization_ result for
$\\{D_{\omega}\\}_{\omega\in\Omega}$.

##  2\. Proof of the localization criteria

The proofs of Theorem 1.9 and Theorem 1.10 consist of several steps, following
the strategy already used in [23, 13, 7, 8].

The first step is to prove that, under the hypotheses of the theorems,
Assumption $\mathbf{(L^{(N)})}$ or $\mathbf{(L^{(N-1)})}$ is satisfied
depending on the parity of $N$. As we mentioned in Section 2, this is already
known for a group which is $J$-$L_{p}$-strongly irreducible. We provide a
proof in the case of a $(J,S)$-$L_{p}$-strongly irreducible group.

Then, we prove that, under Assumption $\mathbf{(L^{(N)})}$ or
$\mathbf{(L^{(N-1)})}$ , the sum of all nonnegative Lyapunov exponents is
Hölder continuous.

######  Theorem 2.1.

Let $I$ be a compact interval on which Assumption $\mathbf{(L^{(N)})}$ holds
or on which Assumption $\mathbf{(L^{(N-1)})}$ holds and $\gamma_{N}$ is
identically $0$. Then, the sum of the Lyapunov exponents associated with
$\\{D_{\omega}\\}_{\omega\in\Omega}$ is Hölder continuous on $I$, _i.e._ there
exist two real numbers $\alpha>0$ and $C>0$ such that

(2.1) $\forall E,E^{\prime}\in
I,\left|\sum_{p=1}^{N}\left(\gamma_{p}(E)-\gamma_{p}(E^{\prime})\right)\right|\leq
C|E-E^{\prime}|^{\alpha}.$

We prove this first result in Section 2.2. Note that (2.1) holds not only on
intervals where the hypotheses of Theorem 2.1 are satisfied but also trivially
on intervals where $\sum_{p=1}^{N}\gamma_{p}$ is identically 0. A singularity
can only occur at the boundary between such intervals.

As in [8, 35], the next step towards localization is to prove Hölder
regularity of the integrated density of states. We get this regularity from
the one of the Lyapunov exponents. We define the density of states in the same
way as in [35].

Given $x\in\mathbb{R}$ and $L>0$, we define the operator $D_{\omega}^{(x,L)}$,
called the _restricted operator with Dirichlet boundary condition_ to the
interval $\Lambda_{L}(x):=(x-\ell L,x+\ell L)$, as the operator acting as
$D_{\omega}$ on the domain

$\mathrm{Dom}\left(D_{\omega}^{(x,L)}\right):=\left\\{\psi=\left(\begin{smallmatrix}\psi^{\uparrow}\\\
\psi^{\downarrow}\end{smallmatrix}\right)\in
H^{1}\left(\Lambda_{L}(x),\mathbb{R}^{2N}\right)\text{ such that
}\psi^{\uparrow}(x-\ell L)=\psi^{\uparrow}(x+\ell L)=0\right\\}.$

We will use the notation $D_{\omega}^{(L)}$ for $D_{\omega}^{(0,L)}$.

We define the density of states in the following way.

######  Definition 2.2.

For all compactly supported continuous functions $\phi$, the _density of
states_ is

$\nu(\phi)=\lim_{L\to\infty}\frac{1}{2\ell
L}{\mathbb{E}}(\operatorname{tr}(\phi(D_{\omega}^{(L)}))).$

The well-definedness of this limit and its independence from $\omega$,
consequence of the ergodic theorem, are proven in [35, Appendix B] in the case
$N=1$. The generalization to bigger $N$ is straightforward.

The function $\nu$ is a positive linear functional on the space of compactly
supported continuous functions. By the Riesz-Markov theorem, it can be seen as
an integral with respect to some Borel measure on $\mathbb{R}$, which will be
denoted by $\nu$ too. The integrated density of states is defined on
$\mathbb{R}$ by

(2.2) $F(E):=\left\\{\begin{array}[]{l}-\nu((E,0])\text{ if }E<0\\\
\nu((0,E])\text{ if }E\geq 0\end{array}\right.$

We prove that, when the sum of the Lyapunov exponents is Hölder continous, the
integrated density of states is Hölder continuous as well:

######  Theorem 2.3.

Let $I$ be a compact interval and $\tilde{I}$ an open interval containing $I$.
If the sum $\sum_{i=p}^{N}\gamma_{p}$ is Hölder continuous on $\tilde{I}$,
then there exist $C^{\prime}>0$ and $\alpha^{\prime}>0$ such that

(2.3) $\forall E,E^{\prime}\in I,\ |F(E)-F(E^{\prime})|\leq
C^{\prime}|E-E^{\prime}|^{\alpha^{\prime}}.$

The proof of Theorem 2.3 is a combination of arguments of [35], [7] and [29].
We give the main steps of it in Section 2.3.

The previous results will enable us to prove Anderson and dynamical
localization for random Dirac operators, under the hypotheses of Theorem 1.9
or 1.10. We use a method called _bootstrap multiscale analysis_ , which has
been developed by Germinet and Klein in [16] for a large class of operators.
We present it in detail in Section 2.4.1. In Sections 2.4.2 and 2.4.3, we
explain how we get the conditions needed for the multiscale analysis from what
we have proved before.

###  2.1. Lyapunov exponents associated with subgroups of
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$

We want to prove here that if a group $G$ is $2p$-contracting and
$(J,S)$-$L_{2p}$-strongly irreducible for every $2p$ in $\\{1,\dots,N\\}$,
then for all sequence of i.i.d. random matrices in $G$,

  1. (1)

if $N$ is even, then $\mathbf{(L^{(N})}$ holds and the $N$-th Lyapunov
exponent associated with the sequence is positive;

  2. (2)

if $N$ is odd, then $\mathbf{(L^{(N-1)})}$ holds and the $N$-th Lyapunov
exponent associated with the sequence is zero.

We will use the properties of $\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ proven
in Appendix A. First, we prove that when $G$ is included in
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, all Lyapunov exponents are twice
degenerate.

######  Proposition 2.4.

Let $(A^{\omega}_{n})_{n\in\mathbb{Z}}$ be an i.i.d. sequence of matrices in
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ and $(\gamma_{1},\dots,\gamma_{2N})$
the associated family of Lyapunov exponents. Then, for all $p\leq N$,
$\gamma_{2p}=\gamma_{2p-1}$.

###### Proof.

We see from Proposition A.1 (4) that the singular values of a matrix in
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ always have multiplicity at least 2.
This together with [5, Proposition A.III.5.6] implies the degeneracy of the
Lyapunov exponents. ∎

We denote, for even $N$, $N=:2d$ and, for odd $N$, $N=:2d+1$. The space
$(J,S)$-$L_{p}$, introduced in Section 2, can be described in the following
way. For $p\leq d$, we define

(2.4)
$f_{1}:=\frac{1}{2^{p}}(e_{1}+e_{2})\wedge\dots\wedge(e_{2p-1}+e_{2p})\wedge(e_{N+1}-e_{N+2})\wedge\dots\wedge(e_{N+2p-1}-e_{N+2p})$

and

(2.5)
$(J,S)\mbox{-}L_{2p}=\operatorname{span}\\{\Lambda^{2p}Mf_{1}\,|\,M\in\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})\\}.$

We see in particular that $(J,S)$-$L_{2d}$ is stable under the action of
$\Lambda^{2d}g$: $\mathbf{(L_{1}^{(2d)}})$ is satisfied. We prove
$\mathbf{(L_{2}^{(2d)}})$–$\mathbf{(L_{4}^{(2d)}})$ with the following result.

######  Proposition 2.5.

Let $(A_{n}^{\omega})$ be a sequence of i.i.d. random matrices in
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, of common law $\mu$, and $p$ an
integer between 1 and $d$. We denote by $G_{\mu}$ the Furstenberg group
associated with the sequence $(A_{n}^{\omega})$. We assume that it is
$2p$-contracting and $(J,S)$-$L_{2p}$-strongly irreducible and that
${\mathbb{E}}(\log\|A_{0}^{\omega}\|)<\infty$. Then,

  1. (1)

for all $x\neq 0$ in $L_{2p}$,

(2.6) $\lim_{n\to\infty}\frac{1}{n}\log\|\Lambda^{2p}(A_{n-1}^{\omega}\dots
A_{0}^{\omega})x\|=\sum_{i=1}^{2p}\gamma_{i}\text{ almost surely};$

  2. (2)

there exists a unique probability measure $\nu_{2p}$ on $\mathbb{P}(L_{2p})$
which is $\mu$-invariant, such that

(2.7)
$\gamma_{1}+\dots+\gamma_{2p}=\int_{\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})\times\mathbb{P}(L_{2p})}\log\frac{\|(\Lambda^{2p}g)x\|}{\|x\|}{\rm
d}\mu(g){\rm d}\nu_{2p}(\bar{x});$

  3. (3)

(2.8) $\gamma_{2p}>\gamma_{2p+1}.$

Points (1) and (2) of Proposition 2.5 for $p=d$ directly give
$\mathbf{(L_{2}^{(2d)}})$ and $\mathbf{(L_{3}^{(2d)}})$. Moreover, Point (3)
implies that $\gamma_{2d}>\gamma_{2d+1}$. If $N=2d$ is even, we have
$\gamma_{N+1}=-\gamma_{N}$ and if $N=2d+1$ is odd, $\gamma_{N}=0$ by
Proposition 2.4. In both cases this implies that $\gamma_{2d}>0$ and _a
fortiori_ $\sum_{p=1}^{2d}\gamma_{p}>0$ which is $\mathbf{(L_{4}^{(2d)}})$.

###### Proof of Proposition 2.5.

Let $k$ be the dimension of $(J,S)$-$L_{2p}$. Given an orthonormal basis
$(f_{1},\dots,f_{k})$ of $(J,S)$-$L_{2p}$ ($f_{1}$ being defined by (2.4)),
for each $M\in\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, we denote by
$\widehat{M}$ the matrix of $\operatorname{GL}_{k}(\mathbb{R})$ such that

(2.9) $\widehat{M}_{ij}=\langle f_{i},\Lambda^{2p}Mf_{j}\rangle.$

Let us now denote by $\widehat{G}_{\mu}$ the subgroup of
$\operatorname{GL}_{k}(\mathbb{R})$ which is generated by all
$(\widehat{A}_{0}^{\omega})$. We want to prove that $\widehat{G}_{\mu}$
satisfies the hypotheses of theorems given in Chapter A.III of [5], which will
give us information on the two top Lyapunov exponents associated with the
sequence $(\widehat{A}_{n}^{\omega})$. In a second time, we will establish a
relation between the Lyapunov exponents associated with the sequence
$(\widehat{A}_{n}^{\omega})$ of matrices of
$\operatorname{GL}_{k}(\mathbb{R})$ and partial sums of the Lyapunov exponents
associated with the sequence $(A_{n}^{\omega})$ of matrices of
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$.

Since $G_{\mu}$ is $(J,S)$-$L_{2p}$-strongly irreducible, $\widehat{G}_{\mu}$
is strongly irreducible, as a subset of $\operatorname{GL}_{k}(\mathbb{R})$.
Similarly, as $G_{\mu}$ is $2p$-contracting, $\widehat{G}_{\mu}$ is
contracting. We can thus apply [5, Theorem A.III.6.1] and find, denoting by
$\hat{\gamma}_{1}$ and $\hat{\gamma}_{2}$ the two highest Lyapunov exponents
associated with the sequence $(\widehat{A}_{n}^{\omega})$, that

(2.10) $\hat{\gamma}_{1}>\hat{\gamma}_{2}.$

Moreover, Corollary A.III.3.4 of [5] gives that, for all $x\neq 0$ in
$\mathbb{R}^{k}$,

(2.11)
$\lim_{n\to\infty}\frac{1}{n}\log\|(\widehat{A}_{n-1}^{\omega}\dots\widehat{A}_{0}^{\omega})x\|=\hat{\gamma}_{1}\text{
almost surely}$

and, together with Theorem A.III.4.3 of the same [5], that there exists a
unique probability measure $\hat{\nu}$ on $\mathbb{P}(\mathbb{R}^{k})$ which
is $\mu$-invariant, such that

(2.12)
$\hat{\gamma_{1}}=\int_{\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})\times\mathbb{P}(\mathbb{R}^{k})}\log\frac{\|\hat{g}x\|}{\|x\|}{\rm
d}\mu(g){\rm d}\hat{\nu}(\bar{x}).$

We now link the two Lyapunov exponents $\hat{\gamma}_{1}$ and
$\hat{\gamma}_{2}$ with the Lyapunov exponents associated with the sequence
$(A_{n}^{\omega})$. In view of Definition (1.8), we have to estimate
$\|\widehat{M}\|$ and $\|\Lambda^{2}\widehat{M}\|$ for matrices
$\widehat{M}\in\widehat{G}_{\mu}$.

For such a matrix, we consider $t_{1}\geq\dots\geq t_{p}$ the values given in
the decomposition of Proposition A.1 (v). Let us prove that
$\|\widehat{M}\|=\|\Lambda^{2p}M\|={\rm e}^{2t_{1}}\dots\rm e^{2t_{p}}$: this
will imply that $\hat{\gamma_{1}}=\sum_{i=1}^{2p}\gamma_{i}$. Since
$(J,S)$-$L_{2p}$ is a subspace of $\Lambda^{2p}\mathbb{R}^{2N}$, we have that
$\|\widehat{M}\|\leq\|\Lambda^{2p}M\|$. To prove the other inequality, recall
that $f_{1}$, defined in (2.4), is in $(J,S)$-$L_{2p}$. As a consequence, we
have, using the notation of Proposition A.1 (5),

$\|\Lambda^{2p}M\|={\rm e}^{2t_{1}}\dots\rm
e^{2t_{p}}=\|\Lambda^{2p}Rf_{1}\|=\|\widehat{R}f_{1}\|\leq\|\widehat{R}\|\leq\|\Lambda^{2p}R\|=\|\Lambda^{2p}M\|.$

We used [5, Proposition A.III.5.3] for the first equality and the fact that
$(J,S)$-$L_{2p}$ is a subspace of $\Lambda^{2p}\mathbb{R}^{2N}$ for the last
inequality. This implies that $\|\Lambda^{2p}M\|=\|\widehat{R}\|$. But, since
the matrices $K$ and $U$ are orthogonal and in
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$,
$\|\widehat{R}\|=\|\widehat{K}\widehat{R}\widehat{U}\|=\|\widehat{M}\|$. As a
consequence,

(2.13) $\hat{\gamma}_{1}=\sum_{i=1}^{2p}\gamma_{i}.$

This last equality together with (2.11) implies (2.6). Together with (2.12),
(2.13) implies (2.7).

In order to prove (2.8), we now estimate $\|\Lambda^{2}\widehat{M}\|$. We
first notice that, since $(J,S)$-$L_{2p}$ is a vector subspace of
$\Lambda^{2p}\mathbb{R}^{2N}$,
$\|\Lambda^{2}[\Lambda^{2p}M]\|\geq\|\Lambda^{2}\widehat{M}\|$.

We begin with the case $p<d$. We choose

(2.14) $\displaystyle f_{2}:=\frac{1}{2^{p}}$
$\displaystyle(e_{1}+e_{2})\wedge\dots\wedge(e_{2p-3}+e_{2p-2})\wedge(e_{2p+1}+e_{2p+2})$
$\displaystyle\wedge(e_{N+1}-e_{N+2})\wedge\dots\wedge(e_{N+2p-3}-e_{N+2p-2})\wedge(e_{N+2p+1}-e_{N+2p+2}).$

Let

(2.15) $M_{0}:=\left(\begin{matrix}\begin{matrix}I_{\mathrm{2p-2}}&&\\\
&\left(\begin{matrix}0&I_{2}\\\ I_{2}&0\end{matrix}\right)&\\\
&&I_{\mathrm{N-2p-2}}\end{matrix}&\vline&0\\\ \hline\cr
0&\vline&\begin{matrix}I_{\mathrm{2p-2}}&&\\\ &\left(\begin{matrix}0&I_{2}\\\
I_{2}&0\end{matrix}\right)&\\\
&&I_{\mathrm{N-2p-2}}\end{matrix}\end{matrix}\right).$

Then $M_{0}\in\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ and
$\wedge^{2p}M_{0}f_{1}=f_{2}$, hence $f_{2}\in(J,S)$-$L_{2p}$ and it is easy
to see that $f_{1}$ and $f_{2}$ are orthogonal. Consequently, we can complete
$(f_{1},f_{2})$ into an orthonormal basis $(f_{1},\dots,f_{k})$ of
$\operatorname{GL}_{k}(\mathbb{R})$ and use it in the definition of
$\widehat{M}$.

We find that

(2.16)
$\|\Lambda^{2}\widehat{M}\|=\|\Lambda^{2}\widehat{R}\|\geq\|\Lambda^{2}\widehat{R}(f_{1}\wedge
f_{2})\|=\|\widehat{M}\|{\rm e}^{2t_{1}}\dots\rm e^{2t_{p-1}}{\rm
e}^{2t_{p+1}}.$

In terms of Lyapunov exponents, this implies that
$\hat{\gamma_{1}}+\hat{\gamma}_{2}\geq\hat{\gamma_{1}}+\sum_{i=1}^{2(p-1)}\gamma_{i}+2\gamma_{2p+1}$.
This together with (2.10) and (2.13) implies that $\gamma_{2p}>\gamma_{2p+1}$.

When $p=d$, we prove that $\gamma_{2d}>0$. This implies (2.8) since, when $N$
is even, $\gamma_{2d+1}=\gamma_{N+1}\leq 0$ and, when $N$ is odd,
$\gamma_{2d+1}=\gamma_{N}=0$ as we have already seen. We define

(2.17) $\displaystyle f_{2}:=\frac{1}{2^{d}}$
$\displaystyle(e_{1}+e_{2})\wedge\dots\wedge(e_{2d-3}+e_{2d-2})\wedge(e_{2d-1}-e_{2d})$
$\displaystyle\wedge(e_{N+1}-e_{N+2})\wedge\dots\wedge(e_{N+2d-3}-e_{N+2d-2})\wedge(e_{N+2d-1}+e_{N+2d}).$

If we introduce

(2.18) $M_{0}^{\prime}:=\left(\begin{matrix}\begin{matrix}I_{\mathrm{N-1}}&\\\
&-1\end{matrix}&\vline&0\\\ \hline\cr
0&\vline&\begin{matrix}I_{\mathrm{N-1}}&\\\
&-1\end{matrix}\end{matrix}\right)\text{ (even $N$) or
}M_{0}^{\prime}:=\left(\begin{matrix}\begin{matrix}I_{\mathrm{N-2}}&&\\\
&-1&\\\ &&1\end{matrix}&\vline&0\\\ \hline\cr
0&\vline&\begin{matrix}I_{\mathrm{N-2}}&&\\\ &-1&\\\
&&1\end{matrix}\end{matrix}\right)\text{ (odd $N$)}$

then $M_{0}^{\prime}\in\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ and
$f_{2}=\Lambda^{2d}f_{1}$ so $f_{2}\in(J,S)$-$L_{2p}$ with $f_{2}$ orthogonal
to $f_{1}$. As a consequence,

(2.19)
$\|\Lambda^{2}\widehat{M}\|=\|\Lambda^{2}\widehat{R}\|\geq\|\Lambda^{2}\widehat{R}(f_{1}\wedge
f_{2})\|=\|\widehat{M}\|e^{2t_{1}}\dots e^{2t_{d-1}}.$

In terms of Lyapunov exponents, this means that
$\hat{\gamma}_{2}\geq\sum_{i=1}^{2(d-1)}\gamma_{i}$, which, together with
(2.10) and (2.13), implies that $\gamma_{2d}>0$. ∎

###  2.2. Regularity of the Lyapunov exponents

The goal of this section is to prove Theorem 2.1. It is a consequence of the
following result. First, we note that the definitions of Lyapunov exponents
and Furstenberg group can be extended to the case of a sequence of i.i.d.
random matrices in $\mathrm{GL}_{\mathrm{2N}}(\mathbb{C})$. Hence one can
state hypothesis $\mathbf{(L^{(N)})}$ not only for transfer matrices
associated with an ergodic family of operators.

######  Theorem 2.6.

We fix a compact interval $I\subset\mathbb{R}$. Let
$(A_{n}^{\omega}(E))_{n\in\mathbb{Z}}$ be a sequence of i.i.d. random matrices
in $\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$ depending on a parameter $E$ in
$I$. For each $E$, let $\mu_{E}$ be the common distribution of the
$A_{n}^{\omega}(E)$. We assume that, for all $E\in I$,
${\mathbb{E}}(\log\|A_{0}^{\omega}(E)\|)<\infty$. We also assume that
Assumption $\mathbf{(L^{(N)})}$ holds on $I$, or that Assumption
$\mathbf{(L^{(N-1)})}$ holds and $\gamma_{N}$ is identically $0$ on $I$.
Moreover, we assume that,

  1. (1)

there exists $C_{1}>0$ independent of $E$, $\omega$ and $n$ such that, for
every $E\in I$,

(2.20) $\|\Lambda^{N}A_{n}^{\omega}(E)\|^{2}\leq C_{1};$

  2. (2)

there exists $C_{2}>0$ independent of $E$, $\omega$ and $n$ such that for
every $E,E^{\prime}\in I$

(2.21)
$\|\Lambda^{N}A_{n}^{\omega}(E)-\Lambda^{N}A_{n}^{\omega}(E^{\prime})\|^{2}\leq
C_{2}|E-E^{\prime}|.$

Then, there exist $C>0$ and $\alpha>0$ such that

(2.22) $\forall E,E^{\prime}\in
I,\quad|\gamma_{1}(E)+\dots+\gamma_{N}(E)-\gamma_{1}(E^{\prime})-\dots-\gamma_{N}(E^{\prime})|\leq
C|E-E^{\prime}|^{\alpha}.$

This theorem is basically a restatement of Theorem 1 of [7]. A detailed proof
can be found in [6, Section 6.3]. One difference is that we only prove Hölder
regularity for the sum $\gamma_{1}+\dots+\gamma_{N}$, while the result of [7]
provides regularity for each of the $\gamma_{p}$’s. For this reason, it needs
hypotheses for all $p$ while we need them only for $p=N$. Then, our hypotheses
(1) and (2) correspond to hypotheses (ii) and (iii) in [7]. His hypothesis (i)
is the fact that the Furstenberg group is $p$-contracting and
$J$-$L_{p}$-strongly irreducible for all $p$ in $\\{1,\dots,N\\}$. If we
carefully look at the proof of the theorem of [7], we see that the only reason
for which this is needed is because it implies Assumption $\mathbf{(L^{(N)})}$
on $I$. As a consequence, we can use this hypothesis instead of the one of
$p$-contractivity and $J$-$L_{p}$-strong irreducibilty for all $p$, which
makes it possible to apply the theorem in a more general setting.

We want to apply Theorem 2.6 to the sequence
$(A_{n}^{\omega}(E))_{n\in\mathbb{Z}}=(T_{\omega^{(n)}}(E))_{n\in\mathbb{Z}}$,
the transfer matrices as defined in (1.7), to get Theorem 2.1. Therefore, we
have to prove that this sequence satisfies the hypotheses of Theorem 2.6. We
already assumed that Assumption $\mathbf{(L^{(N)})}$ holds on $I$, or that
Assumption $\mathbf{(L^{(N-1)})}$ holds and $\gamma_{N}$ is identically $0$ on
$I$ in the hypotheses of Theorem 2.1. The boundedness of the potential implies
that ${\mathbb{E}}(\log\|T_{\omega^{(0)}}(E)\|)<\infty$ for all $E$. Finally,
we are left with proving that the estimates (2.20) and (2.21) are always
satisfied for the sequence $(T_{\omega^{(n)}}(E))_{n\in\mathbb{Z}}$. We begin
with two lemmas which provides estimates on solutions of Dirac equations. Both
have been proven in the case $N=1$ in [35] and can easily be generalized to
$N\geq 1$. As a consequence, we give only sketches of the proofs.

######  Lemma 2.7.

Let $\psi=\begin{pmatrix}\psi^{\uparrow}\\\ \psi^{\downarrow}\end{pmatrix}$ be
a solution to $D_{0}^{(N)}\psi+V\psi=0$, where $V$ is a $L_{loc}^{1}$ function
with values in $2N$-by-$2N$ matrices. Then, for all $x$, $y\in\mathbb{R}$,

(2.23)
$|\psi^{\uparrow}(x)|^{2}+|\psi^{\downarrow}(x)|^{2}\leq(|\psi^{\uparrow}(y)|^{2}+|\psi^{\downarrow}(y)|^{2})\exp\left(2\int_{\min(x,y)}^{\max(x,y)}|V(t)|\mathrm{d}t\right).$

###### Proof.

This is a straightforward adaptation of [35, Lemma 4.3]. The key argument is
Grönwall’s lemma applied to the function
$|\psi^{\uparrow}(x)|^{2}+|\psi^{\downarrow}(x)|^{2}$.∎

######  Lemma 2.8.

Let $\psi_{1}=\begin{pmatrix}\psi_{1}^{\uparrow}\\\
\psi_{1}^{\downarrow}\end{pmatrix}$ and
$\psi_{2}=\begin{pmatrix}\psi_{2}^{\uparrow}\\\
\psi_{2}^{\downarrow}\end{pmatrix}$ be solutions to
$D_{0}^{(N)}\psi_{i}+V_{i}\psi_{i}=0$, where, for $i=1,2$, $V_{i}$ is a
$L_{loc}^{1}$ function with values in $2N$-by-$2N$ matrices, and such that,
for some $y$ in $\mathbb{R}$, $\psi_{1}(y)=\psi_{2}(y)$. Then, we have, for
all $x$ in $\mathbb{R}$,

(2.24)
$|\psi_{1}(x)-\psi_{2}(x)|\leq|\psi_{2}(y)|\exp\left(\int_{\min(y,x)}^{\max(y,x)}|V_{1}(t)|+|V_{2}(t)|\mathrm{d}t\right)\times\int_{\min(y,x)}^{\max(y,x)}|V_{1}(s)-V_{2}(s)|\mathrm{d}s.$

###### Proof.

The proof is similar to the one of Lemma 4.4 in [35]. It relies once again on
Grönwall’s lemma, applied here to the function $\psi_{1}-\psi_{2}$. ∎

We get (2.20) and (2.21) applying the arguments of [6, Section 6.2.2.1]. We
prove that $\|A_{n}^{\omega}(E)\|^{2}\leq C_{1}$ and that
$\|A_{n}^{\omega}(E)-A_{n}^{\omega}(E^{\prime})\|^{2}\leq C_{2}|E-E^{\prime}|$
by applying respectively Lemmas 2.7 and 2.8 to each column of the transfer
matrix, which is a solution of $D_{0}^{(N)}\psi+(V_{\omega}-E)\psi=0$ (_cf._
Lemmas 6.2.3 and 6.2.5 of [6]). We get the estimates with the external power
as in Lemma 6.2.6 of [6]: (2.20) directly comes from the fact that, for all
invertible matrix $M$, $\|\Lambda^{N}M\|\leq\|M\|^{N}$, while the proof of
(2.21) uses that, for all invertible matrices $M_{1}$ and $M_{2}$,

$\|\Lambda^{N}M_{1}-\Lambda^{N}M_{2}\|\leq\|M_{1}-M_{2}\|\left(\|M_{1}\|^{N-1}+\|M_{1}\|^{N-2}\|M_{2}\|+\dots+\|M_{2}\|^{N-1}\right).$

This concludes the proof of (2.20) and (2.21) and thus of Theorem 2.1.

###  2.3. Hölder continuity of the integrated density of states

We prove in this section Theorem 2.3. The link between the Lyapunov exponents
and the integrated density of states is given by the following Thouless
formula, similar to [35, Proposition 5.2] and [7, Theorem 3].

######  Proposition 2.9.

Let $F$ and $F_{0}$ be the integrated densities of states, respectively of
$\\{D_{\omega}\\}_{\omega\in\Omega}$ and $D_{0}^{(N)}$, and
$\gamma_{1},\dots,\gamma_{N}$ be the Lyapunov exponents of
$\\{D_{\omega}\\}_{\omega\in\Omega}$. Then, there exists $a\in\mathbb{R}$ such
that, for every almost-every $E\in\mathbb{R}$,

(2.25)
$\sum_{i=1}^{N}\gamma_{i}(E)=-a+\int_{\mathbb{R}}\log\left|\frac{E-t}{t-\mathrm{i}}\right|\mathrm{d}(F-F_{0})(t).$

Moreover, if $I\subset\mathbb{R}$ is an interval on which
$E\mapsto(\gamma_{1}+\ldots+\gamma_{N})(E)$ is continuous then (2.25) holds
for every $E\in I$.

Note that, since the spectrum of $D_{0}^{(N)}$ is purely absolutely continuous
with multiplicity $2N$ on the whole real line, all its Lyapunov exponents are
identically zero for real energies (cf. [13, Corollary VII.3.4]). That is why
they do not appear in the formulas. The integrated density of states $F_{0}$
of the free Dirac operator $D_{0}^{(N)}$ can be explicitly computed. Indeeed,
the eigenvalues of $D_{0}^{(N)}$ restricted to $[-\ell L,\ell L]$ with
Dirichlet boundary conditions are the $(k\ell L)_{k\in\mathbb{Z}}$, each of
them with multiplicity $N$. As a consequence, $F_{0}(E)=NE/\pi$.

The following lemma, which has been proven in the case $N=1$ in [35], holds in
fact for all $N$. It makes it possible to control the difference
$F(E)-F_{0}(E)$.

######  Lemma 2.10 ([35], Lemma 5.3).

There exists a constant $C$, depending only on $\|V_{\omega}\|_{\infty}$, such
that for all $E\in\mathbb{R}$, we have

(2.26) $|F(E)-F_{0}(E)|\leq C.$

The next step towards the Thouless formula is to introduce a tool named Kotani
$w$-function. We use it in the way it has been introduced by Sadel and Schulz-
Baldes for Dirac operators in [29, Section 5]. The first step is to introduced
the so-called Weyl-Titchmarsh matrix. Existence and uniqueness of this matrix
are proven in [29, Theorem 2].

######  Definition 2.11.

Let $z\in\mathbb{C}\backslash\mathbb{R}$. Then, there exists a unique
$M_{\omega}(z)\in\mathcal{M}_{N}(\mathbb{C})$, called the _Weyl-Titchmarsh
matrix_ such that the space of solutions of $D_{\omega}\psi=z\psi$ which are
square-integrable on $(0,+\infty)$ is spanned by the solutions whose value in
0 is one of the column of $\left(\begin{smallmatrix}I_{N}\\\
M_{\omega}(z)\end{smallmatrix}\right)$.

In order to be able to use the results of Sadel and Schulz-Baldes, we need to
have an $\mathbb{R}$-ergodic operator, which is _a priori_ stronger than being
$\mathbb{Z}$-ergodic. Nevertheless, in [20], Kirsch proves that, given a
$\mathbb{Z}$-ergodic operator $H_{\omega}$, we can construct an
$\mathbb{R}$-ergodic operator $\overline{H}_{\overline{\omega}}$, on a wider
probability space, with, for each $\overline{\omega}$,
$\overline{H}_{\overline{\omega}}$ is unitarily equivalent to $H_{\omega}$ for
some $\omega$. We apply in our setting Kirsch’s suspension procedure.
Therefore, we introduce a new expectation $\overline{\mathbb{E}}$ which must
here be understood as both expectation on $\Omega$ and average value on
$[0,\ell]$ for the potentials.

As in [29], we introduce a block decomposition for the potential:

$V_{\omega}=:\begin{pmatrix}P_{\omega}&R_{\omega}\\\
R_{\omega}^{*}&Q_{\omega}\end{pmatrix}.$

The $w$ function is defined on $\mathbb{C}\setminus\mathbb{R}$ (with values in
$\mathbb{C}$) as follows:

(2.27)
$w(z):=-\overline{\mathbb{E}}\left[\operatorname{tr}(R_{\omega}+M_{\omega}(z)(Q_{\omega}-z))\right].$

This function will provide us a link between the Lyapunov exponent and the
integrated density of states through the Green’s function. We recall that the
Green’s function $G(z,\cdot,\cdot)$ is the integral kernel of the resolvent
$(D_{\omega}-z)^{-1}$. We start by the following theorem of Sadel and Schulz-
Baldes, which links the Green’s function and the Lyapunov exponent. Here
$G(z):=G(z,x,x)$, and we drop the dependence in $x$ since we take the
expectation $\overline{\mathbb{E}}$.

######  Theorem 2.12 ([29], Theorem 5).

Let $\operatorname{Im}(z)\neq 0$. Then,

  1. (1)

$\sum_{i=1}^{N}\gamma_{i}(z)=-\operatorname{Re}(w(z))$.

  2. (2)

$\partial_{z}w(z)=\overline{\mathbb{E}}(\operatorname{tr}~{}G(z))$.

Once we have this, proving Proposition 2.9 can be done as in [35, Section 5]
(which considers the case $N=1$) and [7, Section 4.2] (which considers
Schrödinger operators for general $N$). As a consequence, we give here only
the main steps of the proof and let the interested reader refer to the cited
papers for details. We first establish a relation between the density of
states measure $\nu$ and the Green’s function.

######  Lemma 2.13 ([35], Lemma 5.5,[7], Proposition 9).

Let $z\in\mathbb{C}\backslash\mathbb{R}$. Then,

$\overline{\mathbb{E}}(\operatorname{tr}(G(z)))=\int_{\mathbb{R}}\frac{\mathrm{d}\nu(E)}{E-z}.$

This together with the second point of Theorem 2.12 give a relation between
the imaginary part of the $w$ function and the integrated density of states.

######  Lemma 2.14 ([35], Lemma 5.6, [7], Proposition 10).

There exists $c\in\mathbb{R}$ such that, for all $E\in\mathbb{R}$, we have

(2.28) $\lim_{a\to
0^{+}}\operatorname{Im}\left(w(E+\mathrm{i}a)-w_{0}(E+\mathrm{i}a)\right)=\pi(F(E)-F_{0}(E))+c.$

We have an analogous result for the real part of $w$.

######  Lemma 2.15.

For Lebesgue-almost every $E$ in $\mathbb{R}$, we have :

(2.29) $\lim_{a\to 0^{+}}\mathrm{Re}\
(w(E+\mathrm{i}a)-w_{0}(E+\mathrm{i}a))=-(\gamma_{1}+\ldots+\gamma_{N})(E).$

Moreover, if $I\subset\mathbb{R}$ is an interval on which
$E\mapsto-(\gamma_{1}+\ldots+\gamma_{N})(E)$ is continuous then (2.29) holds
for every $E\in I$.

We can thus conclude the proof of Proposition 2.9 with the arguments of [7]
and [35]. We then use the Hilbert transformation as in [7] and [35] to get,
through the Thouless formula proven in Proposition 2.9, the Hölder regularity
of the integrated density of states from the one of the sum of the $N$
positive Lyapunov exponents. This achieve the proof of Theorem 2.3.

###  2.4. Multiscale analysis and localization

In this last subsection, we explain how we can apply the multiscale analysis
to get Anderson and dynamical localization in our setting.

####  2.4.1. Conditions for the multiscale analysis

We state here the hypotheses and the theorems on the multiscale analysis. We
follow the presentation of the lecture notes [22]. Then, we explain how we can
apply them to our case. In the following paragraphs,
$\\{H_{\omega}\\}_{\omega\in\Omega}$ will be any self-adjoint random ergodic
operator on $L^{2}(\mathbb{R},\mathbb{C}^{2N})$.

The first property gives the possibility to use generalized eigenfunctions.

Let $\mathcal{H}:=L^{2}(\mathbb{R},\mathrm{d}x;\mathbb{C}^{2N})$. Given
$\nu>1/4$, we put, for $x\in\mathbb{R}$, $\langle x\rangle:=\sqrt{1+x^{2}}$
and we define the weighted spaces $\mathcal{H}_{\pm}$ as

$\mathcal{H}_{\pm}:=L^{2}(\mathbb{R},\langle x\rangle^{\pm
4\nu}\mathrm{d}x;\mathbb{C}^{2N}).$

We define a duality map between $\mathcal{H}_{+}$ and $\mathcal{H}_{-}$ by the
following sesquilinear form, where $\phi_{1}\in\mathcal{H}_{+}$ and
$\phi_{2}\in\mathcal{H}_{-}$:

$\langle\phi_{1},\phi_{2}\rangle_{\mathcal{H}_{+},\mathcal{H}_{-}}:=\int\bar{\phi_{1}}(x)\cdot\phi_{2}(x)\mathrm{d}x.$

We set $T$ to be the self-adjoint operator on $\mathcal{H}$ given by
multiplication by the function $\langle x\rangle^{2\nu}$; note that $T^{-1}$
is bounded.

######  Property (SGEE).

We say that an ergodic random operator $\\{H_{\omega}\\}_{\omega\in\Omega}$
satisfies the strong property of generalized eigenfunction expansion (SGEE) in
some open interval $I$ if, for some $\nu>1/4$,

  1. (1)

The set

$\mathrm{Dom}^{\omega}_{+}:=\\{\phi\in\mathrm{Dom}(H_{\omega})\cap\mathcal{H}_{+};H_{\omega}\phi\in\mathcal{H}_{+}\\}$

is dense in $\mathcal{H}_{+}$ and is an operator core for $H_{\omega}$ with
probability one.

  2. (2)

There exists a bounded, continuous function $f$ on $\mathbb{R}$, strictly
positive on the spectrum of $H_{\omega}$, such that

${\mathbb{E}}\left\\{[\operatorname{tr}(T^{-1}f(H_{\omega})\Pi_{\omega}(I)T^{-1})]^{2}\right\\}<\infty,$

$\Pi_{\omega}$ being the spectral projection associated with $H_{\omega}$.

######  Definition 2.16.

A measurable function $\psi:\mathbb{R}\to\mathbb{C}^{2N}$ is said to be a
_generalized eigenfunction_ of $H_{\omega}$ with _generalized eigenvalue_ $E$
if $\psi\in\mathcal{H}_{-}\backslash\\{0\\}$ and

$\langle
H_{\omega}\phi,\psi\rangle_{\mathcal{H}_{+},\mathcal{H}_{-}}=E\langle\phi,\psi\rangle_{\mathcal{H}_{+},\mathcal{H}_{-}},\
\mbox{ for all }\phi\in\mathrm{Dom}_{+}^{\omega}.$

As explained in [22], when (SGEE) holds, a generalized eigenfunction which is
in $\mathcal{H}$ is a _bona fide_ eigenfunction. Moreover, if $\mu_{\omega}$
is the spectral measure for the restriction of $H_{\omega}$ to the Hilbert
space $E_{\omega}(I)\mathcal{H}$, then $\mu_{\omega}$-almost every $E$ is a
generalized eigenvalue of $H_{\omega}$.

The following properties are about finite volume operators, restricted to
intervals of the form $\Lambda_{L}(x):=[x-\ell L,x+\ell L]$.

######  Definition 2.17.

We say that an ergodic random family of operators
$\\{H_{\omega}\\}_{\omega\in\Omega}$ is _standard_ if for each
$x\in\mathbb{Z}$, $L\in\mathbb{N}$ there is a measurable map $H_{\cdot,x,L}$
from $\Omega$ to self-adjoint operators on
$L^{2}(\Lambda_{L}(x),\mathbb{C}^{2N})$ such that

$U(y)H_{\omega,x,L}U(-y)=H_{\tau_{y}(\omega),x+y,L}$

where $\tau$ and $U$ define the ergodicity:

$U(y)H_{\omega}U(y)^{*}=H_{\tau_{y}(\omega)}.$

We can then define $R_{\omega,x,L}(z):=(H_{\omega,x,L}-z)^{-1}$ as the
resolvent of $H_{\omega,x,L}$ and $\Pi_{\omega,x,L}(\cdot)$ as its spectral
projection.

We now enumerate the properties which are needed for multiscale analysis to be
performed, yielding thus various localization properties.

######  Definition 2.18.

An event is said to be _based in a box_ $\Lambda_{L}(x)$ if it is determined
by conditions on the finite volume operators
$(H_{\omega,x,L})_{\omega\in\Omega}$.

######  Property (IAD).

Events based in disjoint boxes are independent.

The following properties are to hold in a fixed open interval $I$. We will
denote by $\chi_{x,L}$ the characteristic function of $\Lambda_{L}(x)$ and
$\chi_{x}:=\chi_{x,1}$. We also denote by $\Gamma_{x,L}$ the characteristic
function of the union of two regions near the boundary of $\Lambda_{L}(x)$:
$[x-\ell(L-1),x-\ell(L-3)]\cup[x+\ell(L-3),x+\ell(L-1)]$.

######  Property (SLI).

For any compact interval $J\subset I$ there exists a finite constant
$\kappa_{J}$ such that, given $L$, $l^{\prime}$, $l^{\prime\prime}\in
2\mathbb{N}$, $x$, $y$, $y^{\prime}\in\mathbb{Z}$ with
$\Lambda_{l^{\prime\prime}}(y)\subset\Lambda_{l^{\prime}-3}(y^{\prime})\subset\Lambda_{L-6}(x)$,
then, for $\mathbb{P}$-almost every $\omega$, if $E\in J$ with
$E\notin\sigma(H_{\omega,x,L})\cup\sigma(H_{\omega,y^{\prime},l^{\prime}})$ we
have

(2.30)
$\|\Gamma_{x,L}R_{\omega,x,L}(E)\chi_{y,l^{\prime\prime}}\|\leqslant\kappa_{J}\|\Gamma_{y^{\prime},l^{\prime}}R_{\omega,y^{\prime},l^{\prime}}(E)\chi_{y,l^{\prime\prime}}\|\|\Gamma_{x,L}R_{\omega,x,L}(E)\Gamma_{y^{\prime},l^{\prime}}\|.$

######  Property (EDI).

For any compact interval $J\subset I$ there exists a finite constant
$\tilde{\kappa}_{J}$ such that for $\mathbb{P}$-almost every $\omega$, given a
generalized eigenfunction $\psi$ of $H_{\omega}$ with generalized eigenvalue
$E\in J$, we have, for any $x\in\mathbb{Z}$ and $L\in 2\mathbb{N}$ with
$E\notin\sigma(H_{\omega,x,L})$, that

$\|\chi_{x}\psi\|\leqslant\tilde{\kappa}_{J}\|\Gamma_{x,L}R_{\omega,x,L}(E)\chi_{x}\|\|\Gamma_{x,L}\psi\|.$

######  Property (NE).

For any compact interval $J\subset I$ there exists a finite constant $C_{J}$
such that, for all $x\in\mathbb{Z}$ and $L\in 2\mathbb{N}$,

${\mathbb{E}}\left(\operatorname{tr}\left(\Pi_{\omega,x,L}(J)\right)\right)\leqslant
C_{J}L.$

######  Property (W).

For every $\beta\in(0,1)$ and every $\sigma>0$, there exists
$L_{0}\in\mathbb{N}$ and $\alpha>0$ such that

(2.31) ${\mathbb{P}}(\operatorname{dist}(E,\sigma(H_{\omega,x,L}))\leq{\rm
e}^{-\sigma L^{\beta}})\leq{\rm e}^{-\alpha L^{\beta}}$

for all $E\in I$, $x\in\mathbb{Z}$ and $L\geq L_{0}$.

This Wegner estimate is not exactly similar to the one stated in [22], but it
is possible to use this version for multiscale analysis, as explained in [12]
and [22, Remark 4.6].

The last property depends on several parameters: $\theta>1$,
$E_{0}\in\mathbb{R}$ and $L_{0}\in 6\mathbb{N}$.

######  Property (ILSE($\theta$, $E_{0}$, $L_{0}$)).

(2.32)
${\mathbb{P}}\left\\{\left\|\Gamma_{0,L_{0}}R_{\omega,0,L_{0}}(E_{0})\chi_{0,L_{0}/3}\right\|\leqslant\frac{1}{L_{0}^{\theta}}\right\\}>1-\frac{1}{841}.$

These properties are the hypotheses of the bootstrap multiscale analysis.

######  Definition 2.19.

Given $E\in\mathbb{R}$, $x\in\mathbb{Z}$ and $L\in 6\mathbb{N}$ with
$E\notin\sigma(H_{\omega,x,L})$, we say that the box $\Lambda_{L}(x)$ is
$(\omega,m,E)$-regular for a given $m>0$ if

(2.33) $\left\|\Gamma_{x,L}R_{\omega,x,L}(E)\chi_{x,L/3}\right\|\leqslant{\rm
e}^{-mL}.$

In the following, we denote

$[L]_{3\mathbb{N}}=\sup\\{n\in 3\mathbb{N}|n\leqslant L\\}.$

######  Definition 2.20.

For $x$, $y\in\mathbb{Z}$, $L\in 3\mathbb{N}$, $m>0$ and $I\subset\mathbb{R}$
an interval, we denote

$R(m,L,I,x,y)=\left\\{\omega;\text{for every }E^{\prime}\in I\text{ either
}\Lambda_{L}(x)\text{ or }\Lambda_{L}(y)\text{ is
}(\omega,m,E^{\prime})\text{-regular.}\right\\}.$

The multiscale analysis region $\Sigma_{MSA}$ for
$\\{H_{\omega}\\}_{\omega\in\Omega}$ is the set of $E\in\Sigma$ for which
there exists some open interval $I\ni E$ such that, given any $\zeta$,
$0<\zeta<1$ and $\alpha$, $1<\alpha<\zeta^{-1}$, there is a length scale
$L_{0}\in 3\mathbb{N}$ and a mass $m>0$ so if we set
$L_{k+1}=[L_{k}^{\alpha}]_{3\mathbb{N}}$, $k=0,1,\ldots$, we have

$\mathbb{P}\left\\{R(m,L_{k},I,x,y)\right\\}\geqslant 1-e^{-L_{k}^{\zeta}}$

for all $k\in\mathbb{N}$, $x$, $y\in\mathbb{Z}^{d}$ with $|x-y|>L_{k}$.

######  Theorem 2.21 (Multiscale analysis - Theorem 5.4 p136 of [22]).

Let $\\{H_{\omega}\\}_{\omega\in\Omega}$ be a standard ergodic random operator
with (IAD) and properties (SLI), (NE) and (W) fulfilled in an open interval
$I$ and let $\Sigma$ be the almost sure spectrum of
$\\{H_{\omega}\\}_{\omega\in\Omega}$. Given $\theta>1$, for each $E\in I$
there exists a finite scale
$\mathcal{L}_{\theta}(E)=\mathcal{L}_{\theta}(E)>0$ bounded on compact
subintervals of $I$ such that, if for a given $E_{0}\in\Sigma\cap I$ we have
(ILSE)($\theta$, $E_{0}$, $L_{0}$) at some scale $L_{0}\in 3\mathbb{N}$ with
$L_{0}>\mathcal{L}_{\theta}(E_{0})$, then $E_{0}\in\Sigma_{MSA}$.

######  Theorem 2.22 (Localization - Theorem 6.1 p139 of [22]).

Let $\\{H_{\omega}\\}_{\omega\in\Omega}$ be a standard ergodic operator with
(IAD) and properties (SGEE) and (EDI) in an open interval $I$. Then,
$\\{H_{\omega}\\}_{\omega\in\Omega}$ exhibits both dynamical and Anderson
localization on $\Sigma_{MSA}\cap I$.

We can summarize in the following figure, the ingredients of a proof of
localization using multiscale analysis.

(2.34)
$\underbrace{\mathrm{(IAD)}+\mathrm{(SLI)}+\mathrm{(NE)}+\mathrm{(W)}+\mathrm{(ILSE)}}_{\Downarrow}$
$\qquad\qquad\qquad\qquad\
\underbrace{\mathrm{(MSA)}+\mathrm{(SGEE)}+\mathrm{(EDI)}}_{\Downarrow}$
$\qquad\qquad\qquad\qquad\ \overbrace{\mathrm{(AL)}+\mathrm{(DL)}}$

These theorems make it possible to prove Theorem 1.9 and Theorem 1.10, by
applying the multiscale analysis to the random family of operators
$\\{D_{\omega}\\}_{\omega\in\Omega}$ defined by (1.2).

####  2.4.2. The Wegner estimate

We begin with the Wegner estimate: we will get it from the regularity of the
integrated density of states. The proof is a straightforward adaptation of
what we have already done in [8] and [35]. As a consequence, we only give a
sketch of it.

We prove the following proposition.

######  Proposition 2.23.

Let $I$ be a compact interval included in an open interval $\tilde{I}$ on
which Assumption $\mathbf{(L^{(N)})}$ holds or on which Assumption
$\mathbf{(L^{(N-1)})}$ holds and $\gamma_{N}$ is identically $0$. Then (W) is
satisfied on $I$.

If $I$ is such an interval, we know from the previous sections that (2.3) is
satisfied on $I$. This implies the following lemma. It is proven in [35] in
the case $N=1$, but its generalization to higher $N$ is straightforward.

######  Lemma 2.24 ([35], Lemma 6.16).

There exists $\rho>0$ and $C<\infty$ such that, for every $E\in I$ and every
$\epsilon>0$, we have for $L\geq L_{0}$

(2.35) $\begin{split}&{\mathbb{P}}\\{\text{There exists
}E^{\prime}\in(E-\epsilon,E+\epsilon)\text{ and
}\phi\in\mathrm{Dom}\left(D_{\omega}^{(L)}\right),\|\phi\|=1,\text{ such that
}\\\ &(D_{\omega}^{(L)}-E^{\prime})\phi=0,|\phi^{\downarrow}(-\ell
L)|^{2}+|\phi^{\downarrow}(\ell L)|^{2}\leq\epsilon^{2}\\}\leq
CL\epsilon^{\rho}.\end{split}$

Moreover, hypotheses of Proposition 2.23 also implies Lemma 3 of [8], which is
_a priori_ only applicable random _Schrödinger_ operators, but can also be
applied to our setting since its proof only consists of algebraic
manipulations of transfer matrices.

######  Lemma 2.25 ([8], Lemma 3).

For all $p=1,\dots,N$, there exist $\xi_{1}>0$, $\delta>0$ and
$n_{1}\in\mathbb{N}$ such that, for every $E\in I$, $n\geq n_{1}$ and
$x\in\Lambda^{p}\mathbb{C}^{2N}$ with $\|x\|=1$, we have

(2.36)
${\mathbb{E}}\left(\|\Lambda^{p}\Phi_{E}(n,\omega)x\|^{-\delta}\right)\leq{\rm
e}^{-\xi_{1}n}.$

From these two lemmas, it is possible to prove Proposition 2.23 following line
by line the arguments of [8, Section 5] or [35, Section 6.2].

####  2.4.3. The Initial Length Scale Estimate

The last ingredient in order to apply a multiscale analysis scheme to our
models is an ILSE which allows to start the induction proof. We denote by
$\Lambda_{L}$ the interval $\Lambda_{L}(0)$. We will prove an estimate
stronger that (2.32), namely that there exists $L_{0}\in 3\mathbb{N}$ and
$C,c,\delta>0$ such that, for all $L\geq L_{0}$ and all $E\in I$,

(2.37)
${\mathbb{P}}\left\\{\left\|\Gamma_{0,L}R_{\omega,0,L}(E_{0})\chi_{0,L/3}\right\|\leqslant{\rm
e}^{-\delta L}\right\\}\geq 1-C{\rm e}^{-cL}.$

As in most previous works, we will use the fact that the resolvent of the
restricted operator has an integral kernel. We prove as in Equation (6.11) of
[35] that the resolvent $R_{\omega,0,L}(E)$ has an integral kernel, _i.e._
there exists a function $G_{\Lambda_{L}}^{\omega}$ such that for all $\psi$ in
$L^{2}(\Lambda_{L})$ and almost every $y\in\Lambda_{L}$

(2.38) $(D_{\omega}^{(L)}-E)\int_{-\ell L}^{\ell
L}G_{\Lambda_{L}}^{\omega}(E,x,y)\psi(x)\mathrm{d}x=\psi(y).$

Note that [35] provides an explicit formula for this integral kernel, but we
will not use it here. We know from the Schur test that

(2.39) $\left\|\Gamma_{0,L}R_{\omega,0,L}(E)\chi_{0,L/3}\right\|\leq
L\sup_{x,y\in\Lambda_{L}}|\Gamma_{0,L}(y)G_{\Lambda_{L}}^{\omega}(E,x,y)\chi_{0,L/3}(x)|.$

Our goal is therefore to bound $|G_{\Lambda_{L}}^{\omega}(E,x,y)|$ for $x$ and
$y$ respectively in the support of $\chi_{0,L/3}$ and $\Gamma_{0,L}$, with
probability exponentially close to $1$.

To this purpose, we give an explicit formula for $G_{\Lambda_{L}}^{\omega}$ at
these points. For $E\in I$, we consider the two functions
$\Phi_{\pm}=\left(\begin{smallmatrix}\Phi_{\pm}^{\uparrow}\\\
\Phi_{\pm}^{\downarrow}\end{smallmatrix}\right)$ with values in
$\mathcal{M}_{2\mathrm{N},\mathrm{N}}(\mathbb{C})$, satisfying
$D_{\omega}\Phi_{\pm}=E\Phi_{\pm}$ with

(2.40) $\Phi_{-}(-\ell L)=\left(\begin{smallmatrix}0\\\
I_{\mathrm{N}}\end{smallmatrix}\right)\ \mbox{ and }\ \Phi_{+}(\ell
L)=\left(\begin{smallmatrix}0\\\ I_{\mathrm{N}}\end{smallmatrix}\right).$

######  Lemma 2.26.

Let $\omega\in\Omega$ and let $x,y\in\Lambda_{L}$. Assume that
$\Phi_{+}^{\uparrow}(x)$, $\Phi_{+}^{\downarrow}(x)$,
$\Phi_{-}^{\uparrow}(x)$, and $\Phi_{-}^{\downarrow}(x)$ are invertible as
well as
$\Phi_{+}^{\downarrow}(x)\Phi_{+}^{\uparrow}(x)^{-1}-\Phi_{-}^{\downarrow}(x)\Phi_{-}^{\uparrow}(x)^{-1}$
and
$\Phi_{+}^{\uparrow}(x)\Phi_{+}^{\downarrow}(x)^{-1}-\Phi_{-}^{\uparrow}(x)\Phi_{-}^{\downarrow}(x)^{-1}$.
The Green kernel of $D_{\omega}^{(L)}$ is given by

(2.41) $G_{\Lambda_{L}}^{\omega}(E,x,y)=\left\\{\begin{array}[]{r l
}\Phi_{+}(y)\alpha_{+}(x)&\text{ for }x\leq y\\\
\Phi_{-}(y)\alpha_{-}(x)&\text{ for }x>y\end{array}\right.$

where

(2.42)
$\alpha_{+}(x):=\left(\begin{smallmatrix}\Phi_{+}^{\uparrow}(x)^{-1}\left(\Phi_{+}^{\downarrow}(x)\Phi_{+}^{\uparrow}(x)^{-1}-\Phi_{-}^{\downarrow}(x)\Phi_{-}^{\uparrow}(x)^{-1}\right)^{-1},\Phi_{+}^{\downarrow}(x)^{-1}\left(\Phi_{-}^{\uparrow}(x)\Phi_{-}^{\downarrow}(x)^{-1}-\Phi_{+}^{\uparrow}(x)\Phi_{+}^{\downarrow}(x)^{-1}\right)^{-1}\end{smallmatrix}\right)$

and

(2.43)
$\alpha_{-}(x):=\left(\begin{smallmatrix}\Phi_{-}^{\uparrow}(x)^{-1}\left(\Phi_{+}^{\downarrow}(x)\Phi_{+}^{\uparrow}(x)^{-1}-\Phi_{-}^{\downarrow}(x)\Phi_{-}^{\uparrow}(x)^{-1}\right)^{-1},\Phi_{-}^{\downarrow}(x)^{-1}\left(\Phi_{-}^{\uparrow}(x)\Phi_{-}^{\downarrow}(x)^{-1}-\Phi_{+}^{\uparrow}(x)\Phi_{+}^{\downarrow}(x)^{-1}\right)^{-1}\end{smallmatrix}\right)$

###### Proof.

By definition, (2.38) has to be satisfied for all $\psi\in L^{2}(\Lambda_{L})$
and almost all $x\in\Lambda_{L}$. We compute this expression explicitly,
assuming that $G_{\Lambda_{L}}^{\omega}$ has the form given by (2.41) with
$\alpha_{\pm}=(\alpha_{\pm}^{\leftarrow},\alpha_{\pm}^{\rightarrow})$. Using
the fact that $(D_{\omega}^{(L)}-E)\Phi_{\pm}=0$, we find that for all
$x\in\Lambda_{L}$

(2.44)
$\begin{pmatrix}\phi_{+}^{\uparrow}(x)\alpha_{+}^{\leftarrow}(x)-\phi_{-}^{\uparrow}(x)\alpha_{-}^{\leftarrow}(x)&\phi_{+}^{\uparrow}(x)\alpha_{+}^{\rightarrow}(x)-\phi_{-}^{\uparrow}(x)\alpha_{-}^{\rightarrow}(x)\\\
\phi_{+}^{\downarrow}(x)\alpha_{+}^{\leftarrow}(x)-\phi_{-}^{\downarrow}(x)\alpha_{-}^{\leftarrow}(x)&\phi_{+}^{\downarrow}(x)\alpha_{+}^{\rightarrow}(x)-\phi_{-}^{\downarrow}(x)\alpha_{-}^{\rightarrow}(x)\end{pmatrix}=\begin{pmatrix}0&-I_{N}\\\
I_{N}&0\end{pmatrix}$

Assuming the invertibility conditions given in the statement of the lemma, we
can solve this linear system explicitly, which gives the expressions (2.42)
and (2.43). ∎

We now introduce two events on which we will be able to bound the integral
kernel. Their probability will be controlled by either Proposition 2.28 or
Proposition 2.30. In order to estimate blocks of the products of transfer
matrices as in [24], we introduce, given a vector subspace $F$ of
$\mathbb{C}^{2N}$, the orthogonal projection onto $F$
$\pi_{F}:\mathbb{C}^{2N}\to F$ and we set

$\pi_{F}^{*}\ :\ \begin{array}[]{ccl}F&\to&\mathbb{C}^{2N}\\\
x&\mapsto&x\end{array}.$

For any integer $n\in[-L/3,L/3]$, any
$T\in\mathcal{M}_{2\mathrm{N}}(\mathbb{C})$ and any vector subspace
$F\subset\mathbb{C}^{2N}$, we define

(2.45) $\displaystyle\Omega_{\epsilon}^{F}[T]:=$
$\displaystyle\left\\{\max_{1\leq p\leq N}\left(\left|\frac{1}{\ell|n-L|}\log
s_{p}\left(T\right)-\gamma_{p}(E)\right|+\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.\left.\left|\frac{1}{\ell|n-L|}\log
s_{p}\left(T\pi_{F}^{*}\right)-\gamma_{p}(E)\right|\right)\leq\frac{\epsilon}{100N}\right\\}.$

where $s_{p}(\cdot)$ denotes the $p$-th singular value of the considered
matrix. Let

(2.46) $F_{+}:=\left\\{\left(\begin{smallmatrix}u\\\
0\end{smallmatrix}\right)\ |u\in\mathbb{C}^{N}\right\\}\subset\mathbb{C}^{2N}\
\mbox{ and }\ F_{-}:=\left\\{\left(\begin{smallmatrix}0\\\
v\end{smallmatrix}\right)\ |v\in\mathbb{C}^{N}\right\\}\subset\mathbb{C}^{2N}$

and for any $n\in\mathbb{Z}$,

(2.47) $F_{n}:=\left\\{\left(\begin{smallmatrix}u\\\
v\end{smallmatrix}\right)\in\mathbb{C}^{2N}\ \big{|}\
u=-\Phi_{+}^{\downarrow}(\ell n)(\Phi_{+}^{\uparrow}(\ell n))^{-1}v\right\\}.$

Then, set

(2.48) $\Omega_{\epsilon}(n):=\Omega_{\epsilon}^{F_{n}}[(T_{\ell L}^{\ell
n}(E))^{*}]\cap\Omega_{\epsilon}^{F_{+}}[(T_{\ell L}^{\ell
n}(E))^{*}]\cap\Omega_{\epsilon}^{F_{+}}[T_{\ell L}^{\ell
n}(E)]\cap\Omega_{\epsilon}^{F_{-}}[(T_{\ell L}^{\ell
n}(E))^{*}]\cap\Omega_{\epsilon}^{F_{-}}[T_{\ell L}^{\ell n}(E)].$

Note that if the transfer matrices are in
$\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$, the vector subspace $F_{n}$ is
$J$-Lagrangian, _i.e._ it is orthogonal to itself for $J$ and of dimension
$N$. If the transfer matrices are in $\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$,
$F_{n}$ is $(J,S)$-Lagrangian, _i.e._ it is orthogonal to itself for $J$ and
for $S$ and it is of dimension $N$. Also note that $F_{+}$ and $F_{-}$ are
always $J$-Lagrangian, but they are not $(J,S)$-Lagrangian. To bypass this
difficulty in the case of $N=2d$ even, we write $F_{+}=F_{+}^{(+)}\oplus
F_{+}^{(-)}$ and $F_{-}=F_{-}^{(+)}\oplus F_{-}^{(-)}$ where,

(2.49) $F_{+}^{(\pm)}:=\left\\{\left(\begin{smallmatrix}x_{1}\\\ \pm x_{1}\\\
\vdots\\\ x_{d}\\\ \pm x_{d}\\\ 0\\\ \vdots\\\ 0\end{smallmatrix}\right)\ ;\
(x_{1},\ldots,x_{d})\in\mathbb{C}^{d}\right\\}\ \mbox{ and }\
F_{-}^{(\pm)}:=\left\\{\left(\begin{smallmatrix}0\\\ \vdots\\\ 0\\\ x_{1}\\\
\pm x_{1}\\\ \vdots\\\ x_{d}\\\ \pm x_{d}\end{smallmatrix}\right)\ ;\
(x_{1},\ldots,x_{d})\in\mathbb{C}^{d}\right\\}.$

Then all the $F_{\pm}^{(\pm)}$ are $(J,S)$-Lagrangian.

One then define the event

$\displaystyle\tilde{\Omega}_{\epsilon}(n):=\Omega_{\epsilon}^{F_{x}}[(T_{\ell
L}^{\ell n}(E))^{*}]$
$\displaystyle\cap\Omega_{\epsilon}^{F_{+}^{(+)}}[(T_{\ell L}^{\ell
n}(E))^{*}]\cap\Omega_{\epsilon}^{F_{+}^{(+)}}[T_{\ell L}^{\ell
n}(E)]\cap\Omega_{\epsilon}^{F_{+}^{(-)}}[(T_{\ell L}^{\ell
n}(E))^{*}]\cap\Omega_{\epsilon}^{F_{+}^{(-)}}[T_{\ell L}^{\ell n}(E)]$ (2.50)
$\displaystyle\cap\Omega_{\epsilon}^{F_{-}^{(+)}}[(T_{\ell L}^{\ell
n}(E))^{*}]\cap\Omega_{\epsilon}^{F_{-}^{(+)}}[T_{\ell L}^{\ell
n}(E)]\cap\Omega_{\epsilon}^{F_{-}^{(-)}}[(T_{\ell L}^{\ell
n}(E))^{*}]\cap\Omega_{\epsilon}^{F_{-}^{(-)}}[T_{\ell L}^{\ell n}(E)].$

We can control $||G_{\Lambda_{L}}^{\omega}(E,x,y)||$ on these two events.

######  Proposition 2.27.

Let $I\subset\mathbb{R}$ a compact interval on which for every $E\in I$,
$\inf_{E\in I}\gamma_{N}(E)>0$. Assume either that

  1. (1)

$\forall E\in I$, $G(E)\in\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$ and
$\omega\in\Omega_{\epsilon}:=\cap_{n\in[-L/3,L/3]}\Omega_{\epsilon}(n)$ or

  2. (2)

$\forall E\in I$, $G(E)\in\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, $N$ is even
and
$\omega\in\tilde{\Omega}_{\epsilon}:=\cap_{n\in[-L/3,L/3]}\tilde{\Omega}_{\epsilon}(n).$

Let us choose $\epsilon\in\left(0,\inf_{E\in
I}\frac{\gamma_{N}(E)}{3}\right)$. Then, there exists $C>0$ such that, for $L$
large enough, for all $E\in I$, $x\in\operatorname{supp}\chi_{0,L/3}$ and
$y\in\operatorname{supp}\Gamma_{0,L}$,

(2.51) $||G_{\Lambda_{L}}^{\omega}(E,x,y)||\leq C{\rm
e}^{-2(\gamma_{N}(E)-\frac{7}{4}\epsilon)\ell L}.$

###### Proof.

We give a complete proof in case (1). Then at the end we will explain the
slight differences in the proof when we assume that we are in case (2).

We will prove the proposition only for $y$ in $[\ell L-3\ell,\ell L-\ell]$.
The proof for $y$ in $[-\ell L+\ell,-\ell L+3\ell]$ is similar. Since, in this
case, $x<y$, the Green kernel will be given by the first line of (2.41).

We begin with proving that on $\Omega_{\epsilon}$, for all $x\in[-\ell
L/3,\ell L/3]$, the matrices $\Phi^{\uparrow}_{+}(x)$ and
$\Phi_{+}^{\downarrow}(x)$ are invertible and we estimate their inverse. It
corresponds to bounding from below their $N$-th singular value.The reader who
is not familiar with inequalities on singular values can refer to Chapter 3 of
[18].

By definition (recall (1.5)), for any $x$ in $\Lambda_{L}$,

$\begin{pmatrix}\Phi_{+}^{\uparrow}(x)\\\
\Phi_{+}^{\downarrow}(x)\end{pmatrix}=T_{\ell L}^{x}(E)\begin{pmatrix}0\\\
I_{N}\end{pmatrix}.$

We write the singular value decomposition of $T_{\ell L}^{x}(E)$:

(2.52) $T_{\ell L}^{x}(E)=U^{(x)}\Sigma^{(x)}V^{(x)},$

where $U^{(x)}$ and $V^{(x)}$ are unitary and, since we have a symplectic
matrix, we can write
$\Sigma^{(x)}=\left(\begin{smallmatrix}\Sigma_{+}^{(x)}&0\\\
0&\Sigma_{-}^{(x)}\end{smallmatrix}\right)$ with
$\Sigma_{+}^{(x)}=\operatorname{diag}(s_{1}(T_{\ell
L}^{x}),\dots,s_{N}(T_{\ell L}^{x}))$ and
$\Sigma_{-}^{(x)}=\operatorname{diag}(1/s_{1}(T_{\ell
L}^{x}),\dots,1/s_{N}(T_{\ell L}^{x}))$ with $s_{1}(T_{\ell
L}^{x})\geq\dots\geq s_{N}(T_{\ell L}^{x})\geq 1$. We can write a block
decomposition for $U^{(x)}$ and $V^{(x)}$:
$U^{(x)}=\left(\begin{smallmatrix}U_{11}^{(x)}&U_{12}^{(x)}\\\
U_{21}^{(x)}&U_{22}^{(x)}\end{smallmatrix}\right)$ and
$V^{(x)}=\left(\begin{smallmatrix}V_{11}^{(x)}&V_{12}^{(x)}\\\
V_{21}^{(x)}&V_{22}^{(x)}\end{smallmatrix}\right)$. We have then

$\begin{pmatrix}\Phi_{+}^{\uparrow}(x)\\\
\Phi_{+}^{\downarrow}(x)\end{pmatrix}=\begin{pmatrix}U_{11}^{(x)}\Sigma_{+}^{(x)}V_{12}^{(x)}+U_{12}^{(x)}\Sigma_{-}^{(x)}V_{22}^{(x)}\\\
U_{21}^{(x)}\Sigma_{+}^{(x)}V_{12}^{(x)}+U_{22}^{(x)}\Sigma_{-}^{(x)}V_{22}^{(x)}\end{pmatrix}.$

As a consequence, we can write that

(2.53) $\displaystyle s_{N}\left(\Phi_{+}^{\uparrow}(x)\right)$
$\displaystyle\geq
s_{N}\left(U_{11}^{(x)}\Sigma_{+}^{(x)}V_{12}^{(x)}\right)-\|U_{12}^{(x)}\Sigma_{-}^{(x)}V_{22}^{(x)}\|$
(2.54) $\displaystyle\geq
s_{N}\left(U_{11}^{(x)}\right)s_{N}\left(\Sigma_{+}^{(x)}\right)s_{N}\left(V_{12}^{(x)}\right)-\|\Sigma_{-}^{(x)}\|,$

where we used in the last line that the blocks $U_{ij}^{(x)}$, $V_{ij}^{(x)}$
have norm less than 1.

We now estimate $s_{N}\left(\Sigma_{+}^{(x)}\right)$, which is also
$\|\Sigma_{-}^{(x)}\|^{-1}$, on the event $\Omega_{\epsilon}$. Let $n$ be the
unique integer such that $x\in[(n-1)\ell,n\ell)$. We first remark that, for
all $p=1,\dots,2N$,

(2.55) $s_{p}\left(T_{\ell L}^{x}\right)\geq s_{p}\left(T_{\ell L}^{\ell
n}\right)s_{2N}\left(T_{\ell n}^{x}\right)=s_{p}\left(T_{\ell L}^{\ell
n}\right)\|T_{\ell n}^{x}\|^{-1}.$

But we know from Lemma 2.7 that there exists a constant $C>0$, independent of
$E$, $x$, $\omega$ and $L$, such that $\|T_{\ell n}^{x}\|\leq C$. As a
consequence, $s_{p}\left(T_{\ell L}^{x}\right)\geq C^{-1}s_{p}\left(T_{\ell
L}^{\ell n}\right)$. Consequently, on the event $\Omega_{\epsilon}(n)$,
$s_{N}\left(T_{\ell L}^{x}\right)\geq C^{-1}{\rm
e}^{(\gamma_{N}(E)-\epsilon)\ell|n-L|}$ and $\|\Sigma_{-}^{(x)}\|\leq C{\rm
e}^{-(\gamma_{N}(E)-\epsilon)\ell|n-L|}$, so

(2.56) $s_{N}\left(\Phi_{+}^{\uparrow}(x)\right)\geq C^{-1}{\rm
e}^{(\gamma_{N}(E)-\epsilon)\ell|n-L|}s_{N}\left(U_{11}^{(x)}\right)s_{N}\left(V_{12}^{(x)}\right)-C{\rm
e}^{-(\gamma_{N}(E)-\epsilon)\ell|n-L|}$

and, similarly,

(2.57) $s_{N}\left(\Phi_{+}^{\downarrow}(x)\right)\geq C^{-1}{\rm
e}^{(\gamma_{N}(E)-\epsilon)\ell|n-L|}s_{N}\left(U_{21}^{(x)}\right)s_{N}\left(V_{12}^{(x)}\right)-C{\rm
e}^{-(\gamma_{N}(E)-\epsilon)\ell|n-L|}.$

We are left with controlling $s_{N}\left(U_{11}^{(x)}\right)$,
$s_{N}\left(U_{21}^{(x)}\right)$ and $s_{N}\left(V_{12}^{(x)}\right)$.
Recalling the notation introduced in (2.46), we have that
$U_{11}^{(x)}=\pi_{F_{+}}^{*}U^{(x)}\pi_{F_{+}}$,
$U_{21}^{(x)}=\pi_{F_{-}}^{*}U^{(x)}\pi_{F_{+}}$ and
$V_{12}^{(x)}=\pi_{F_{+}}^{*}V^{(x)}\pi_{F_{-}}$. Since the sets $F_{+}$ and
$F_{-}$ as defined in (2.46) are $J$-Lagrangian, we can prove as in Claim 3.4
and Remark 3.5 of [24] that, on $\Omega_{\epsilon}(n)$, we have for $L$ large
enough (depending only on $\epsilon$ and $\ell$)

(2.58) $s_{N}\left(V_{12}^{(x)}\right)\geq{\rm
e}^{-\frac{\epsilon}{25}\ell|n-L|}\text{,
}s_{N}\left(U_{21}^{(x)}\right)\geq{\rm
e}^{-\frac{\epsilon}{25}\ell|n-L|}\text{ and
}s_{N}\left(U_{11}^{(x)}\right)\geq{\rm e}^{-\frac{\epsilon}{25}\ell|n-L|}.$

More precisely, the first inequality corresponds to the event
$\Omega_{\epsilon}^{F_{-}}[T_{\ell L}^{\ell n}(E)]$, the second to the event
$\Omega_{\epsilon}^{F_{-}}[(T_{\ell L}^{\ell n}(E))^{*}]$ and the third to
$\Omega_{\epsilon}^{F_{+}}[(T_{\ell L}^{\ell n}(E))^{*}]$.

As a consequence, on $\Omega_{\epsilon}(n)$,

(2.59) $s_{N}\left(\Phi_{+}^{\uparrow\downarrow}(x)\right)\geq C^{-1}{\rm
e}^{(\gamma_{N}(E)-\frac{27\epsilon}{25})\ell|n-L|}-C{\rm
e}^{-(\gamma_{N}(E)-\epsilon)\ell|n-L|}.$

For $L$ large enough (depending here again only on $C$, $\epsilon$ and
$\ell$), one gets on $\Omega_{\epsilon}(n)$, using that $|n-L|\geq 2L/3$,

(2.60) $s_{N}\left(\Phi_{+}^{\uparrow\downarrow}(x)\right)\geq{\rm
e}^{(\gamma_{N}(E)-2\epsilon)\tfrac{2\ell L}{3}}>0.$

In particular, $\Phi_{+}^{\uparrow}(x)$ and $\Phi_{+}^{\downarrow}(x)$ are
invertible.

The next step to be able to apply Lemma 2.26 is to prove that
$\Phi_{+}^{\downarrow}(x)\Phi_{+}^{\uparrow}(x)^{-1}-\Phi_{-}^{\downarrow}(x)\Phi_{-}^{\uparrow}(x)^{-1}$
and
$\Phi_{+}^{\uparrow}(x)\Phi_{+}^{\downarrow}(x)^{-1}-\Phi_{-}^{\uparrow}(x)\Phi_{-}^{\downarrow}(x)^{-1}$
are invertible. By (2.40),
$\Phi_{+}^{\downarrow}(x)\Phi_{+}^{\uparrow}(x)^{-1}=(T_{\ell
L}^{x})_{22}((T_{\ell L}^{x})_{12})^{-1}$ and
$\Phi_{-}^{\downarrow}(x)\Phi_{-}^{\uparrow}(x)^{-1}=(T_{-\ell
L}^{x})_{22}((T_{-\ell L}^{x})_{12})^{-1}$. It does not exactly correspond to
the blocks involved in $X^{+}$ and $X^{-}$ in Equation (33) of [24] since we
did not use the same boundary conditions, but we will be able to carry on the
same proof as [24]. Since the sets $F_{n}$, $F_{+}$ and $F_{-}$ are
$J$-Lagrangian, we prove as in Equation (33) of [24] that there exists $C>0$
such that on $\Omega_{\epsilon}(n)$

(2.61)
$s_{N}\left(\Phi_{+}^{\downarrow}(x)\Phi_{+}^{\uparrow}(x)^{-1}-\Phi_{-}^{\downarrow}(x)\Phi_{-}^{\uparrow}(x)^{-1}\right)\geq
C^{-1}{\rm e}^{-\frac{\epsilon}{2}\ell L}$

as well as the same result for
$\Phi_{+}^{\uparrow}(x)\Phi_{+}^{\downarrow}(x)^{-1}-\Phi_{-}^{\uparrow}(x)\Phi_{-}^{\downarrow}(x)^{-1}$.
Note that the constant $C>0$ appears as we estimate the $N$-th singular value
of the matrix at point $x$ by its value at $n$ its integer part and the norm
of the difference which is bounded in $L$ times $s_{N}(U_{21})$ which is lower
bounded by an exponentially small term. Note also that in the proof appears
$s_{N}\left(V_{11}^{(x)}\right)$ which is controlled on
$\Omega_{\epsilon}^{F_{+}}[T_{\ell L}^{\ell n}(E)]$.

Consequently, these two matrices are invertible and the norm of their inverse
is smaller than $C{\rm e}^{\frac{\epsilon}{2}\ell L}$.

Finally, we remark that, for $y\in[\ell L-3\ell,\ell L-\ell]$, we have by
Lemma 2.7,

(2.62) $||\Phi_{+}^{\uparrow}(y)||^{2}+||\Phi_{+}^{\downarrow}(y)||^{2}\leq
N\exp\left(2\int_{y}^{\ell L}|V_{\omega}(t)|\mathrm{d}t\right)\leq C,$

with $C$ independent of $E$, $\omega$ and $L$.

Together with (2.60) and (2.61), it gives that on $\Omega_{\epsilon}$

(2.63) $||G_{\Lambda_{L}}^{\omega}(E,x,y)||\leq C{\rm
e}^{-2(\gamma_{N}(E)-\frac{7}{4}\epsilon)\ell L},$

which concludes the proof of the proposition under the assumptions of Theorem
1.9.

Now let us briefly explain how to adapt the proof when we are in case (2).
First, remark that if the transfer matrices are in
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, then $F_{n}$ is a $(J,S)$-Lagrangian
since it depends on the transfer matrices through two of their blocks.
Moreover, all the $F_{\pm}^{(\pm)}$ are $(J,S)$-Lagrangian. Since,
$\pi_{F_{+}}^{*}=\pi_{F_{+}^{(+)}}^{*}+\pi_{F_{+}^{(-)}}^{*}$, one has
inequalities such as :

(2.64)
$s_{N}(U_{11}^{(x)})=s_{N}(\pi_{F_{+}}U^{(x)}\pi_{F_{+}}^{*})=s_{N}(\pi_{F_{+}}U^{(x)}\pi_{F_{+}^{(+)}}^{*}+\pi_{F_{+}}U^{(x)}\pi_{F_{+}^{(-)}}^{*})\geq
s_{d}(\pi_{F_{+}}U^{(x)}\pi_{F_{+}^{(+)}}^{*})\geq{\rm
e}^{-\frac{\epsilon}{25}\ell|n-L|}.$

The first inequality comes from the fact that all the singular values are of
multiplicity $2$ using the decomposition $(4)$ of Proposition A.1 hence
$s_{d}(\pi_{F_{+}}U^{(x)}\pi_{F_{+}^{(+)}}^{*})$ is the $2d$-th singular value
of $\pi_{F_{+}}U^{(x)}\pi_{F_{+}^{(+)}}^{*}$. The second inequality comes from
the fact that $F_{+}^{(+)}$ is a subspace of $F_{+}$ for which we can use
Claim 3.4 and Remark 3.5 of [24]. The same kind of inequalities allow to
obtain a lower bound for
$s_{N}(U_{21}^{(x)}-\Phi_{+}^{\downarrow}(x)(\Phi_{+}^{\uparrow}(x))^{-1}U_{11}^{(x)})$
which is needed to obtain a lower bound as (2.61) under the assumptions of
case (2). Having in mind these specific properties when the transfer matrices
are in $\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, one proves (2.51) in this
case, the same way as it was done when they are in
$\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$. ∎

In order to estimate the probability of $\Omega_{\epsilon}$ and
$\tilde{\Omega}_{\epsilon}$, we prove a Large Deviation Property for the
singular values of the products of transfer matrices. We need to do this
separately for $\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$ and
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$.

######  Proposition 2.28.

We fix a compact interval $I\subset\mathbb{R}$. We assume that for every $E\in
I$:

  1. (1)

the Furstenberg group $G(E)$ is included in
$\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$ ;

  2. (2)

for every $p\in\\{1,\ldots,N\\}$, $G(E)$ is $J$-$L_{p}$-strongly irreducible.

Then for all $\epsilon>0$ and all $E\in I$, there exist $C(\epsilon,E)>0$ and
$c(\epsilon,E)>0$ such that, for all $p\in\\{1,\dots,N\\}$, any $J$-Lagrangian
subspace $F$ and all integers $m,n$,

(2.65) $\mathbb{P}\left(\left\\{\left|\frac{1}{\ell(n-m)}\log
s_{p}\left(T_{\ell m}^{\ell
n}(E)\right)-\gamma_{p}(E)\right|\geq\epsilon\right\\}\right)\leq
C(\epsilon,E){\rm e}^{-c(\epsilon,E)\ell|n-m|}$

and

(2.66) $\mathbb{P}\left(\left\\{\left|\frac{1}{\ell(n-m)}\log
s_{p}\left(T_{\ell m}^{\ell
n}(E)\pi_{F}^{*}\right)-\gamma_{p}(E)\right|\geq\epsilon\right\\}\right)\leq
C(\epsilon,E){\rm e}^{-c(\epsilon,E)\ell|n-m|}.$

######  Remark 2.29.

The constants $C(\epsilon,E)$ and $c(\epsilon,E)$ depend _a priori_ on $E$ and
$\epsilon$ but they can be taken uniform in $\epsilon$ as it tends to $0$ and
uniform in $E$ on the compact interval $I$. This is one of the reasons why we
need to take the interval $I$ compact.

###### Proof.

First of all, recall that $\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$ can be
identified to $\mathrm{Sp}_{2\mathrm{N}}(\mathbb{R})$ using the following
application which split the real and imaginery parts of the matrices in
$\mathcal{M}_{\mathrm{2N}}(\mathbb{C})$:

$\begin{array}[]{cll}\mathcal{M}_{\mathrm{2N}}(\mathbb{C})&\to&\mathcal{M}_{\mathrm{4N}}(\mathbb{R})\\\\[5.69054pt]
A+iB&\mapsto&\left(\begin{smallmatrix}A&-B\\\
B&A\end{smallmatrix}\right).\end{array}$

In this identification, all the multiplicities of the Lyapunov exponents and
the singular values are doubled. Hence, we can use freely all the results
concerning i.i.d. sequences of random matrices in the real symplectic group.

Recall that for a symplectic matrix
$M\in\mathrm{Sp}_{\mathrm{N}}^{*}(\mathbb{C})$, $s_{p}(M^{-1})=s_{p}(M)$ for
every $p\in\\{1,\ldots 2N\\}$. Hence one can assume that $m\leq n$ without
loss of generality since $(T_{\ell m}^{\ell n}(E))^{-1}=T_{\ell n}^{\ell
m}(E)$.

Let $p\in\\{1,\dots,N\\}$. As in the proof of Proposition 2.5, for each
$M\in\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$, we denote by $\widehat{M}$ the
matrix of $\operatorname{GL}_{k}(\mathbb{R})$ such that

(2.67) $\widehat{M}_{ij}=\langle f_{i},\Lambda^{p}Mf_{j}\rangle.$

where $k$ is the dimension of $J\mbox{-}L_{p}$ and $(f_{1},\dots,f_{k})$ is an
orthonormal basis of $J$-$L_{p}$, with $f_{1}=e_{1}\wedge\cdots\wedge e_{p}$.
Let us now denote by $\widehat{G(E)}$ the subgroup of
$\operatorname{GL}_{k}(\mathbb{R})$ which is generated by the matrices
$\widehat{M}$ for $M\in G(E)$. Then, since $G(E)$ is $J$-$L_{p}$-strongly
irreducible, $\widehat{G(E)}$ is strongly irreducible. Hence, applying [5,
Theorem A.V.6.2], one gets the existence of $\alpha>0$ such that for any
$\epsilon>0$ and any $\bar{x}\in\mathsf{P}(J\mbox{-}L_{p})$,

(2.68) $\limsup_{|n-m|\to+\infty}\frac{1}{\ell|n-m|}\log\
\mathbb{P}\left(\left|\frac{1}{\ell|n-m|}\log(||\Lambda^{p}T_{\ell m}^{\ell
n}(E)\bar{x}||)-(\gamma_{1}+\cdots+\gamma_{p})(E)\right|>\epsilon\right)\leq-\alpha$

and

(2.69) $\limsup_{|n-m|\to+\infty}\frac{1}{\ell|n-m|}\log\
\mathbb{P}\left(\left|\frac{1}{\ell|n-m|}\log(||\Lambda^{p}T_{\ell m}^{\ell
n}(E)||)-(\gamma_{1}+\cdots+\gamma_{p})(E)\right|>\epsilon\right)\leq-\alpha.$

Indeed, since the function $x\mapsto V_{\omega}^{(n)}(x)$ is uniformly bounded
in $x$, $n$ and $\omega$, and since the support of the common law of the
transfer matrices is bounded, the assumption of finiteness of the integral in
[5, Theorem A.V.6.2] is satisfied.

Now, let us take $F$ a $J$-Lagrangian subspace of $\mathbb{C}^{2N}$. Then,

$\displaystyle||\Lambda^{p}(T_{\ell m}^{\ell n}(E)\pi_{F}^{*})||$
$\displaystyle=\sup_{\begin{subarray}{c}u_{1}\wedge\cdots\wedge
u_{p}\in\mathsf{P}(J\mbox{-}L_{p})\\\ u_{i}\in
F\end{subarray}}||(\Lambda^{p}T_{\ell m}^{\ell n}(E))(u_{1}\wedge\cdots\wedge
u_{p})||$ $\displaystyle=||\Lambda^{p}T_{\ell m}^{\ell n}(E)\ \bar{u}||,\
\mbox{ for some }\bar{u}\in\mathsf{P}(J\mbox{-}L_{p})$

since the supremum is attained by compactness of $\mathsf{P}(J\mbox{-}L_{p})$.
Hence, (2.68) rewrites,

(2.70) $\limsup_{|n-m|\to+\infty}\frac{1}{\ell|n-m|}\log\
\mathbb{P}\left(\left|\frac{1}{\ell|n-m|}\log(||\Lambda^{p}(T_{\ell m}^{\ell
n}(E)\pi_{F}^{*})||)-(\gamma_{1}+\cdots+\gamma_{p})(E)\right|>\epsilon\right)\leq-\alpha$

for any $J$-Lagrangian $F$. Let, for $n,m\in\mathbb{Z}$,
$p\in\\{1,\ldots,N\\}$, $\epsilon>0$ and $F$ a $J$-Lagrangian,

$A_{n,m,p}(\epsilon,F)=\left\\{\left|\frac{1}{\ell|n-m|}\log(||\Lambda^{p}(T_{\ell
m}^{\ell
n}(E)\pi_{F}^{*})||)-(\gamma_{1}+\cdots+\gamma_{p})(E)\right|>\epsilon\right\\},$

$A_{n,m,p}(\epsilon)=\left\\{\left|\frac{1}{\ell|n-m|}\log(||\Lambda^{p}T_{\ell
m}^{\ell n}(E)||)-(\gamma_{1}+\cdots+\gamma_{p})(E)\right|>\epsilon\right\\},$

$B_{n,m,p}(\epsilon,F)=\left\\{\left|\frac{1}{\ell|n-m|}\left(\log(||\Lambda^{p}(T_{\ell
m}^{\ell n}(E)\pi_{F}^{*})||)-\log(||\Lambda^{p-1}(T_{\ell m}^{\ell
n}(E)\pi_{F}^{*})||)\right)-\gamma_{p}(E)\right|>\epsilon\right\\}$

and

$B_{n,m,p}(\epsilon)=\left\\{\left|\frac{1}{\ell|n-m|}\left(\log(||\Lambda^{p}T_{\ell
m}^{\ell n}(E)||)-\log(||\Lambda^{p-1}T_{\ell m}^{\ell
n}(E)||)\right)-\gamma_{p}(E)\right|>\epsilon\right\\}.$

Then, one has

(2.71) $B_{n,m,p}(2\epsilon)\subset A_{n,m,p}(\epsilon)\cap
A_{n,m,p-1}(\epsilon)\ \mbox{ and }B_{n,m,p}(2\epsilon,F)\subset
A_{n,m,p}(\epsilon,F)\cap A_{n,m,p-1}(\epsilon,F).$

Since for any $p\in\\{1,\ldots,N\\}$, $||\Lambda^{p}T_{\ell m}^{\ell
n}(E)||=s_{1}(T_{\ell m}^{\ell n}(E))\cdots s_{p}(T_{\ell m}^{\ell n}(E))$,
combining (2.71) and (2.69) one gets (2.65). One also has, for any
$J$-Lagrangian $F$, $||\Lambda^{p}(T_{\ell m}^{\ell
n}(E)\pi_{F}^{*})||=s_{1}(T_{\ell m}^{\ell n}(E)\pi_{F}^{*})\cdots
s_{p}(T_{\ell m}^{\ell n}(E)\pi_{F}^{*})$. Hence, combining (2.71) and (2.70)
one gets (2.66). This achieves the proof. ∎

We then prove the counterpart of Proposition 2.28 in the case covered by
Theorem 1.10.

######  Proposition 2.30.

We fix a compact interval $I\subset\mathbb{R}$. We assume that for every $E\in
I$:

  1. (1)

the Furstenberg group $G(E)$ is included in
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ ;

  2. (2)

for every $2p\in\\{1,\ldots,N\\}$, $G(E)$ is $(J,S)$-$L_{2p}$-strongly
irreducible.

Then for all $\epsilon>0$ and all $E\in I$, there exist $C(\epsilon,E)>0$ and
$c(\epsilon,E)>0$ such that, for all $p\in\\{1,\dots,N\\}$, any
$(J,S)$-Lagrangian subspace $F$ and all integers $m,n$,

(2.72) $\mathbb{P}\left(\left\\{\left|\frac{1}{\ell|n-m|}\log
s_{p}\left(T_{\ell m}^{\ell
n}(E)\right)-\gamma_{p}(E)\right|\geq\epsilon\right\\}\right)\leq
C(\epsilon,E){\rm e}^{-c(\epsilon,E)\ell|n-m|}$

and

(2.73) $\mathbb{P}\left(\left\\{\left|\frac{1}{\ell|n-m|}\log
s_{p}\left(T_{\ell m}^{\ell
n}(E)\pi_{F}^{*}\right)-\gamma_{p}(E)\right|\geq\epsilon\right\\}\right)\leq
C(\epsilon,E){\rm e}^{-c(\epsilon,E)\ell|n-m|}.$

###### Proof.

The proof is very similar to the proof of Proposition 2.28. First, we
introduce $\widetilde{G(E)}$ defined as in the proof of Proposition 2.5. Then,
since $G(E)$ is $(J,S)$-$L_{2p}$-strongly irreducible, $\widehat{G(E)}$ is
strongly irreducible and we can again apply [5, Theorem A.V.6.2] to get for
each $2p\in\\{1,\ldots,N\\}$, the analogous of (2.69) for $\Lambda^{2p}$
instead of $\Lambda^{p}$ and the same for (2.68) for any
$\bar{x}\in\mathsf{P}((J,S)\mbox{-}L_{2p})$. Hence, one gets (2.70) for any
$(J,S)$-Lagrangian $F$ and for $2p$ instead of $p$. Then, one has, for any
$n,m\in\mathbb{Z}$, $2p\in\\{1,\ldots,N\\}$, $\epsilon>0$ and $F$ a
$(J,S)$-Lagrangian,

(2.74) $B_{n,m,2p}(2\epsilon)\subset A_{n,m,2p}(\epsilon)\cap
A_{n,m,2p-2}(\epsilon)\ \mbox{ and }B_{n,m,2p}(2\epsilon,F)\subset
A_{n,m,2p}(\epsilon,F)\cap A_{n,m,2p-2}(\epsilon,F).$

One conclude as in the proof of Proposition 2.28 using that for each
$2p\in\\{1,\ldots,N\\}$ and each $(J,S)$-Lagrangian $F$, $s_{2p}(T_{\ell
m}^{\ell n}(E))=s_{2p+1}(T_{\ell m}^{\ell n}(E))$, $s_{2p}(T_{\ell m}^{\ell
n}(E)\pi_{F}^{*})=s_{2p+1}(T_{\ell m}^{\ell n}(E)\pi_{F}^{*})$ and
$\gamma_{2p}(E)=\gamma_{2p+1}(E)$ since the transfer matrices are in
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$. ∎

###### Proof of the ILSE (2.37).

Assume either the assumptions of Theorem 1.9 or those of Theorem 1.10.
Applying Proposition 2.28 for the $J$-Lagrangian spaces $F_{n}$, $F_{+}$ and
$F_{-}$ one gets that there exist $C(\epsilon,E)>0$ and $c(\epsilon,E)>0$ such
that,

(2.75) $\forall\epsilon>0,\ \forall n\in\Lambda_{\frac{L}{3}},\
\mathbb{P}\left(\Omega_{\epsilon}(n)\right)\leq C(\epsilon,E){\rm
e}^{-c(\epsilon,E)\ell|n-L|}$

from which we deduce, since there is only a finite number of real numbers
$n\ell$ in $\Lambda_{\frac{L}{3}}$, that there exist $C>0$ and $c>0$ such
that,

(2.76) $\forall\epsilon>0,\ \mathbb{P}\left(\Omega_{\epsilon}\right)\leq C{\rm
e}^{-c\frac{4}{3}\ell L}$

remembering that we can choose the constants $C(\epsilon,E)$ and
$c(\epsilon,E)$ uniform in $\epsilon$ when it tends to $0$ and uniform in
$E\in I$ by compactness of $I$.

Applying Proposition 2.30 for the $(J,S)$-Lagrangian spaces $F_{n}$,
$F_{+}^{\pm}$ and $F_{-}^{\pm}$ one gets also that there exist $C>0$ and $c>0$
such that,

(2.77) $\forall\epsilon>0,\
\mathbb{P}\left(\tilde{\Omega}_{\epsilon}\right)\leq C{\rm
e}^{-c\frac{4}{3}\ell L}$

Proposition 2.27 together with (2.39) and (2.76) or (2.77) (up to a passage to
the complementary sets of $\Omega_{\epsilon}$ and $\tilde{\Omega}_{\epsilon}$)
lead to (2.37) under either assumptions of Theorem 1.9 or Theorem 1.10. ∎

###### Proof of Theorem 1.9 and Theorem 1.10.

In view of Theorems 2.21 and 2.22, proving Theorem 1.9 and Theorem 1.10
reduces to prove all the hypotheses stated in Section 2.4.1 on interval $I$.
It is easy to see that the restriction of the operators to intervals with
Dirichlet boundary conditions makes $\\{D_{\omega}\\}_{\omega\in\Omega}$ a
standard family of operators.

As in [35], we first remark that, for all families of operators of the form
given by (1.2), (SGEE), (SLI), (IAD) and (EDI) have already been proven in [3,
Proof of Theorem 4.1] on $\mathbb{R}$. Similarly, (NE) is proven in the same
paper, in the proof of Theorem 4.2. Even if there are extra hypotheses in that
paper, they are not used in the proof of these specific assumptions. We
already proven the Wegner estimate (W) under the hypotheses of Theorem 1.9 or
Theorem 1.10.

Finally, we have have just proven in this Section the ILSE under both
hypotheses of Theorem 1.9 or Theorem 1.10, which achieves the proof of these
two theorems. ∎

##  3\. Application for potentials splitting in a sum of two Pauli matrices

The goal of this section is to establish the properties of Lyapunov exponents
for the models introduced in Section 1.4. We prove that the hypotheses of
either Theorem 1.9, 1.10 or 1.12 are satisfied, which respectively leads to
the localization and delocalization results given by Theorem 1.11. The fact
that the hypotheses of the localization criterion are satisfied depends in
fact not really on the family of measures $\mu_{E}$, but only on the
Furstenberg groups $G(E)$. Indeed, the Furstenberg group can be independent of
$E$ even if the measure $\mu_{E}$ depends on $E$.

Therefore, our first task is to determine the Furstenberg groups in the five
different cases. Since all transfer matrices are symplectic and have real
elements, $G(E)$ is always included in the real symplectic group
$\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$. In some of the cases, we can have
more. Let $\Delta$, the tridiagonal matrix with 0 on the diagonal and 1 on the
upper and lower diagonal.

######  Proposition 3.1.

In all cases, $G(E)\subset\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$. Moreover,

  * •

in Case 1, $G(E)\subset\mathrm{SO}_{\mathrm{N}}(\mathbb{R})$;

  * •

in Case 5, if $V_{\mathrm{per}}=\Delta$,
$G(E)\subset\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$.

###### Proof.

We have already proved in Section 1.2 that the Furstenberg group is always
included in $\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$. In some of the cases
defined in Table 1, we can prove stronger inclusions. First, we have that

$\frac{\mathrm{d}}{\mathrm{d}y}\left[(T_{x}^{y}(E))^{*}T_{x}^{y}(E)\right]=(T_{x}^{y}(E))^{*}[V_{\omega}+V_{\mathrm{per}},J]T_{x}^{y}(E).$

When $V_{\omega}+V_{\mathrm{per}}=\sigma_{0}\otimes V_{0}$, _i.e._ in case 1,
$[V_{\omega}+V_{\mathrm{per}},J]=0$ so $(T_{x}^{y}(E))^{*}T_{x}^{y}(E)=I_{2N}$
for all $x$ and $y$: $G(E)\subset\mathrm{SO}_{2N}(\mathbb{R})$.

If $V_{\mathrm{per}}=\Delta$, in Case 5, a direct computation shows that for
any $x,y\in\mathbb{R}$ and any $E\in\mathbb{R}$,
$T_{x}^{y}(E)\in\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ hence the result on
$G(E)$. ∎

This has already consequences on the Lyapunov exponents in some cases. We
prove that some Lyapunov exponents vanish.

######  Theorem 3.2.

In Case 1, all Lyapunov exponents are identically 0.

In Case 5 for odd $N$ and $V_{\mathrm{per}}=\Delta$, the lowest Lyapunov
exponent $\gamma_{N}$ is identically 0.

###### Proof.

The fact that all Lyapunov exponents associated with a sequence of matrices
included in the compact group $\mathrm{SO}_{2N}(\mathbb{R})$ are identically
zero is a direct consequence of Proposition A.III.5.6 of [5].

The result for Case 5 when $N$ is odd, is a consequence of Proposition 2.4
combined with $\gamma_{p}=-\gamma_{2N-p+1}$ for all $p$, which leads to
$\gamma_{N}=\gamma_{N+1}=0$. ∎

From Theorems 3.2 and 1.12, we directly get the _delocalization_ results in
Theorem 1.11.

To get localization results, we need more information. In Section 3.1, we
compute the Furstenberg groups in the different case when $\ell$ is close to
0, which means that the system is strongly disordered.

######  Theorem 3.3.

Recall the different cases given in Table 1. In case 1, $G(E)$ is a subgroup
of $\mathrm{SO}_{2N}(\mathbb{R})$. Moreover, there exists $\ell_{C}>0$ such
that, for all $\ell\in(0,\ell_{C})$, there exists a compact interval
$I(N,\ell)$ such that for all $E\in I(N,\ell)\setminus\\{0\\}$,

  * •

in cases 2, 3 and 4, $G(E)=\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$.

  * •

in Case 5, if $V_{\mathrm{per}}=\Delta$,
$G(E)=\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$.

We then get that the hypotheses of either Theorem 1.9 or Theorem 1.10 are
satisfied.

######  Theorem 3.4.

Fix $\ell<\ell_{C}$, for the $\ell_{C}$ introduced in Theorem 3.3. Then, for
all $E\in I(N,\ell,V)\setminus\\{0\\}$,

  * •

in case 2, 3 or 4, $G(E)$ is $p$-contracting and $J$-$L_{p}$-strongly
irreducible for all $p$ in $\\{1,\dots,N\\}$.

  * •

in case 5, if $V_{\mathrm{per}}=\Delta$, $G(E)$ is $2p$-contracting and
$(J,S)$-$L_{2p}$-strongly irreducible for every $2p$ in $\\{1,\dots,N\\}$.

###### Proof.

The case where $G(E)=\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$ is treated in the
proof of Proposition A.IV.3.5 of [5].

The remaining case is studied in section 3.2 of the present paper. ∎

Combining Theorems 3.4 and 1.9 or 1.10, we get the _localization_ results of
Theorem 1.11, at least for $V_{\mathrm{per}}=\Delta$. We give the complete
proof of Theorem 1.11 in Section 3.3.

###  3.1. Determination of the Furstenberg groups

In this section, we prove Theorem 3.3. For models of the type introduced in
Section 1.4, the common distribution of the $T_{\omega^{(n)}}(E)$’s is
$\mu_{E}:=(T_{\omega^{(0)}}(E))_{*}\,(\nu)$ and we have the internal
description of $G(E)$:

(3.1) $G(E)=\overline{<T_{\omega^{(0)}}(E)\ ;\
\omega^{(0)}\in\operatorname{supp}\nu>}\ \mbox{for all}\ E\in\mathbb{R}.$

Since we consider potentials which are constant on each interval
$(n\ell,(n+1)\ell)$, one has:

(3.2) $\forall\ell>0,\ \forall n\in\mathbb{Z},\ \forall E\in\mathbb{R},\
T_{\omega^{(n)}}(E)=\exp\left(\ell X_{\omega^{(n)}}(E)\right),$

where, using the decomposition (1.12)–(1.13),

(3.3) $\begin{split}\forall E\in\mathbb{R},\ \forall n\in\mathbb{Z},\
X_{\omega^{(n)}}(E)&:=J(V_{\mathrm{per}}+V_{\omega^{(n)}}-E)\\\
&=\left(\begin{smallmatrix}-\alpha_{1}\hat{V}_{\mathrm{per}}-\beta_{1}\hat{V}_{\omega^{(n)}}&E-(\alpha_{0}-\alpha_{3})\hat{V}_{\mathrm{per}}-(\beta_{0}-\beta_{3})\hat{V}_{\omega^{(n)}}\\\
-E+(\alpha_{0}+\alpha_{3})\hat{V}_{\mathrm{per}}+(\beta_{0}+\beta_{3})\hat{V}_{\omega^{(n)}}&\alpha_{1}V+\beta_{1}\hat{V}_{\omega^{(n)}}\end{smallmatrix}\right)\end{split}$

The explicit formula for transfer matrices, given in (3.2), is not easy to
handle since it is a matrix exponential in which the random parameters and the
energy parameter are mixed. As a consequence, we would like to determine the
Furstenberg groups directly from the matrices $X_{\omega^{(n)}}(E)$. This is
possible when the disorder is large enough thanks to the following result due
to Breuillard and Gelander [11].

######  Theorem 3.5 ([11], Theorem 2.1).

Let $G$ be a real, connected, semisimple Lie group, whose Lie algebra is
$\mathfrak{g}$. Then, there is a neighborhood $\tilde{\mathcal{O}}$ of $1$ in
$G$, on which $\log=\exp^{-1}$ is a well defined diffeomorphism, such that
$g_{1},\ldots,g_{m}\in\tilde{\mathcal{O}}$ generate a dense subgroup whenever
$\log g_{1},\ldots,\log g_{m}$ generate $\mathfrak{g}$.

Note that this result holds only when we consider a finite number of
generators. For any $(i,j)$ in $\\{1,\dots,N\\}$ we denote by $E_{ij}$ the
$N$-by-$N$ matrix with a coefficient 1 in position $(i,j)$ and 0 everywhere
else. For any $P\in\\{0,1\\}^{N}$, we denote by $X_{J}(E)$ the matrix
$X_{\omega^{(0)}}(E)$ for $\hat{V}_{\omega^{(0)}}=\sum_{P_{i}=1}E_{ii}$. Let
us introduce

$G_{\\{0,1\\}}(E):=<\exp\left(\ell X_{P}(E)\right)\ ;\
P\in\\{0,1\\}^{N}>\subset G(E)$

since $\\{0,1\\}^{N}\subset\operatorname{supp}\nu$. The group
$G_{\\{0,1\\}}(E)$ has $2^{N}$ generators. We will use Theorem 3.5 to prove
that $G_{\\{0,1\\}}(E)$ is dense in $\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$ or
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, depending on the case. Together with
Proposition 3.1, it will conclude the proof of Theorem 3.3.

We first need to ensure that the groups $\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$
and $\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ are connected and semisimple. It
is well-known that the symplectic group has these properties. In Appendix A,
we prove that they hold for the group $\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$
as well.

We next prove that, under good conditions on $E$ and $\ell$, the transfer
matrices are in an appropriate neighborhood of the unit. We prove it in a
general result holding for all operators of the form given by (1.2) such that
the potential $V_{\omega}^{(n)}$ is constant on $[0,\ell]$ and has values in
the space of real symmetric matrices.

######  Lemma 3.6.

For all neighborhood $\mathcal{O}$ of 1 in
$\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$, there exists $\ell_{C}>0$,
$c_{1},c_{2}\in\mathbb{R}$ and $d>0$ such that if $\ell<\ell_{C}$, then
$c_{1}-\frac{d}{\ell}<c_{2}+\frac{d}{\ell}$ and
$T_{\omega^{(0)}}(E)\in\mathcal{O}$ for all $\omega^{(0)}\in\\{0,1\\}^{N}$ and
all $E\in[c_{1}-\frac{d}{\ell},c_{2}+\frac{d}{\ell}]$.

Note that we do not need to prove an analogous result for
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$ since, for any neighborhood
$\mathcal{O}$ of 1 in $\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, there exists a
neighborhood $\mathcal{O}^{\prime}$ of 1 in
$\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$ such that
$\mathcal{O}=\mathcal{O}^{\prime}\cap\mathrm{SpO}_{\mathrm{N}}(\mathbb{R}).$

###### Proof.

Given a neighborhood $\mathcal{O}$ of 1 in
$\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$, we set:

$d_{\log\,\mathcal{O}}:=\sup\\{R>0\ |\
\exp\left[\overline{B}(0,R)\right]\subset\mathcal{O}\\},$

where $\overline{B}(0,R)$ is the closed ball, centered on $0$ and of radius
$R>0$, for the topology induced on the Lie algebra
$\mathfrak{sp}_{\mathrm{N}}(\mathbb{R})$ by the matrix norm induced by the
euclidean norm on $\mathbb{R}^{2N}$. Then, $T_{\omega^{(0)}}(E)$ will be in
$\mathcal{O}$ as soon as $\ell\|X_{\omega^{(0)}}(E)\|\leq
d_{\log\,\mathcal{O}}$.

We remark that $X_{\omega^{(0)}}(E)=J^{-1}(E-V_{\omega^{(0)}})$. Since
$J^{-1}$ is an orthogonal matrix, we have, denoting by
$(\lambda_{i}^{\omega^{(0)}})_{1\leq i\leq 2N}$ the eigenvalues of the
symmetric matrix of $V_{\omega^{(0)}}$,

$\|X_{\omega^{(0)}}(E)\|=\|E-V_{\omega^{(0)}}\|=\max_{1\leq i\leq
2N}|\lambda_{i}^{\omega^{(0)}}-E|.$

We want to find an interval of values of $E$ on which
$\ell\|X_{\omega^{(0)}}(E)\|<d_{\log\,\mathcal{O}}$. In other words, we want
to characterize the set

$I(\ell,\mathcal{O}):=\left\\{E\in\mathbb{R},\max_{\omega\in\\{0,1\\}^{N}}\max_{1\leq
i\leq
N}|\lambda_{i}^{\omega}-E|\leq\frac{d_{\log\,\mathcal{O}}}{\ell}\right\\}.$

We see that

$I(\ell,\mathcal{O})=\bigcap_{\omega\in\\{0,1\\}^{N}}\bigcap_{1\leq i\leq
N}\left[\lambda_{i}^{\omega}-\frac{d_{\log\,\mathcal{O}}}{\ell},\lambda_{i}^{\omega}+\frac{d_{\log\,\mathcal{O}}}{\ell}\right].$

We define

$\lambda_{\max}:=\max_{\omega\in\\{0,1\\}^{N}}\max_{1\leq i\leq
N}\lambda_{i}^{\omega}\text{ and
}\lambda_{\min}:=\min_{\omega\in\\{0,1\\}^{N}}\min_{1\leq i\leq
N}\lambda_{i}^{\omega}.$

If $(\lambda_{\max}-\lambda_{\min})/2<d_{\log\,\mathcal{O}}/\ell$, then
$I(\ell,\mathcal{O})=[\lambda_{\max}-d_{\log\,\mathcal{O}}/\ell,\lambda_{\min}+d_{\log\,\mathcal{O}}/\ell]$
which is a nonempty interval. As a consequence, we have proved that if
$\ell<\ell_{C}:=2d_{\log\,\mathcal{O}}/(\lambda_{\max}-\lambda_{\min})$, then
for all $E\in I(\ell,N)$ $\ell\|X_{\omega^{(0)}}(E)\|<d_{\log\,\mathcal{O}}$.
As a consequence, $T_{\omega^{(0)}}(E)=\exp(\ell
X_{\omega}^{(0)}(E))\in\mathcal{O}$. ∎

Finally, we prove that, in each case, the Lie algebra generated by the
$X_{\omega^{(0)}}(E)$’s corresponds with the Lie algebra associated with the
group appearing in Theorem 3.3. Let us denote, for every $E\in\mathbb{R}$,

(3.4) $\mathfrak{g}(E):=\mathrm{Lie}\left\\{X_{P}(E);\
P\in\\{0,1\\}^{N}\right\\}.$

We explicitly compute $\mathfrak{g}(E)$ in the different cases described in
Table 1. In order to state the result, we introduce the following algebras:

  * •

$\mathfrak{sp}_{\mathrm{N}}(\mathbb{R})$ is the algebra associated with the
symplectic group $\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$, it consists of all
matrices of the shape $\begin{pmatrix}A&B\\\ C&-^{t}A\end{pmatrix}$, where $B$
and $C$ are symmetric.

  * •

Let us introduce the operator $s$ defined on $\mathcal{M}_{N}(\mathbb{R})$ by
$(^{s}M)_{ij}=(-1)^{i-j+1}M_{ji}$. We define the Lie algebra

(3.5) $\mathfrak{spo}_{\mathrm{N}}(\mathbb{R}):=\left\\{\begin{pmatrix}A&B\\\
{}^{s}B&-^{t}A\end{pmatrix}\text{ with }^{s}A=A,^{t}B=B\right\\}.$

This is the Lie algebra associated with the group
$\mathrm{SpO}_{\mathrm{N}}(\mathbb{R})$, as we prove in Appendix A.

We have the following result.

######  Proposition 3.7.

Let us recall the 5 cases defined in Table 1 and assume that
$V_{\mathrm{per}}=\Delta$. Then, for all $E\neq 0$,

  * •

in cases 2, 3 and 4, $\mathfrak{g}(E)=\mathfrak{sp}_{\mathrm{N}}(\mathbb{R})$;

  * •

in case 5, $\mathfrak{g}(E)=\mathfrak{spo}_{\mathrm{N}}(\mathbb{R})$.

In the rest of the section, we prove this proposition in the different cases.
We fix $E\neq 0$. We use the following lemma.

######  Lemma 3.8.

We introduce the following notations:

(3.6) $Z_{ij}:=\begin{pmatrix}0&E_{ij}+E_{ji}\\\
E_{ij}+E_{ji}&0\end{pmatrix}\quad Y_{ij}:=\begin{pmatrix}0&E_{ij}+E_{ji}\\\
-E_{ij}-E_{ji}&0\end{pmatrix}.$

If some subalgebra $\mathfrak{g}$ of $\mathfrak{sp}_{\mathrm{N}}(\mathbb{R})$
contains all the $Z_{ij}$’s and $Y_{ij}$’s for $|i-j|\leq 1$ , then
$\mathfrak{g}=\mathfrak{sp}_{\mathrm{N}}(\mathbb{R})$.

A proof of this result can be found in the proof of Lemma 1 of [8], even if it
is not really stated.

#### Case 2: potentials both on $\sigma_{3}$

Recall that, according to Table 1, this case covers the case where potentials
are both on $\sigma_{1}$ as well. We consider the family of matrices
$\\{X_{P}(E)\\}_{P\in\\{0,1\\}^{N}}$, as defined in (3.3), for
$\alpha_{3}=\beta_{3}=1$ and all other coefficients equal to 0. We have

$X_{P}(E)=\left(\begin{smallmatrix}0&E+\Delta+\sum_{P_{i}=1}E_{ii}\\\
-E+\Delta+\sum_{P_{i}=1}E_{ii}&0\end{smallmatrix}\right).$

In view of Lemma 3.8, we will prove that, for such a family
$\\{X_{P}(E)\\}_{P\in\\{0,1\\}^{N}}$, all matrices $Z_{ij}$ and $Y_{ij}$ for
$|i-j|\leq 1$ are in $\mathfrak{g}(E)$. First, for any
$P,P^{\prime}\in\\{0,1\\}^{N}$,

$X_{P}(E)-X_{P^{\prime}}(E)=\left(\begin{smallmatrix}0&\sum_{P_{i}=1}E_{ii}-\sum_{P^{\prime}_{i}=1}E_{ii}\\\
\sum_{P_{i}=1}E_{ii}-\sum_{P^{\prime}_{i}=1}E_{ii}&0\end{smallmatrix}\right).$

For all $i\in\\{1,\dots,N\\}$, we can choose $P=(0,\dots,1,\dots,0)$ with the
1 in position $i$ and $P^{\prime}=(0,\dots,0)$. We find that $Z_{ii}$ is in
$\mathfrak{g}(E)$.

Then, we have that for all $P\in\\{0,1\\}^{N}$

$\left[X_{P}(E),\begin{pmatrix}0&I_{N}\\\
I_{N}&0\end{pmatrix}\right]=-2E\begin{pmatrix}I_{N}&0\\\
0&-I_{N}\end{pmatrix}.$

Therefore, since $E\neq 0$, $\begin{pmatrix}I_{N}&0\\\
0&-I_{N}\end{pmatrix}\in\mathfrak{g}(E)$. We can then see that for all $i$ in
$\\{1,\dots,N\\}$

$\left[X_{ii},\begin{pmatrix}I_{N}&0\\\
0&-I_{N}\end{pmatrix}\right]=-2Y_{ii},$

and thus all the $Y_{ii}$ are in $\mathfrak{g}(E)$. We have thus that

$\\{\left(\begin{smallmatrix}0&D_{1}\\\
D_{2}&0\end{smallmatrix}\right),D_{1},\ D_{2}\text{
diagonal}\\}\subset\mathfrak{g}(E).$

Therefore,

$X_{P}(E)-\left(\begin{smallmatrix}0&E+\sum_{P_{i}=1}E_{ii}\\\
-E+\sum_{P_{i}=1}E_{ii}&0\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&\Delta\\\
\Delta&0\end{smallmatrix}\right)\in\mathfrak{g}(E).$

We compute, for $i$ in $\\{1,\dots,N\\}$,

$\left[\begin{pmatrix}0&\Delta\\\
\Delta&0\end{pmatrix},Y_{ii}\right]=-2\begin{pmatrix}B_{i}&0\\\
0&-B_{i}\end{pmatrix},$

where $B_{i}:=\Delta
E_{ii}+E_{ii}\Delta=E_{i,i-1}+E_{i,i+1}+E_{i-1,i}+E_{i+1,i}$ with the
convention that $E_{ij}=0$ if $i$ or $j$ is not in $[1,N]$. We remark that

$\text{Span}(\\{B_{i},i=1,\dots,N-1\\})=\text{Span}(\\{E_{i,i+1}+E_{i+1,i},i=1,\dots,N-1\\}),$

which is the space of symmetric tridiagonal matrices with zeros on the
diagonal

Now, for any $i$ in $\\{1,\dots,N-1\\}$,

$\left[\begin{pmatrix}E_{i,i+1}+E_{i+1,i}&0\\\
0&-(E_{i,i+1}+E_{i+1,i})\end{pmatrix},\begin{pmatrix}0&I_{N}\\\
I_{N}&0\end{pmatrix}\right]=2Y_{i,i+1}.$

As a consequence, all the $Y_{ij}$’s with $|i-j|=1$ are in $\mathfrak{g}(E)$.
Last,

$\left[\begin{pmatrix}E_{i,i+1}+E_{i+1,i}&0\\\
0&-(E_{i,i+1}+E_{i+1,i})\end{pmatrix},\begin{pmatrix}0&I_{N}\\\
-I_{N}&0\end{pmatrix}\right]=2Z_{i,i+1}.$

We have that all the $Z_{ij}$’s with $|i-j|=1$ are in $\mathfrak{g}(E)$. With
Lemma 3.8, this concludes the proof that
$\mathfrak{g}(E)=\mathfrak{sp}_{\mathrm{N}}(\mathbb{R})$

#### Case 3: deterministic potential on $\sigma_{0}$, random potential on
$\sigma_{3}$.

Recall that, according to Table 1, this covers as well the case with the
deterministic potential on $\sigma_{0}$ and the random potential on
$\sigma_{1}$. We consider the family of matrices
$\\{X_{P}(E)\\}_{P\in\\{0,1\\}^{N}}$, as defined in (3.3), with
$\alpha_{0}=\beta_{3}=1$ and all the other coefficients equal to 0. In other
words,

$X_{P}(E)=\left(\begin{smallmatrix}0&E-\Delta+\sum_{P_{i}=1}E_{ii}\\\
-E+\Delta+\sum_{P_{i}=1}E_{ii}&0\end{smallmatrix}\right).$

We prove that all the $Z_{ij}$’s and $Y_{ij}$’s, for $|i-j|\leq 1$, are in
$\mathfrak{g}(E)$. First, we see that

(3.7)
$X_{P}(E)-X_{P^{\prime}}(E)=\left(\begin{smallmatrix}0&\sum_{P_{i}=1}E_{ii}-\sum_{P^{\prime}_{i}=1}E_{ii}\\\
\sum_{P_{i}=1}E_{ii}-\sum_{P^{\prime}_{i}=1}E_{ii}&0\end{smallmatrix}\right)$

so, for all $i$, $Z_{ii}\in\mathfrak{g}(E)$.

Then,

$\left(\begin{smallmatrix}0&E-\Delta\\\
-E+\Delta&0\end{smallmatrix}\right)\in\mathfrak{g}(E).$

Taking the bracket, we find that for all $i$ in $\\{1,\dots,N\\}$,

(3.8) $\left[\left(\begin{smallmatrix}0&E-\Delta\\\
-E+\Delta&0\end{smallmatrix}\right),Z_{ii}\right]=\left(\begin{smallmatrix}2EE_{ii}-\\{E_{ii},\Delta\\}&0\\\
0&-2EE_{ii}+\\{E_{ii},\Delta\\}\end{smallmatrix}\right)\in\mathfrak{g}(E).$

Recall that we have defined the matrix $K=\sum_{i=1}^{N}(-1)^{i+1}E_{ii}$. We
find that $\\{K,\Delta\\}=0$. As a consequence, since $E\neq 0$, we find that

(3.9) $\sum_{i=1}^{N}(-1)^{i+1}\left[\left(\begin{smallmatrix}0&E-\Delta\\\
-E+\Delta&0\end{smallmatrix}\right),Z_{ii}\right]=2E\left(\begin{smallmatrix}K&0\\\
0&-K\end{smallmatrix}\right)\in\mathfrak{g}(E).$

Taking another bracket, we find that, for all $i\in\\{1,\dots,N\\}$,

(3.10) $\left[\left(\begin{smallmatrix}K&0\\\
0&-K\end{smallmatrix}\right),\left(\begin{smallmatrix}0&E_{ii}\\\
E_{ii}&0\end{smallmatrix}\right)\right]=\left(\begin{smallmatrix}0&2(-1)^{i+1}E_{ii}\\\
-2(-1)^{i+1}E_{ii}&0\end{smallmatrix}\right)=(-1)^{i+1}Y_{ii}\in\mathfrak{g}(E).$

On the other hand, we find that

(3.11) $\Theta:=\sum_{i=1}^{N}\left[\left(\begin{smallmatrix}0&E-\Delta\\\
-E+\Delta&0\end{smallmatrix}\right),Z_{ii}\right]=\left(\begin{smallmatrix}E-\Delta&0\\\
0&-E+\Delta\end{smallmatrix}\right)\in\mathfrak{g}(E).$

For $i$ between 2 and $N-1$,

(3.12)
$\frac{1}{2}[\Theta,Z_{ii}]=\left(\begin{smallmatrix}0&2EE_{ii}-\\{E_{ii},\Delta\\}\\\
-2EE_{ii}+\\{E_{ii},\Delta\\}&0\end{smallmatrix}\right)=EY_{ii}+Y_{i,i+1}+Y_{i,i-1}$

and

(3.13)
$\frac{1}{2}[\Theta,Z_{11}]=\left(\begin{smallmatrix}0&2EE_{11}-\\{E_{11},\Delta\\}\\\
-2EE_{11}+\\{E_{11},\Delta\\}&0\end{smallmatrix}\right)=EY_{11}+Y_{12}.$

This, together with the fact that all $Y_{ii}$’s are in $\mathfrak{g}(E)$,
implies that, for all $i=1,\dots,N-1$, $Y_{i,i+1}\in\mathfrak{g}(E)$.
Similarly, we prove, computing the commutators $[\Theta,Y_{ii}]$ that all
$Z_{i,i+1}$ are in $\mathfrak{g}(E)$. As a consequence, by Lemma 3.8,
$\mathfrak{g}(E)=\mathfrak{sp}_{N}(\mathbb{R})$.

#### Case 4: deterministic potential on $\sigma_{1}$, random potential on
$\sigma_{3}$

Recall that, according to Table 1, this covers as well the case where the
deterministic potential is on $\sigma_{3}$ and the random potential is on
$\sigma_{1}$. We consider the family of matrices
$\\{X_{P}(E)\\}_{P\in\\{0,1\\}^{N}}$, as defined in (3.3), with
$\alpha_{1}=\beta_{3}=1$ and all other coefficients equal to 0. As a
consequence,

$X_{P}(E)=\left(\begin{smallmatrix}-\Delta&E+\sum_{P_{i}=1}E_{ii}\\\
-E+\sum_{P_{i}=1}E_{ii}&\Delta\end{smallmatrix}\right).$

As in the previous cases, we will use Lemma 3.8 and only prove that all the
$Z_{ij}$’s and $Y_{ij}$’s, for $|i-j|\leq 1$, are in $\mathfrak{g}(E)$. First,
we see that for all $P,P^{\prime}$

(3.14)
$X_{P}(E)-X_{P^{\prime}}(E)=\sum_{P_{i}=1}Z_{ii}-\sum_{P^{\prime}_{i}=1}Z_{ii}\in\mathfrak{g}(E)$

so, for all $i$, $Z_{ii}\in\mathfrak{g}(E)$.

Then,

$X_{P}(E)-\sum_{P_{i}=1}Z_{ii}=\left(\begin{smallmatrix}-\Delta&E\\\
-E&\Delta\end{smallmatrix}\right)\in\mathfrak{g}(E).$

Taking the bracket, we find that for all $i$ in $\\{1,\dots,N\\}$,

(3.15) $\left[\left(\begin{smallmatrix}-\Delta&E\\\
-E&-\Delta\end{smallmatrix}\right),Z_{ii}\right]=\left(\begin{smallmatrix}-2EE_{ii}&-\\{E_{ii},\Delta\\}\\\
\\{E_{ii},\Delta\\}&2EE_{ii}\end{smallmatrix}\right)\in\mathfrak{g}(E).$

Similarly to the previous case, we can write

(3.16) $\sum_{i=1}^{N}(-1)^{i+1}\left[\left(\begin{smallmatrix}0&E-\Delta\\\
-E+\Delta&0\end{smallmatrix}\right),X_{ii}\right]=\left(\begin{smallmatrix}2EK&0\\\
0&-2EK\end{smallmatrix}\right)\in\mathfrak{g}(E).$

Taking another bracket, we find that, for all $i$,

(3.17) $\left[\left(\begin{smallmatrix}K&0\\\
0&-K\end{smallmatrix}\right),Z_{ii}\right]=(-1)^{i+1}Y_{ii}\in\mathfrak{g}(E).$

This implies that $\Theta^{\prime}:=\left(\begin{smallmatrix}-\Delta&0\\\
0&\Delta\end{smallmatrix}\right)\in\mathfrak{g}(E).$ We can conclude as in the
previous case.

#### Case 5: deterministic potential on $\sigma_{3}$, random potential on
$\sigma_{0}$.

Recall that, according to Table 1, this covers as well the case with the
deterministic potential on $\sigma_{1}$ and the random potential on
$\sigma_{0}$. We consider the family of matrices
$\\{X_{P}(E)\\}_{P\in\\{0,1\\}^{N}}$, as defined in (3.3), with
$\alpha_{3}=\beta_{0}=1$ and all other coefficients equal to 0. We have

(3.18) $X_{P}(E)=\left(\begin{smallmatrix}0&E+\Delta-\sum_{P_{i}=1}E_{ii}\\\
-E+\Delta+\sum_{P_{i}=1}E_{ii}&0\end{smallmatrix}\right)$

Here, the Lie algebra $\mathfrak{g}(E)$ will not be
$\mathfrak{sp}_{\mathrm{N}}(\mathbb{R})$ but its subalgebra
$\mathfrak{spo}_{\mathrm{N}}(\mathbb{R})$. We have the following result,
similar to Lemma 3.8.

######  Lemma 3.9.

We adopt the same notation as in Lemma 3.8. If some subalgebra $\mathfrak{g}$
of $\mathfrak{spo}_{\mathrm{N}}(\mathbb{R})$ contains all the $Y_{ii}$’s as
well as the $X_{ij}$’s for $|i-j|=1$ , then
$\mathfrak{g}=\mathfrak{spo}_{\mathrm{N}}(\mathbb{R})$.

###### Proof.

We first remark that all the $Y_{ii}$’s as well as the $X_{ij}$’s for
$|i-j|=1$ are in $\mathfrak{spo}_{\mathrm{N}}(\mathbb{R})$. To prove the other
inclusion, let $\mathfrak{g}$ be a subalgebra of
$\mathfrak{spo}_{\mathrm{N}}(\mathbb{R})$ which contains all these matrices.
Recall that a basis of $\\{A\in\mathcal{M}_{N}(\mathbb{R})\ |\ ^{s}A=A\\}$ is
given by $\\{E_{ij}+(-1)^{i-j+1}E_{ji}\\}_{1\leq i\leq j\leq N}$ and one of
$\\{B\in\mathcal{M}_{N}(\mathbb{R})\ |\ ^{t}B=B\\}$ is given by
$\\{E_{ij}+E_{ji}\\}_{1\leq i\leq j\leq N}$. We introduce, for every
$i,j\in\\{1,\ldots N\\}$, the matrices

$V_{ij}=\left(\begin{smallmatrix}E_{ij}+(-1)^{i-j+1}E_{ji}&0\\\
0&(-1)^{i-j}E_{ij}-E_{ji}\end{smallmatrix}\right)\ \mbox{ and }\
W_{ij}=\left(\begin{smallmatrix}0&E_{ij}+E_{ji}\\\
(-1)^{i-j+1}(E_{ij}+E_{ji})&0\end{smallmatrix}\right).$

By (3.5), $\\{V_{ij},W_{ij}\\}_{1\leq i\leq j\leq N}$ is a basis of
$\mathfrak{spo}_{\mathrm{N}}(\mathbb{R})$. Hence, to prove Lemma 3.9, it
suffices to prove that $\mathfrak{g}$ contains all the $V_{ij}$’s and
$W_{ij}$’s for $1\leq i\leq j\leq N$. A direct computation shows that for
every $i,j,k,r\in\\{1,\ldots,N\\}$,

$\displaystyle[V_{ij},W_{kr}]$
$\displaystyle=\delta_{jk}\left(\begin{smallmatrix}\mathrm{I}_{\mathrm{N}}&0\\\
0&(-1)^{k-j}\mathrm{I}_{\mathrm{N}}\end{smallmatrix}\right)W_{ir}+\delta_{ik}(-1)^{i-j+1}\left(\begin{smallmatrix}\mathrm{I}_{\mathrm{N}}&0\\\
0&(-1)^{i+k}\mathrm{I}_{\mathrm{N}}\end{smallmatrix}\right)W_{jr}$
$\displaystyle\qquad\qquad+\delta_{ir}(-1)^{i-j+1}\left(\begin{smallmatrix}\mathrm{I}_{\mathrm{N}}&0\\\
0&(-1)^{i-r}\mathrm{I}_{\mathrm{N}}\end{smallmatrix}\right)W_{jk}+\delta_{jr}\left(\begin{smallmatrix}\mathrm{I}_{\mathrm{N}}&0\\\
0&(-1)^{-j-r}\mathrm{I}_{\mathrm{N}}\end{smallmatrix}\right)W_{ik}$ (3.19)
$\displaystyle=(-1)^{i-j+1}(\delta_{ik}W_{jr}+\delta_{ir}W_{jk})+\delta_{jk}W_{ir}+\delta_{jr}W_{ik}$

and if $(-)_{ijkr}=(-1)^{i-j+k-r+1}$,

$\displaystyle[W_{ij},W_{kr}]$
$\displaystyle=(-1)^{k-r+1}\left[\delta_{jk}\left(\begin{smallmatrix}E_{ir}+(-)_{ijkr}E_{ri}&0\\\
0&-((-)_{ijkr}E_{ir}+E_{ri})\end{smallmatrix}\right)+\delta_{jr}\left(\begin{smallmatrix}E_{ik}+(-)_{ijkr}E_{ki}&0\\\
0&-((-)_{ijkr}E_{ik}+E_{ki})\end{smallmatrix}\right)\right.$
$\displaystyle\qquad\left.+\delta_{ik}\left(\begin{smallmatrix}E_{jr}+(-)_{ijkr}E_{rj}&0\\\
0&-((-)_{ijkr}E_{jr}+E_{rj})\end{smallmatrix}\right)+\delta_{ir}\left(\begin{smallmatrix}E_{jk}+(-)_{ijkr}E_{kj}&0\\\
0&-((-)_{ijkr}E_{jk}+E_{kj})\end{smallmatrix}\right)\right]$ (3.20)
$\displaystyle=(-1)^{k-r+1}(\delta_{jk}V_{ir}+\delta_{jr}V_{ik}+\delta_{ik}V_{jr}+\delta_{ir}V_{jk})$

where $\delta_{ij}$ is the Kronecker’s symbol :

$\delta_{ij}=\left\\{\begin{array}[]{ccl}1&\mathrm{if}&i=j\\\
0&\mathrm{if}&i\neq j.\end{array}\right.$

The hypothesis made on $\mathfrak{g}$ implies that for every
$i\in\\{1,\ldots,N\\}$, $W_{ii}\in\mathfrak{g}$ and for $|i-j|=1$,
$W_{ij}\in\mathfrak{g}$. Let $i\in\\{1,\ldots,N\\}$. Then, using (3.20),
$[W_{ii},W_{i,i+1}]=2V_{i,i+1}$ and $V_{i,i+1}\in\mathfrak{g}$. This proves
that $\mathfrak{g}$ contains all the $V_{ij}$ and $W_{ij}$ for $|i-j|=1$.
Using (3.19), $[V_{i,i+1},W_{i+1,i+2}]=W_{i,i+2}$ and
$W_{i,i+2}\in\mathfrak{g}$. Using (3.20) this implies that
$[W_{ii},W_{i,i+2}]=-2V_{i,i+2}$ and $V_{i,i+2}\in\mathfrak{g}$. Hence,
$\mathfrak{g}$ contains all the $V_{ij}$ and $W_{ij}$ for $|i-j|=2$. By
induction, using (3.19) and (3.20), we prove that $\mathfrak{g}$ contains all
the $V_{ij}$ and $W_{ij}$ for $i\leq j$, hence
$\mathfrak{spo}_{\mathrm{N}}(\mathbb{R})\subset\mathfrak{g}$ and
$\mathfrak{g}=\mathfrak{spo}_{\mathrm{N}}(\mathbb{R})$. ∎

To compute $\mathfrak{g}(E)$, we begin as usual by taking
$X_{P}(E)-X_{P^{\prime}}(E)$ to find that all the $Y_{ii}$’s are in
$\mathfrak{g}(E)$. Then,

$\begin{pmatrix}0&\Delta\\\ \Delta&0\end{pmatrix}\in\mathfrak{g}(E).$

This implies that

$\left[\begin{pmatrix}0&-\Delta\\\
\Delta&0\end{pmatrix},\begin{pmatrix}0&I_{N}\\\
-I_{N}&0\end{pmatrix}\right]=-2\begin{pmatrix}\Delta&0\\\
0&-\Delta\end{pmatrix}\in\mathfrak{g}(E).$

Taking commutators between this last matrix and the $Y_{ii}$’s, we find that
all the $Z_{ij}$’s with $|i-j|=1$ are in $\mathfrak{g}(E)$. ∎

###  3.2. $p$-contractivity and $L_{p}$-strong irreducibility

As we already explained, $\mathrm{Sp}_{\mathrm{N}}(\mathbb{R})$ is