• in Case 5, if $V_{\text{per}} = \Delta$, $G(E) \subset \text{SpO}_N(\mathbb{R})$.

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

$$ \frac{d}{dy} [(T_x^y(E))^* T_x^y(E)] = (T_x^y(E))^* [V_\omega + V_{\text{per}}, J] T_x^y(E). $$

When $V_\omega + V_{\text{per}} = \sigma_0 \otimes V_0$, i.e. in case 1, $[V_\omega + V_{\text{per}}, J] = 0$ so $(T_x^y(E))^* T_x^y(E) = I_{2N}$ for all $x$ and $y: G(E) \subset \text{SO}_{2N}(\mathbb{R})$.

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

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 $V_{\text{per}} = \Delta$, 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 $\text{SO}_{2N}(\mathbb{R})$ 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 $\gamma_N = \gamma_{N+1} = 0$. $\square$

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 $\text{SO}_{2N}(\mathbb{R})$. 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, $G(E) = \text{Sp}_N(\mathbb{R})$.

• in Case 5, if $V_{\text{per}} = \Delta$, $G(E) = \text{SpO}_N(\mathbb{R})$.

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

*Proof.* The case where $G(E) = \text{Sp}_N(\mathbb{R})$ 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. $\square$

Combining Theorems 3.4 and 1.9 or 1.10, we get the *localization* results of Theorem 1.11, at least for $V_{\text{per}} = \Delta$. We give the complete proof of Theorem 1.11 in Section 3.3.
• in Case 5, if $V_{\text{per}} = \Delta$, $G(E) \subset \text{SpO}_N(\mathbb{R})$.

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

$$ \frac{d}{dy} [(T_x^y(E))^* T_x^y(E)] = (T_x^y(E))^* [V_\omega + V_{\text{per}}, J] T_x^y(E). $$

When $V_\omega + V_{\text{per}} = \sigma_0 \otimes V_0$, i.e. in case 1, $[V_\omega + V_{\text{per}}, J] = 0$ so $(T_x^y(E))^* T_x^y(E) = I_{2N}$ for all $x$ and $y: G(E) \subset \text{SO}_{2N}(\mathbb{R})$.

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

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 $V_{\text{per}} = \Delta$, 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 $\text{SO}_{2N}(\mathbb{R})$ 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 $\gamma_N = \gamma_{N+1} = 0$. $\square$

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 $\text{SO}_{2N}(\mathbb{R})$. 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, $G(E) = \text{Sp}_N(\mathbb{R})$.

• in Case 5, if $V_{\text{per}} = \Delta$, $G(E) = \text{SpO}_N(\mathbb{R})$.

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

*Proof.* The case where $G(E) = \text{Sp}_N(\mathbb{R})$ 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. $\square$

Combining Theorems 3.4 and 1.9 or 1.10, we get the *localization* results of Theorem 1.11, at least for $V_{\text{per}} = \Delta$. 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 $G(E)$:

$$ (3.1) \qquad G(E) = \langle T_{\omega^{(0)}}(E) ; \omega^{(0)} \in \text{supp } \nu \rangle \quad \text{for all } E \in \mathbb{R}. $$

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

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

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

$$ \begin{gathered} \forall E \in \mathbb{R}, \forall n \in \mathbb{Z}, X_{\omega^{(n)}}(E) := J(V_{\text{per}} + V_{\omega^{(n)}} - E) \\ (3.3) \qquad = \begin{pmatrix} -\alpha_1 \hat{V}_{\text{per}} - \beta_1 \hat{V}_{\omega^{(n)}} & E - (\alpha_0 - \alpha_3) \hat{V}_{\text{per}} - (\beta_0 - \beta_3) \hat{V}_{\omega^{(n)}} \\ -E + (\alpha_0 + \alpha_3) \hat{V}_{\text{per}} + (\beta_0 + \beta_3) \hat{V}_{\omega^{(n)}} & \alpha_1 V + \beta_1 \hat{V}_{\omega^{(n)}} \end{pmatrix} \end{gathered} $$

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 $\tilde{O}$ of 1 in $G$, on which $\log = \exp^{-1}$ is a well defined diffeomorphism, such that $g_1, \dots, g_m \in \tilde{O}$ generate a dense subgroup whenever $\log g_1, \dots, \log g_m$ 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

$$ G_{\{0,1\}}(E) := \langle \exp (\ell X_P(E)) ; P \in \{0,1\}^N \rangle < G(E) $$

since $\{0,1\}^N \subset \text{supp } \nu$. 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 $\text{Sp}_N(\mathbb{R})$ or $\text{SpO}_N(\mathbb{R})$, 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 $\text{Sp}_N(\mathbb{R})$ and $\text{SpO}_N(\mathbb{R})$ 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 $\text{SpO}_N(\mathbb{R})$ 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 $\text{I}_{2N}$ in $\text{Sp}_N(\mathbb{R})$, 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 $\text{SpO}_N(\mathbb{R})$ since, for any neighborhood $\mathcal{O}$ of $\text{I}_{2N}$ in $\text{SpO}_N(\mathbb{R})$, there exists a neighborhood $\mathcal{O}'$ of $\text{I}_{2N}$ in $\text{Sp}_N(\mathbb{R})$ such that $\mathcal{O} = \mathcal{O}' \cap \text{SpO}_N(\mathbb{R})$.
**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 $G(E)$:

$$ (3.1) \qquad G(E) = \langle T_{\omega^{(0)}}(E) ; \omega^{(0)} \in \text{supp } \nu \rangle \quad \text{for all } E \in \mathbb{R}. $$

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

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

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

$$ (3.3) \qquad \forall E \in \mathbb{R}, \forall n \in \mathbb{Z}, X_{\omega^{(n)}}(E) := J(V_{\text{per}} + V_{\omega^{(n)}} - E) \\ = \begin{pmatrix} -\alpha_1 \hat{V}_{\text{per}} - \beta_1 \hat{V}_{\omega^{(n)}} & E - (\alpha_0 - \alpha_3) \hat{V}_{\text{per}} - (\beta_0 - \beta_3) \hat{V}_{\omega^{(n)}} \\ -E + (\alpha_0 + \alpha_3) \hat{V}_{\text{per}} + (\beta_0 + \beta_3) \hat{V}_{\omega^{(n)}} & \alpha_1 V + \beta_1 \hat{V}_{\omega^{(n)}} \end{pmatrix} $$

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 $\tilde{O}$ of 1 in $G$, on which $\log = \exp^{-1}$ is a well defined diffeomorphism, such that $g_1, \dots, g_m \in \tilde{O}$ generate a dense subgroup whenever $\log g_1, \dots, \log g_m$ 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

$$ G_{\{0,1\}}(E) := \langle \exp (\ell X_P(E)) ; P \in \{0,1\}^N \rangle \subset G(E) $$

since $\{0, 1\}^N \subset \text{supp } \nu$. 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 $\text{Sp}_N(\mathbb{R})$ or $\text{SpO}_N(\mathbb{R})$, 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 $\text{Sp}_N(\mathbb{R})$ and $\text{SpO}_N(\mathbb{R})$ 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 $\text{SpO}_N(\mathbb{R})$ 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 $\text{I}_{2N}$ in $\text{Sp}_N(\mathbb{R})$, 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 $\text{SpO}_N(\mathbb{R})$ since, for any neighborhood $\mathcal{O}$ of $\text{I}_{2N}$ in $\text{SpO}_N(\mathbb{R})$, there exists a neighborhood $\mathcal{O}'$ of $\text{I}_{2N}$ in $\text{Sp}_N(\mathbb{R})$ such that $\mathcal{O} = \mathcal{O}' \cap \text{SpO}_N(\mathbb{R})$.
*Proof.* Given a neighborhood $\mathcal{O}$ of 1 in $\text{Sp}_N(\mathbb{R})$, we set:

$$d_{\log \mathcal{O}} := \sup\{R > 0 \mid \exp[\bar{B}(0, R)] \subset \mathcal{O}\},$$

where $\bar{B}(0, R)$ is the closed ball, centered on 0 and of radius $R > 0$, for the topology induced on the Lie algebra $\mathfrak{sp}_N(\mathbb{R})$ 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 $\ell \|X_{\omega^{(0)}}(E)\| \le d_{\log \mathcal{O}}$.

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

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

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

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

We see that

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

We define

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

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, $T_{\omega^{(0)}}(E) = \exp(\ell X_{\omega^{(0)}}(E)) \in \mathcal{O}$. $\square$

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 $E \in \mathbb{R}$,

$$(3.4) \qquad \mathfrak{g}(E) := \text{Lie}\{X_P(E); P \in \{0,1\}^N\}.$$

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:

* $\mathfrak{sp}_N(\mathbb{R})$ is the algebra associated with the symplectic group $\text{Sp}_N(\mathbb{R})$, it consists of all matrices of the shape $\begin{pmatrix} A & B \\ C & -tA \end{pmatrix}$, 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

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

This is the Lie algebra associated with the group $\text{SpO}_N(\mathbb{R})$, 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 $V_{\text{per}} = \Delta$. Then, for all $E \neq 0$,

* in cases 2, 3 and 4, $\mathfrak{g}(E) = \mathfrak{sp}_N(\mathbb{R})$;
*Proof.* Given a neighborhood $\mathcal{O}$ of 1 in $\text{Sp}_N(\mathbb{R})$, we set:

$$d_{\log \mathcal{O}} := \sup\{R > 0 \mid \exp[\bar{B}(0, R)] \subset \mathcal{O}\},$$

where $\bar{B}(0, R)$ is the closed ball, centered on 0 and of radius $R > 0$, for the topology induced on the Lie algebra $\mathfrak{sp}_N(\mathbb{R})$ 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 $\ell \|X_{\omega^{(0)}}(E)\| \le d_{\log \mathcal{O}}$.

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

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

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

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

We see that

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

We define

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

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, $T_{\omega^{(0)}}(E) = \exp(\ell X_{\omega^{(0)}}(E)) \in \mathcal{O}$. $\square$

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 $E \in \mathbb{R}$,

$$(3.4) \qquad \mathfrak{g}(E) := \text{Lie}\{X_P(E); P \in \{0,1\}^N\}.$$

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:

* $\mathfrak{sp}_N(\mathbb{R})$ is the algebra associated with the symplectic group $\text{Sp}_N(\mathbb{R})$, 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

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

This is the Lie algebra associated with the group $\text{SpO}_N(\mathbb{R})$, 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 $V_{\text{per}} = \Delta$. Then, for all $E \neq 0$,

* in cases 2, 3 and 4, $\mathfrak{g}(E) = \mathfrak{sp}_N(\mathbb{R})$;
• in case 5, $\mathfrak{g}(E) = \mathfrak{spo}_N(\mathbb{R})$.

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:

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

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

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

$$ X_P(E) = \begin{pmatrix} 0 & E+\Delta+\sum_{P_i=1} E_{ii} \\ -E+\Delta+\sum_{P_i=1} E_{ii} & 0 \end{pmatrix}. $$

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 $|i-j| \le 1$ are in $\mathfrak{g}(E)$. First, for any $P, P' \in \{0,1\}^N$,

$$ X_P(E) - X_{P'}(E) = \begin{pmatrix} 0 & \sum_{P_i=1} E_{ii} - \sum_{P'_i=1} E_{ii} \\ \sum_{P_i=1} E_{ii} - \sum_{P'_i=1} E_{ii} & 0 \end{pmatrix}. $$

For all $i \in \{1, \dots, N\}$, we can choose $P = (0, \dots, 1, \dots, 0)$ with the 1 in position $i$ and $P' = (0, \dots, 0)$. 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

$$ \{ \begin{pmatrix} 0 & D_1 \\ D_2 & 0 \end{pmatrix}, D_1, D_2 \text{ diagonal} \} \subset \mathfrak{g}(E). $$

Therefore,

$$ X_P(E) - \begin{pmatrix} 0 & E+\sum_{P_i=1} E_{ii} \\ -E+\sum_{P_i=1} E_{ii} & 0 \end{pmatrix} = \begin{pmatrix} 0 & \Delta \\ \Delta & 0 \end{pmatrix} \in \mathfrak{g}(E). $$

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

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

which is the space of symmetric tridiagonal matrices with zeros on the diagonal.
• in case 5, $\mathfrak{g}(E) = \mathfrak{spo}_N(\mathbb{R})$.

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:

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

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

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

$$ X_P(E) = \begin{pmatrix} 0 & E+\Delta+\sum_{P_i=1} E_{ii} \\ -E+\Delta+\sum_{P_i=1} E_{ii} & 0 \end{pmatrix}. $$

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 $|i-j| \le 1$ are in $\mathfrak{g}(E)$. First, for any $P, P' \in \{0,1\}^N$,

$$ X_P(E) - X_{P'}(E) = \begin{pmatrix} 0 & \sum_{P_i=1} E_{ii} - \sum_{P'_i=1} E_{ii} \\ \sum_{P_i=1} E_{ii} - \sum_{P'_i=1} E_{ii} & 0 \end{pmatrix}. $$

For all $i \in \{1, \dots, N\}$, we can choose $P = (0, \dots, 1, \dots, 0)$ with the 1 in position $i$ and $P' = (0, \dots, 0)$. 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

$$ \{ \begin{pmatrix} 0 & D_1 \\ D_2 & 0 \end{pmatrix}, D_1, D_2 \text{ diagonal} \} \subset \mathfrak{g}(E). $$

Therefore,

$$ X_P(E) - \begin{pmatrix} 0 \\ -E + \sum_{P_i=1} E_{ii} \end{pmatrix}^{E+\sum_{P_i=1} E_{ii}} = \begin{pmatrix} 0 \\ \Delta \end{pmatrix} \in \mathfrak{g}(E). $$

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

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

which is the space of symmetric tridiagonal matrices with zeros on the diagonal.
Now, for any $i$ in $\{1, \dots, N-1\}$,

$$ \begin{bmatrix} (E_{i,i+1} + E_{i+1,i}) & 0 \\ 0 & -(E_{i,i+1} + E_{i+1,i}) \end{bmatrix}, \begin{pmatrix} 0 & I_N \\ I_N & 0 \end{pmatrix} = 2Y_{i,i+1}. $$

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

$$ \begin{bmatrix} (E_{i,i+1} + E_{i+1,i}) & 0 \\ 0 & -(E_{i,i+1} + E_{i+1,i}) \end{bmatrix}, \begin{pmatrix} 0 & I_N \\ -I_N & 0 \end{pmatrix} = 2Z_{i,i+1}. $$

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 $\mathfrak{g}(E) = \mathfrak{sp}_N(\mathbb{R})$.

**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,

$$ X_P(E) = \begin{pmatrix} 0 & E-\Delta + \sum_{P_i=1}^{N} E_{ii} \\ -E+\Delta + \sum_{P_i=1}^{N} E_{ii} & 0 \end{pmatrix}. $$

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

$$ (3.7) \qquad X_P(E) - X_{P'}(E) = \begin{pmatrix} 0 & \sum_{P_i=1}^{N} E_{ii} - \sum_{P'_i=1}^{N} E_{ii} \\ \sum_{P_i=1}^{N} E_{ii} - \sum_{P'_i=1}^{N} E_{ii} & 0 \end{pmatrix} $$

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

Then,

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

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

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

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

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

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

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

On the other hand, we find that

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

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

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

and

$$ (3.13) \qquad \frac{1}{2}[\Theta, Z_{11}] = \begin{pmatrix} 0 & 2E_{11} - \{E_{11}, \Delta\} \\ -2E_{11} + \{E_{11}, \Delta\} & 0 \end{pmatrix} = EY_{11} + Y_{12}. $$
Now, for any $i$ in $\{1, \dots, N-1\}$,

$$ \begin{bmatrix} (E_{i,i+1} + E_{i+1,i}) & 0 \\ 0 & -(E_{i,i+1} + E_{i+1,i}) \end{bmatrix}, \begin{pmatrix} 0 & I_N \\ I_N & 0 \end{pmatrix} = 2Y_{i,i+1}. $$

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

$$ \begin{bmatrix} (E_{i,i+1} + E_{i+1,i}) & 0 \\ 0 & -(E_{i,i+1} + E_{i+1,i}) \end{bmatrix}, \begin{pmatrix} 0 & I_N \\ -I_N & 0 \end{pmatrix} = 2Z_{i,i+1}. $$

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 $\mathfrak{g}(E) = \mathfrak{sp}_N(\mathbb{R})$.

**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,

$$ X_P(E) = \begin{pmatrix} 0 & E-\Delta + \sum_{P_i=1}^{N} E_{ii} \\ -E+\Delta + \sum_{P_i=1}^{N} E_{ii} & 0 \end{pmatrix}. $$

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

$$ (3.7) \qquad X_P(E) - X_{P'}(E) = \begin{pmatrix} 0 & \sum_{P_i=1}^{N} E_{ii} - \sum_{P'_i=1}^{N} E_{ii} \\ \sum_{P_i=1}^{N} E_{ii} - \sum_{P'_i=1}^{N} E_{ii} & 0 \end{pmatrix} $$

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

Then,

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

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

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

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

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

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

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

On the other hand, we find that

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

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

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

and

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

$$X_P(E) = \left( \begin{array}{cc} -\Delta & E+\sum_{P_i=1}^{E+\sum_{P_i=1} E_{ii}} \\ -E+\sum_{P_i=1}^{E+\sum_{P_i=1} E_{ii}} & \Delta \end{array} \right).$$

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 $|i-j| \le 1$, are in $\mathfrak{g}(E)$. First, we see that for all $P, P'$

$$(3.14) \qquad X_P(E) - X_{P'}(E) = \sum_{P_i=1} Z_{ii} - \sum_{P'_i=1} Z_{ii} \in \mathfrak{g}(E)$$

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

Then,

$$X_P(E) - \sum_{P_i=1} Z_{ii} = \left( \begin{array}{c} -\Delta \\ -E \end{array} \right)^E \in \mathfrak{g}(E).$$

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

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

Similarly to the previous case, we can write

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

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

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

This implies that $\Theta' := \begin{pmatrix} -\Delta & 0 \\ 0 & \Delta \end{pmatrix} \in \mathfrak{g}(E)$. 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

$$(3.18) \qquad X_P(E) = \left( \begin{array}{cc} 0 & E+\Delta-\sum_{P_i=1}^{E+\Delta-E_{ii}} \\ -E+\Delta+\sum_{P_i=1}^{E+\Delta-E_{ii}} & 0 \end{array} \right).$$

Here, the Lie algebra $\mathfrak{g}(E)$ will not be $\mathfrak{sp}_N(\mathbb{R})$ but its subalgebra $\mathfrak{spo}_N(\mathbb{R})$. 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 $\mathfrak{spo}_N(\mathbb{R})$ contains all the $Y_{ii}$'s as well as the $X_{ij}$'s for $|i-j| = 1$, then $\mathfrak{g} = \mathfrak{spo}_N(\mathbb{R})$.

*Proof.* We first remark that all the $Y_{ii}$'s as well as the $X_{ij}$'s for $|i-j| = 1$ are in $\mathfrak{spo}_N(\mathbb{R})$. To prove the other inclusion, let $\mathfrak{g}$ be a subalgebra of $\mathfrak{spo}_N(\mathbb{R})$ which contains all these matrices. Recall that a basis of $\{A \in \mathcal{M}_N(\mathbb{R}) | {}^sA = A\}$ is given by $\{E_{ij} + (-1)^{i-j+1}E_{ji}\}_{1 \le i \le j \le N}$ and
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,

$$X_P(E) = \begin{pmatrix} -\Delta & E+\sum_{P_i=1}^{E+\sum_{P_i=1}^{E_{ii}}} E_{ii} \\ -E+\sum_{P_i=1}^{E} E_{ii} & \Delta \end{pmatrix}.$$

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 $|i-j| \le 1$, are in $\mathfrak{g}(E)$. First, we see that for all $P, P'$

$$(3.14) \qquad X_P(E) - X_{P'}(E) = \sum_{P_i=1} Z_{ii} - \sum_{P'_i=1} Z_{ii} \in \mathfrak{g}(E)$$

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

Then,

$$X_P(E) - \sum_{P_i=1} Z_{ii} = \begin{pmatrix} -\Delta & E \\ -E & \Delta \end{pmatrix} \in \mathfrak{g}(E).$$

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

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

Similarly to the previous case, we can write

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

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

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

This implies that $\Theta' := \begin{pmatrix} -\Delta & 0 \\ 0 & \Delta \end{pmatrix} \in \mathfrak{g}(E)$. 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

$$(3.18) \qquad X_P(E) = \begin{pmatrix} 0 & E+\Delta-\sum_{P_i=1}^{E+\Delta-E_{ii}} E_{ii} \\ -E+\Delta+\sum_{P_i=1}^{E} E_{ii} & 0 \end{pmatrix}$$

Here, the Lie algebra $\mathfrak{g}(E)$ will not be $\mathfrak{sp}_N(\mathbb{R})$ but its subalgebra $\mathfrak{spo}_N(\mathbb{R})$. 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 $\mathfrak{spo}_N(\mathbb{R})$ contains all the $Y_{ii}$'s as well as the $X_{ij}$'s for $|i-j| = 1$, then $\mathfrak{g} = \mathfrak{spo}_N(\mathbb{R})$.

*Proof.* We first remark that all the $Y_{ii}$'s as well as the $X_{ij}$'s for $|i-j| = 1$ are in $\mathfrak{spo}_N(\mathbb{R})$. To prove the other inclusion, let $\mathfrak{g}$ be a subalgebra of $\mathfrak{spo}_N(\mathbb{R})$ which contains all these matrices. Recall that a basis of $\{A \in \mathcal{M}_N(\mathbb{R}) | {}^sA = A\}$ is given by $\{E_{ij} + (-1)^{i-j+1}E_{ji}\}_{1 \le i \le j \le N}$ and
one of {$B \in \mathcal{M}_N(\mathbb{R}) | {}^tB = B$} is given by {$E_{ij} + E_{ji} \}_{1\le i\le j\le N}$. We introduce, for every $i, j \in \{1, \dots, N\}$, the matrices

$$V_{ij} = \begin{pmatrix} E_{ij} + (-1)^{i-j+1} E_{ji} & 0 \\ 0 & (-1)^{i-j} E_{ij} - E_{ji} \end{pmatrix} \quad \text{and} \quad W_{ij} = \begin{pmatrix} 0 & E_{ij} + E_{ji} \\ (-1)^{i-j+1} (E_{ij} + E_{ji}) & 0 \end{pmatrix}.$$

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

$$ \begin{align} [V_{ij}, W_{kr}] &= \delta_{jk} \begin{pmatrix} I_N & 0 \\ 0 & (-1)^{k-j} I_{I_N} \end{pmatrix} W_{ir} + \delta_{ik} (-1)^{i-j+1} \begin{pmatrix} I_N & 0 \\ 0 & (-1)^{i+k} I_{I_N} \end{pmatrix} W_{jr} \nonumber \\ &\quad + \delta_{ir} (-1)^{i-j+1} \begin{pmatrix} I_N & 0 \\ 0 & (-1)^{i-r} I_{I_N} \end{pmatrix} W_{jk} + \delta_{jr} \begin{pmatrix} I_N & 0 \\ 0 & (-1)^{-j-r} I_{I_N} \end{pmatrix} W_{ik} \tag{3.19} \\ &\quad = (-1)^{i-j+1} (\delta_{ik} W_{jr} + \delta_{ir} W_{jk}) + \delta_{jk} W_{ir} + \delta_{jr} W_{ik} \nonumber \end{align} $$

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

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

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

$$ \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j. \end{cases} $$

The hypothesis made on $\mathfrak{g}$ implies that for every $i \in \{1, \dots, N\}$, $W_{ii} \in \mathfrak{g}$ and for $|i-j| = 1$, $W_{ij} \in \mathfrak{g}$. Let $i \in \{1, \dots, N\}$. 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 $i \le j$, hence $\mathfrak{spo}_N(\mathbb{R}) \subset \mathfrak{g}$ and $\mathfrak{g} = \mathfrak{spo}_N(\mathbb{R})$. $\square$

To compute $\mathfrak{g}(E)$, we begin as usual by taking $X_P(E) - X_{P'}(E)$ 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 $\mathfrak{g}(E)$. $\square$

**3.2. p-contractivity and $L_p$-strong irreducibility.** As we already explained, $\text{Sp}_N(\mathbb{R})$ is p-contracting and $J-L_p$-strongly irreducible for all $p \in \{1, \dots, N\}$. We prove here the corresponding result for $\text{SpO}_N(\mathbb{R})$. We prove a similar result for $\text{SpO}_N(\mathbb{R})$.

**Proposition 3.10.** The group $\text{SpO}_N(\mathbb{R})$ is $2p$-contracting and $(J, S)-L_{2p}$-strongly irreducible for all $p \in \{1, \dots, N\}$.
one of {$B \in \mathcal{M}_N(\mathbb{R}) | {}^tB = B$} is given by {$E_{ij} + E_{ji} \}_{1\le i\le j\le N}$. We introduce, for every $i, j \in \{1, \dots, N\}$, the matrices

$$V_{ij} = \begin{pmatrix} E_{ij} + (-1)^{i-j+1} E_{ji} & 0 \\ 0 & (-1)^{i-j} E_{ij} - E_{ji} \end{pmatrix} \quad \text{and} \quad W_{ij} = \begin{pmatrix} 0 & E_{ij} + E_{ji} \\ (-1)^{i-j+1} (E_{ij} + E_{ji}) & 0 \end{pmatrix}.$$

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

$$ \begin{align} [V_{ij}, W_{kr}] &= \delta_{jk} \begin{pmatrix} I_N & 0 \\ 0 & (-1)^{k-j} I_{N} \end{pmatrix} W_{ir} + \delta_{ik} (-1)^{i-j+1} \begin{pmatrix} I_N & 0 \\ 0 & (-1)^{i+k} I_{N} \end{pmatrix} W_{jr} \nonumber \\ &\quad + \delta_{ir} (-1)^{i-j+1} \begin{pmatrix} I_N & 0 \\ 0 & (-1)^{i-r} I_{N} \end{pmatrix} W_{jk} + \delta_{jr} \begin{pmatrix} I_N & 0 \\ 0 & (-1)^{-j-r} I_{N} \end{pmatrix} W_{ik} \tag{3.19} \\ &\quad = (-1)^{i-j+1} (\delta_{ik} W_{jr} + \delta_{ir} W_{jk}) + \delta_{jk} W_{ir} + \delta_{jr} W_{ik} \nonumber \end{align} $$

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

$$ \begin{align} [W_{ij}, W_{kr}] &= (-1)^{k-r+1} \Biggl[ \delta_{jk} \begin{pmatrix} E_{ir} + (-1)^{ijkl} E_{ri} & 0 \\ 0 & -((-)_{ijkl} E_{ir} + E_{ri}) \end{pmatrix} + \delta_{jr} \begin{pmatrix} E_{ik} + (-1)^{ijkl} E_{ki} & 0 \\ 0 & -((-)_{ijkl} E_{ik} + E_{ki}) \end{pmatrix} \nonumber \\ &\quad + \delta_{ik} \begin{pmatrix} E_{jr} + (-1)^{ijkl} E_{rj} & 0 \\ 0 & -((-)_{ijkl} E_{jr} + E_{rj}) \end{pmatrix} + \delta_{ir} \begin{pmatrix} E_{jk} + (-1)^{ijkl} E_{kj} & 0 \\ 0 & -((-)_{ijkl} E_{jk} + E_{kj}) \end{pmatrix} \Biggr] \tag{3.20} \\ &\quad = (-1)^{k-r+1} (\delta_{jk} V_{ir} + \delta_{jr} V_{ik} + \delta_{ik} V_{jr} + \delta_{ir} V_{jk}) \nonumber \end{align} $$

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

$$ \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j. \end{cases} $$

The hypothesis made on $\mathfrak{g}$ implies that for every $i \in \{1, \dots, N\}$, $W_{ii} \in \mathfrak{g}$ and for $|i-j| = 1$, $W_{ij} \in \mathfrak{g}$. Let $i \in \{1, \dots, N\}$. 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 $i \le j$, hence $\mathfrak{spo}_N(\mathbb{R}) \subset \mathfrak{g}$ and $\mathfrak{g} = \mathfrak{spo}_N(\mathbb{R})$. $\square$

To compute $\mathfrak{g}(E)$, we begin as usual by taking $X_P(E) - X_{P'}(E)$ 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 $\mathfrak{g}(E)$. $\square$

3.2. **p-contractivity and $L_p$-strong irreducibility.** As we already explained, $\text{Sp}_N(\mathbb{R})$ is $p$-contracting and $J-L_p$-strongly irreducible for all $p \in \{1, \dots, N\}$. We prove here the corresponding result for $\text{SpO}_N(\mathbb{R})$. We prove a similar result for $\text{SpO}_N(\mathbb{R})$.

**Proposition 3.10.** The group $\text{SpO}_N(\mathbb{R})$ is $2p$-contracting and $(J, S)-L_{2p}$-strongly irreducible for all $p \in \{1, \dots, N\}$.
Proof. According to [5, Proposition A.IV.2.1], a subset of GL$_{2N}$(R) is 2p-contracting if there exists in it a sequence (M$_{n}$) such that, if we denote by s$_{1}$(M$_{n}$) ≥ · · · ≥ s$_{2N}$(M$_{n}$) the singular values of M$_{n}$, then lim$_{n}$$_{→∞}$ s$_{2}$$_{p}$$_{+1}$(M$_{n}$) /s$_{2}$$_{p}$(M$_{n}$) = 0. But, if we take a sequence t$_{1}$ > · · · > t$_{d}$ > 0 and construct the matrix R as in Proposition A.1 (iv), then for all n ∈ N the singular values of R n are, for p ≤ d, s$_{2}$$_{p}$$_{−}$$_{1}$(M$_{n}$) = s$_{2}$$_{p}$(M$_{n}$) = e nt$_{p}$ so the sequence satisfies the criterion: SpO$_{N}$(R) is 2p-contracting.

Let us now prove that SpO$_{N}$(R) is L$_{2p}$-strongly irreducible. Since SpO$_{N}$(R) is connected (see Proposition A.1), according to [5, Exercise A.IV.2.9], we only have to prove that there exists no proper subspace V of L$_{2p}$ such that (Λ $^{2p}$M )(V ) ⊂ V for all M in SpO$_{N}$(R). Assume that such a V exists and let us consider a matrix R as in Proposition A.1 (iv), with t$_{1}$ > · · · > t$_{d}$ . We have in particular that (Λ $^{2p}$R )(V ) ⊂ V . For f$_{1}$ defined in (2.4), let us write the unique decomposition f$_{1}$ = v + v ⊥ with v ∈ V and v ⊥ ∈ V $^{⊥}$. Then,

$$
(3.21) \qquad (\Lambda^{2p} R) f_1 = e^{2t_1} \cdots e^{2t_p} f_1 = (\Lambda^{2p} R)v + (\Lambda^{2p} R)v^{\perp}.
$$

Since V is stable by Λ $^{2p}$R and V ⊥ is stable by (Λ $^{2p}$R ) ∗ = Λ $^{2p}$R, (Λ $^{2p}$R ) v ∈ V and (Λ $^{2p}$R ) v ⊥ ∈ V ⊥ . By orthogonality, (Λ $^{2p}$R ) v = e 2 t$_{1}$ · · · e 2 t$_{p}$ v. If v = 0 then f$_{1}$ ∈ V ⊥ . If v ̸ = 0, v is an eigenvector of Λ $^{2p}$R associated with the eigenvalue e 2 t$_{1}$ · · · e 2 t$_{p}$ . But, by construction of R , its eigenspace associated with the eigenvalue e 2 t$_{1}$ · · · e 2 t$_{p}$ is Span( f$_{1}$ ). Hence, if v is an eigenvector of Λ $^{2p}$R n associated to the eigenvalue e 2 nt$_{1}$ · · · e 2 nt$_{p}$ , then v ∈ Span( f$_{1}$ ) and f$_{1}$ ∈ V . Thus, we proved that either f$_{1}$ ∈ V or f$_{1}$ ∈ V ⊥ .

If f$_{1}$ ∈ V , then by definition V = L$_{2p}$ . If f is orthogonal to V , then, for all v ∈ V and M ∈ SpO$_{N}$(R), 〈 Λ $^{2p}$M f$_{1}$, v 〉 = 〈 f$_{1}$, Λ $^{2p}$( t $^{)}$M ) v 〉 = 0 since t $^{)}$M is in SpO$_{N}$(R) as well. Consequently, V = L ⊥ 2 $_{p}$, which contradicts the hypothesis that V is proper. □

3.3. Proof of Theorem 1.11. The proof of Theorem 1.11 comes from the application of one of the localization criteria or the delocalization criterion ( i.e. Theorem 1.9, Theorem 1.10 or Theorem 1.12) to { D$_{ω}$ }$_{ω}$$_{∈}$$_{Ω}$ for the particular choice of V = Δ, the tridiagonal matrix with 0 on the diagonal and 1 on the upper and lower diagonal.

Proof of Theorem 1.11. Once we obtained Theorem 1.11 for V = Δ we apply the genericity argument developed in [9] in the following way. We fix G ∈ { Sp$_{N}$(R) , SpO$_{N}$(R) } and g ∈ { sp$_{N}$(R) , spo$_{N}$(R) } the corresponding Lie algebra. Both Sp$_{N}$(R) and SpO$_{N}$(R) are algebraic groups. The use of Theorem 3.5 to obtain the separability of Lyapunov exponents leads us to prove an algebraic property on a Lie algebra generated by a finite number of matrices. Hence, the n -tuples of elements in G that do not generate a dense subgroup are contained in a closed analytic subvariety which implies that we can perturb the interaction potential Δ into a potential V while keeping the property that the Furstenberg group is equal to G for any energy in an interval I ( N, ℓ, V ) for ℓ ∈ (0 , ℓ$_{C}$ ( N, V )). The transfer matrices are written in exponential form,

$$
T_{\omega^{(0)}}(E) = \exp(\ell X_{\omega^{(0)}}(E, V)),
$$

where the X$_{ω}$ (0) ( E, V ) are define in (3.3) and where we explicit the dependency in V of these matrices since E and V are both important variables in the genericity argument. We denote the family { X$_{ω}$ (0) ( E, V ) }$_{ω}$$_{(0)}$$_{∈{}$$_{0}$$_{,}$$_{1}$$_{}}$ N by X$_{1}$$_{(}$$_{E,}$$_{V}$$_{)}$ , . . . , X$_{2}$$_{N}$$_{(}$$_{E,}$$_{V}$$_{)}$ . For k ∈ N $^{∗}$, let

$$
(3.22) \qquad \mathcal{V}_k = \{ (X_1, \dots, X_k) \in \mathfrak{sp}_N(\mathbb{R})^k \mid (X_1, \dots, X_k) \text{ does not generate } \mathfrak{g} \}.
$$

Since generating the algebra g is an algebraic condition of the type non-vanishing of a finite family of determinants (finite because, for all m ∈ N $^{∗}$, the ring R [ T$_{1}$, . . . , T$_{m}$ ] is Noetherian),
Proof. According to [5, Proposition A.IV.2.1], a subset of GL$_{2N}$(R) is 2p-contracting if there exists in it a sequence (M$_{n}$) such that, if we denote by s$_{1}$(M$_{n}$) ≥ · · · ≥ s$_{2N}$(M$_{n}$) the singular values of M$_{n}$, then lim$_{n}$$_{→∞}$ s$_{2p}$$_{+1}$(M$_{n}$) /s$_{2p}$(M$_{n}$) = 0. But, if we take a sequence t$_{1}$ > · · · > t$_{d}$ > 0 and construct the matrix R as in Proposition A.1 (iv), then for all n ∈ N the singular values of R n are, for p ≤ d, s$_{2p}$$_{−}$$_{1}$(M$_{n}$) = s$_{2p}$(M$_{n}$) = e nt$_{p}$ so the sequence satisfies the criterion: SpO$_{N}$(R) is 2p-contracting.

Let us now prove that SpO$_{N}$(R) is L$_{2p}$-strongly irreducible. Since SpO$_{N}$(R) is connected (see Proposition A.1), according to [5, Exercise A.IV.2.9], we only have to prove that there exists no proper subspace V of L$_{2p}$ such that (Λ $^{2p}$M )(V ) ⊂ V for all M in SpO$_{N}$(R). Assume that such a V exists and let us consider a matrix R as in Proposition A.1 (iv), with t$_{1}$ > · · · > t$_{d}$ . We have in particular that (Λ $^{2p}$R )(V ) ⊂ V . For f$_{1}$ defined in (2.4), let us write the unique decomposition f$_{1}$ = v + v ⊥ with v ∈ V and v ⊥ ∈ V $^{⊥}$. Then,

$$
(3.21) \qquad (\Lambda^{2p} R) f_1 = e^{2t_1} \cdots e^{2t_p} f_1 = (\Lambda^{2p} R)v + (\Lambda^{2p} R)v^{\perp}.
$$

Since V is stable by Λ $^{2p}$R and V ⊥ is stable by (Λ $^{2p}$R) ∗ = Λ $^{2p}$R, (Λ $^{2p}$R ) v ∈ V and (Λ $^{2p}$R ) v ⊥ ∈ V ⊥ . By orthogonality, (Λ $^{2p}$R ) v = e 2 t$_{1}$ · · · e 2 t$_{p}$ v. If v = 0 then f$_{1}$ ∈ V ⊥ . If v ̸ = 0, v is an eigenvector of Λ $^{2p}$R associated with the eigenvalue e 2 t$_{1}$ · · · e 2 t$_{p}$ . But, by construction of R , its eigenspace associated with the eigenvalue e 2 t$_{1}$ · · · e 2 t$_{p}$ is Span( f$_{1}$ ). Hence, if v is an eigenvector of Λ $^{2p}$R n associated to the eigenvalue e 2 nt$_{1}$ · · · e 2 nt$_{p}$ , then v ∈ Span( f$_{1}$ ) and f$_{1}$ ∈ V . Thus, we proved that either f$_{1}$ ∈ V or f$_{1}$ ∈ V ⊥ .

If f$_{1}$ ∈ V , then by definition V = L$_{2p}$ . If f is orthogonal to V , then, for all v ∈ V and M ∈ SpO$_{N}$(R), 〈 Λ $^{2p}$M f$_{1}$, v 〉 = 〈 f$_{1}$, Λ $^{2p}$( t $^{)}$M ) v 〉 = 0 since t $^{)}$M is in SpO$_{N}$(R) as well. Consequently, V = L ⊥ 2 $_{p}$, which contradicts the hypothesis that V is proper. □

3.3. Proof of Theorem 1.11. The proof of Theorem 1.11 comes from the application of one of the localization criteria or the delocalization criterion ( i.e. Theorem 1.9, Theorem 1.10 or Theorem 1.12) to { D$_{ω}$ }$_{ω}$$_{∈}$$_{Ω}$ for the particular choice of V = Δ, the tridiagonal matrix with 0 on the diagonal and 1 on the upper and lower diagonal.

Proof of Theorem 1.11. Once we obtained Theorem 1.11 for V = Δ we apply the genericity argument developed in [9] in the following way. We fix G ∈ { Sp$_{N}$(R) , SpO$_{N}$(R) } and g ∈ { sp$_{N}$(R) , spo$_{N}$(R) } the corresponding Lie algebra. Both Sp$_{N}$(R) and SpO$_{N}$(R) are algebraic groups. The use of Theorem 3.5 to obtain the separability of Lyapunov exponents leads us to prove an algebraic property on a Lie algebra generated by a finite number of matrices. Hence, the n -tuples of elements in G that do not generate a dense subgroup are contained in a closed analytic subvariety which implies that we can perturb the interaction potential Δ into a potential V while keeping the property that the Furstenberg group is equal to G for any energy in an interval I ( N, ℓ, V ) for ℓ ∈ (0 , ℓ$_{C}$ ( N, V )). The transfer matrices are written in exponential form,

$$
T_{\omega^{(0)}}(E) = \exp(\ell X_{\omega^{(0)}}(E, V)),
$$

where the X$_{ω}$ (0) ( E, V ) are define in (3.3) and where we explicit the dependency in V of these matrices since E and V are both important variables in the genericity argument. We denote the family { X$_{ω}$ (0) ( E, V ) }$_{ω}$$_{(0)}$$_{∈{}$$_{0}$$_{,}$$_{1}$$_{}}$ N by X$_{1}$ ( E, V ) , . . . , X$_{2}$$_{N}$ ( E, V ). For k ∈ N $^{∗}$, let

$$
(3.22) \qquad \mathcal{V}_k = \{ (X_1, \dots, X_k) \in \mathfrak{sp}_N(\mathbb{R})^k \mid (X_1, \dots, X_k) \text{ does not generate } \mathfrak{g} \}.
$$

Since generating the algebra g is an algebraic condition of the type non-vanishing of a finite family of determinants (finite because, for all m ∈ N $^{∗}$, the ring R [ T$_{1}$, . . . , T$_{m}$ ] is Noetherian),
there exist $Q_1, \dots, Q_{r_k} \in \mathbb{R}[g^k]$ such that:

$$ (3.23) \qquad \mathcal{V}_k = \{ (X_1, \dots, X_k) \in g^k \mid Q_1(X_1, \dots, X_k) = 0, \dots, Q_{r_k}(X_1, \dots, X_k) = 0 \}. $$

Here, we identify $\mathbb{R}[g^k] \simeq \mathbb{R}[T_1, \dots, T_{k \times \dim g}]$ where $\dim g = 2N^2 + N$ for $g = \text{sp}_N(\mathbb{R})$ and $\dim g = N^2$ for $g = \text{spo}_N(\mathbb{R})$. Let $E \in \mathbb{R}$ and,

$$ (3.24) \qquad \mathcal{V}(E) = \{ V \in \text{S}_N(\mathbb{R}) \mid X_1(E, V), \dots, X_{2^N}(E, V) \text{ does not generate } g \}. $$

We show that $\text{Leb}_{\frac{N(N+1)}{2}}(\mathcal{V}(E)) = 0$. Indeed, let

$$ (3.25) \qquad f_E : \begin{array}{rcl} \text{S}_N(\mathbb{R}) & \to & g^{2^N} \\ V & \mapsto & (X_1(E, V), \dots, X_{2^N}(E, V)) \end{array} . $$

Then, $f_E$ is polynomial in the $\frac{N(N+1)}{2}$ coefficients defining $V$, and we have:

$$ (3.26) \qquad V \in \mathcal{V}(E) \Leftrightarrow (Q_1 \circ f_E)(V) = 0, \dots, (Q_{r_{2^N}} \circ f_E)(V) = 0, $$

where each $Q_i \circ f_E$ is polynomial in the $\frac{N(N+1)}{2}$ coefficients defining $V$. But, we have shown in Section 1.2 that in Cases 2, 3, 4 and 5 for $N$ even, $\Delta \notin \mathcal{V}(E)$. Therefore, there exists $i_0 \in \{1, \dots, r_{2^N}\}$ such that $(Q_{i_0} \circ f_E)(\Delta) \neq 0$, and since the function $Q_{i_0} \circ f_E$ is polynomial and not identically zero,

$$ (3.27) \qquad \text{Leb}_{\frac{N(N+1)}{2}} (\{V \in \text{S}_N(\mathbb{R}) \mid (Q_i \circ f_E)(V) = 0\}) = 0, $$

and, by inclusion,

$$ (3.28) \qquad \forall E \in \mathbb{R}, \text{Leb}_{\frac{N(N+1)}{2}}(\mathcal{V}(E)) = 0. $$

Finally, let $\mathcal{V}(g) = \cap_{E \in \mathbb{R}} \mathcal{V}(E)$. Then $\mathcal{V}(g)$ has Lebesgue measure zero, and if $V \notin \mathcal{V}(g)$, there exists $E_0 \in \mathbb{R}$ such that the family $\{X_1(E_0, V), \dots, X_{2^N}(E_0, V)\}$ generates $g$. Therefore, there exists $i_0 \in \{1, \dots, r_{2^N}\}$ such that $(Q_{i_0} \circ f)(E_0, V) \neq 0$, where:

$$ (3.29) \qquad f : \begin{array}{rcl} \mathbb{R} \times \text{S}_N(\mathbb{R}) & \to & g^{2^N} \\ (E, V) & \mapsto & (X_1(E, V), \dots, X_{2^N}(E, V)) \end{array} . $$

Now, for $V$ fixed, $E \mapsto (Q_{i_0} \circ f_E)(E, V)$ is polynomial and not identically zero, so it has only a finite set $S_V$ of zeros, and for all $E \in \mathbb{R} \setminus S_V$, $\{X_1(E, V), \dots, X_{2^N}(E, V)\}$ does not generate $g$.

Thus we have obtained that $\mathcal{V}(g)$ has Lebesgue measure zero, and if $V \notin \mathcal{V}(g)$, there exists $S_V \subset \mathbb{R}$ finite such that for all $E \in \mathbb{R} \setminus S_V$, $\{X_1(E, V), \dots, X_{2^N}(E, V)\}$ generates $g$.

From there, we finish the proof of Theorem 1.11. In Cases 2, 3, 4, we fix $V \in \text{S}_N(\mathbb{R}) \setminus \mathcal{V}(\text{sp}_N(\mathbb{R}))$ and apply Theorem 3.5, using the real number $\ell_C(N, V)$ and the interval $I(N, V, \ell)$ given by Lemma 3.6. Then we get that the hypotheses of Theorem 1.9 are satisfied for this $V$ on $I(N, V, \ell) \setminus S_V$ since $G(E) = \text{Sp}_N(\mathbb{R})$ on this interval. In Case 5 we fix $V \in \text{S}_N(\mathbb{R}) \setminus \mathcal{V}(\text{spo}_N(\mathbb{R}))$ and we do the same as in Cases 2, 3, 4 to get that the hypotheses of Theorem 1.10 are satisfied since $G(E) = \text{SpO}_N(\mathbb{R})$ on the constructed interval. $\square$

*Remark 3.11.* It is the algebraic nature of the objects involved that allows us to prove a generic result in $V$ and the finiteness of the set of critical energies. We can summarize the ideas used simply by recalling that the set of zeros of a non-zero single-variable polynomial is finite and that more generally, the set of zeros of a non-zero polynomial in several variables is of Lebesgue measure zero.
there exist $Q_1, \dots, Q_{r_k} \in \mathbb{R}[g^k]$ such that:

$$ (3.23) \qquad \mathcal{V}_k = \{ (X_1, \dots, X_k) \in g^k \mid Q_1(X_1, \dots, X_k) = 0, \dots, Q_{r_k}(X_1, \dots, X_k) = 0 \}. $$

Here, we identify $\mathbb{R}[g^k] \simeq \mathbb{R}[T_1, \dots, T_{k \times \dim g}]$ where $\dim g = 2N^2 + N$ for $g = \text{sp}_N(\mathbb{R})$ and $\dim g = N^2$ for $g = \text{spo}_N(\mathbb{R})$. Let $E \in \mathbb{R}$ and,

$$ (3.24) \qquad \mathcal{V}(E) = \{ V \in \text{S}_N(\mathbb{R}) \mid X_1(E, V), \dots, X_{2^N}(E, V) \text{ does not generate } g \}. $$

We show that $\text{Leb}_{\frac{N(N+1)}{2}}(\mathcal{V}(E)) = 0$. Indeed, let

$$ (3.25) \qquad f_E : \begin{array}{ccc} \text{S}_N(\mathbb{R}) & \to & g^{2^N} \\ V & \mapsto & (X_1(E, V), \dots, X_{2^N}(E, V)) \end{array} $$

Then, $f_E$ is polynomial in the $\frac{N(N+1)}{2}$ coefficients defining $V$, and we have:

$$ (3.26) \qquad V \in \mathcal{V}(E) \Leftrightarrow (Q_1 \circ f_E)(V) = 0, \dots, (Q_{r_{2^N}} \circ f_E)(V) = 0, $$

where each $Q_i \circ f_E$ is polynomial in the $\frac{N(N+1)}{2}$ coefficients defining $V$. But, we have shown in Section 1.2 that in Cases 2, 3, 4 and 5 for $N$ even, $\Delta \notin \mathcal{V}(E)$. Therefore, there exists $i_0 \in \{1, \dots, r_{2^N}\}$ such that $(Q_{i_0} \circ f_E)(\Delta) \neq 0$, and since the function $Q_{i_0} \circ f_E$ is polynomial and not identically zero,

$$ (3.27) \qquad \text{Leb}_{\frac{N(N+1)}{2}} (\{V \in \text{S}_N(\mathbb{R}) \mid (Q_i \circ f_E)(V) = 0\}) = 0, $$

and, by inclusion,

$$ (3.28) \qquad \forall E \in \mathbb{R}, \text{Leb}_{\frac{N(N+1)}{2}}(\mathcal{V}(E)) = 0. $$

Finally, let $\mathcal{V}(g) = \cap_{E \in \mathbb{R}} \mathcal{V}(E)$. Then $\mathcal{V}(g)$ has Lebesgue measure zero, and if $V \notin \mathcal{V}(g)$, there exists $E_0 \in \mathbb{R}$ such that the family $\{X_1(E_0, V), \dots, X_{2^N}(E_0, V)\}$ generates $g$. Therefore, there exists $i_0 \in \{1, \dots, r_{2^N}\}$ such that $(Q_{i_0} \circ f)(E_0, V) \neq 0$, where:

$$ (3.29) \qquad f : \begin{array}{ccc} \mathbb{R} \times \text{S}_N(\mathbb{R}) & \to & g^{2^N} \\ (E, V) & \mapsto & (X_1(E, V), \dots, X_{2^N}(E, V)) \end{array} $$

Now, for $V$ fixed, $E \mapsto (Q_{i_0} \circ f_E)(E, V)$ is polynomial and not identically zero, so it has only a finite set $S_V$ of zeros, and for all $E \in \mathbb{R} \setminus S_V$, $\{X_1(E, V), \dots, X_{2^N}(E, V)\}$ does not generate $g$.

Thus we have obtained that $\mathcal{V}(g)$ has Lebesgue measure zero, and if $V \notin \mathcal{V}(g)$, there exists $S_V \subset \mathbb{R}$ finite such that for all $E \in \mathbb{R} \setminus S_V$, $\{X_1(E, V), \dots, X_{2^N}(E, V)\}$ generates $g$.

From there, we finish the proof of Theorem 1.11. In Cases 2, 3, 4, we fix $V \in \text{S}_N(\mathbb{R}) \setminus \mathcal{V}(\text{sp}_N(\mathbb{R}))$ and apply Theorem 3.5, using the real number $\ell_C(N, V)$ and the interval $I(N, V, \ell)$ given by Lemma 3.6. Then we get that the hypotheses of Theorem 1.9 are satisfied for this $V$ on $I(N, V, \ell) \setminus S_V$ since $G(E) = \text{Sp}_N(\mathbb{R})$ on this interval. In Case 5 we fix $V \in \text{S}_N(\mathbb{R}) \setminus \mathcal{V}(\text{spo}_N(\mathbb{R}))$ and we do the same as in Cases 2, 3, 4 to get that the hypotheses of Theorem 1.10 are satisfied since $G(E) = \text{SpO}_N(\mathbb{R})$ on the constructed interval. $\square$

*Remark 3.11.* It is the algebraic nature of the objects involved that allows us to prove a generic result in $V$ and the finiteness of the set of critical energies. We can summarize the ideas used simply by recalling that the set of zeros of a non-zero single-variable polynomial is finite and that more generally, the set of zeros of a non-zero polynomial in several variables is of Lebesgue measure zero.
APPENDIX A. THE GROUP SpO<sub>N</sub>(R)

In order to study the Lyapunov exponents associated with a sequence of i.i.d. matrices of
SpO<sub>N</sub>(R), as defined by (1.10), we state here the most important properties of this group.

**Proposition A.1.** Let $M \in \text{SpO}_N(\mathbb{R})$.

(1) $^tM \in \text{SpO}_N(\mathbb{R})$.

(2) If $Mv = \lambda v$ for $v \in \mathbb{R}^{2N}$ and $\lambda \in \mathbb{R}$, then $^tMJv = \lambda^{-1}Jv$, $^tMSv = \lambda^{-1}Sv$ and $MJ Sv = \lambda JSv$.

(3) For all $v \in \mathbb{R}^{2N}$, the vectors $v$ and $JSv$ are orthogonal.

(4) For $t \in \mathbb{R}$, we denote $B_t := \begin{pmatrix} \cosh t & \sinh t \\ \sinh t & \cosh t \end{pmatrix}$. If $N$ is even, we denote $N =: 2d$ and, if $N$ is odd, $N =: 2d + 1$. There exists $U \in \text{SO}_{2N}(\mathbb{R}) \cap \text{SpO}_N(\mathbb{R})$ and real numbers $t_1 \ge \cdots \ge t_d \ge 0$ such that, if we denote in the even case

$$
R := \begin{pmatrix} \operatorname{diag}(B_{t_1}, \dots, B_{t_d}) & 0 \\ 0 & \operatorname{diag}(B_{-t_1}, \dots, B_{-t_d}) \end{pmatrix},
$$

in the odd case

$$
R := \begin{pmatrix} \operatorname{diag}(B_{t_1}, \dots, B_{t_d}, 1) & 0 \\ 0 & \operatorname{diag}(B_{-t_1}, \dots, B_{-t_d}, 1) \end{pmatrix},
$$

then

$$
(A.1) \qquad U^t M M^t U = R.
$$

(5) There exists $K$, $U \in \text{SO}_{2N}(\mathbb{R}) \cap \text{SpO}_N(\mathbb{R})$ and a matrix $R$ similar to the one introduced above such that

$$
(A.2) \qquad M = KRU.
$$

(6) The group SpO<sub>N</sub>(R) is pathwise connected.

(7) The Lie algebra of the Lie group SpO<sub>N</sub>(R) is denoted by spo<sub>N</sub>(R) and is given by

$$
\mathfrak{spo}_N(\mathbb{R}) := \left\{ \begin{pmatrix} A & B \\ sB & -tA \end{pmatrix} \text{ with } {}^s A = A, {}^t B = B \right\}
$$

where the operator $^s$ is defined on $\mathcal{M}_N(\mathbb{R})$ by $({}^s M)_{ij} = (-1)^{i-j+1} M_{ji}$.

(8) The group SpO<sub>N</sub>(R) is semisimple.

Proof. (1) If we take the inverse of the relations defining SpO<sub>N</sub>(R), we find, since S<sup>-1</sup> = S and J<sup>-1</sup> = -J,

$$
M^{-1}S(^{t}M)^{-1} = S \text{ and } M^{-1}J(^{t}M)^{-1} = J,
$$

so $(^tM)^{-1} \in \text{SpO}_N(\mathbb{R})$, which implies that it is the case for $^tM$ as well, since $\text{SpO}_N(\mathbb{R})$
is a group.

(2) If $Mv = \lambda v$, then $\lambda^t M J v = {}^t M J M v = J v$. The same argument is true if we replace $J$ by $S$. Combining these two results, we find the one with $JS$.

(3) Since $JS = \begin{pmatrix} 0 & -K \\ K & 0 \end{pmatrix}$, for $v = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$, we have

$$
v \cdot JSv = -v_1 \cdot Kv_2 + v_2 \cdot Kv_1 = 0
$$

since the matrix K is symmetric.
APPENDIX A. THE GROUP SpO<sub>N</sub>(R)

In order to study the Lyapunov exponents associated with a sequence of i.i.d. matrices of
SpO<sub>N</sub>(R), as defined by (1.10), we state here the most important properties of this group.

**Proposition A.1.** Let $M \in \text{SpO}_N(\mathbb{R})$.

(1) $^tM \in \text{SpO}_N(\mathbb{R})$.

(2) If $Mv = \lambda v$ for $v \in \mathbb{R}^{2N}$ and $\lambda \in \mathbb{R}$, then $^tMJv = \lambda^{-1}Jv$, $^tMSv = \lambda^{-1}Sv$ and $MJSv = \lambda JSv$.

(3) For all $v \in \mathbb{R}^{2N}$, the vectors $v$ and $JSv$ are orthogonal.

(4) For $t \in \mathbb{R}$, we denote $B_t := \begin{pmatrix} \cosh t & \sinh t \\ \sinh t & \cosh t \end{pmatrix}$. If $N$ is even, we denote $N =: 2d$ and, if $N$ is odd, $N =: 2d + 1$. There exists $U \in \text{SO}_{2N}(\mathbb{R}) \cap \text{SpO}_N(\mathbb{R})$ and real numbers $t_1 \ge \cdots \ge t_d \ge 0$ such that, if we denote in the even case

$$
R := \begin{pmatrix} \operatorname{diag}(B_{t_1}, \dots, B_{t_d}) & 0 \\ 0 & \operatorname{diag}(B_{-t_1}, \dots, B_{-t_d}) \end{pmatrix},
$$

in the odd case

$$
R := \begin{pmatrix} \operatorname{diag}(B_{t_1}, \dots, B_{t_d}, 1) & 0 \\ 0 & \operatorname{diag}(B_{-t_1}, \dots, B_{-t_d}, 1) \end{pmatrix},
$$

then

$$
(A.1) \qquad U^t M M^t U = R.
$$

(5) There exists $K$, $U \in \text{SO}_{2N}(\mathbb{R}) \cap \text{SpO}_N(\mathbb{R})$ and a matrix $R$ similar to the one introduced above such that

$$
(A.2) \qquad M = KRU.
$$

(6) The group SpO<sub>N</sub>(R) is pathwise connected.

(7) The Lie algebra of the Lie group SpO<sub>N</sub>(R) is denoted by spo<sub>N</sub>(R) and is given by

$$
\mathfrak{spo}_N(\mathbb{R}) := \left\{ \begin{pmatrix} A & B \\ sB & -tA \end{pmatrix} \text{ with } {}^s A = A, {}^t B = B \right\}
$$

where the operator $^s$ is defined on $\mathcal{M}_N(\mathbb{R})$ by $({}^s M)_{ij} = (-1)^{i-j+1} M_{ji}$.

(8) The group SpO<sub>N</sub>(R) is semisimple.

Proof. (1) If we take the inverse of the relations defining SpO<sub>N</sub>(R), we find, since S<sup>-1</sup> = S and J<sup>-1</sup> = -J,

$$
M^{-1} S ({}^t M)^{-1} = S \text{ and } M^{-1} J ({}^t M)^{-1} = J,
$$

so $(^tM)^{-1} \in \text{SpO}_N(\mathbb{R})$, which implies that it is the case for $^tM$ as well, since $\text{SpO}_N(\mathbb{R})$
is a group.

(2) If $Mv = \lambda v$, then $\lambda^t M J v = {}^t M J M v = J v$. The same argument is true if we replace $J$ by $S$. Combining these two results, we find the one with $JS$.

(3) Since $JS = \begin{pmatrix} 0 & -K \\ K & 0 \end{pmatrix}$, for $v = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$, we have

$$
v \cdot JSv = -v_1 \cdot Kv_2 + v_2 \cdot Kv_1 = 0
$$

since the matrix K is symmetric.
(4) We first prove that $^tMM$ has an orthonormal basis of eigenvectors which can be written

$$ (v_1, \dots, v_d, Jv_1, \dots, Jv_d, Sv_1, \dots, Sv_d, JSv_1, \dots, JSv_d) $$

when $N = 2d$ and

$$ (v_1, \dots, v_d, v_{d+1}, Jv_1, \dots, Jv_d, Jv_{d+1}Sv_1, \dots, Sv_d, JSv_1, \dots, JSv_d) $$

when $N = 2d + 1$. To this purpose, we split the space $\mathbb{R}^{2N}$ into 2 subspaces: $V_1$ is the eigenspace of $^tMM$ associated with the eigenvalue 1 and $V_0 = V_1^\perp$.

Let us first consider $V_0$: we prove by induction that its dimension is a multiple of 4 and that it has a basis of the desired form. If the dimension is 0, there is nothing to prove. Otherwise, we take an eigenvector $v_1$ of $^tMM$, associated with its largest eigenvalue $e^{t_1} > 1$ which has norm 1. We see by Point (2) that $JSv$ is an eigenvector associated with the same eigenvalue and, by Point (3), it is orthogonal to $v_1$. It has norm 1 as well. Since $^tMM$ is symmetric, $Jv_1$ and $Sv_1$ are by Point (2) eigenvectors of it as well, associated with the eigenvalue $e^{-t_1} \neq e^{t_1}$. They both have norm 1, are orthogonal to each other (since $Jv_1 = JSSv_1$), and orthogonal to $v_1$ and $JSv_1$ since the eigenspaces associated with different eigenvalues are orthogonal. Then, the space spanned by the 4 vectors $v_1, Jv_1, Sv_1$ and $JSv_1$ has dimension 4 and has an orthonormal basis of the desired form. We can now consider the orthogonal of the space spanned by these 4 eigenvectors to apply the induction hypothesis.

We now consider the space $V_1$. Note that this space is stable under $S$. Therefore, $V_1 = V_1^+ \oplus V_1^-$, where the spaces $V_1^\pm := \{w \in V_1, Sw = \pm w\}$, are orthogonal. These 2 spaces are moreover isomorphic since, if $w \in V_1^+$, then $Jw \in V_1^-$ and vice versa. Let us assume that $\dim(V_1) \ge 4$, which implies that $\dim(V_1^+) \ge 2$. We can consider two orthogonal vectors $w_1, w_2 \in V_1^+$, both with norm 1. If we define $v_{r+1} = (w_1+Jw_2)/\sqrt{2}$, we can see that the vectors $v_{r+1}, Jv_{r+1} = (Jw_1-w_2)/\sqrt{2}$, $Sv_{r+1} = (w_1-Jw_2)/\sqrt{2}$ and $JSv_{r+1} = (Jw_1+w_2)/\sqrt{2}$ are orthonormal. We can complete the basis by applying the same process to the orthogonal of the space spanned by these 4 vectors in $V_1$. If $N$ is even, the dimension of $V_1$ is a multiple of 4 so we can construct the whole basis in this way. If $N$ is odd, we are left with a 2-dimensional subspace of $V_1$, of which we can take an orthonormal basis of the form $(v_{d+1}, Jv_{d+1})$.

We now construct the matrix $U$ in the following way. For $i = 1, \dots, d$, we define $Uv_i = 1/\sqrt{2}(e_{2i} + e_{2i-1})$, $UJv_i = 1/\sqrt{2}(e_{2i+N} + e_{2i-1+N})$, $USv_i = 1/\sqrt{2}(e_{2i} - e_{2i-1})$ and $UJSv_i = 1/\sqrt{2}(e_{2i+N} - e_{2i-1+N})$. If $N$ is odd, we define too $Uv_{d+1} = e_N$ and $UJv_{d+1} = e_{2N}$. We easily see that such a $U$ is unitary. Moreover, by construction, $U^tMM^tUe_{2i-1} = \cosh t_i e_{2i-1} + \sinh t_i e_{2i}$, $U^tMM^tUe_{2i} = \cosh t_i e_{2i} + \sinh t_i e_{2i-1}$, $U^tMM^tUe_{2i-1+N} = \cosh t_i e_{2i-1+N} - \sinh t_i e_{2i+N}$ and $U^tMM^tUe_{2i+N} = \cosh t_i e_{2i+N} - \sinh t_i e_{2i-1+N}$, so $U^tMM^tU = R$. We can explicitly compute that $UJ^tUe_i = e_{N+i}$ and $UJ^tUe_{N+i} = -e_i$ so $UJ^tU = J$. Similarly, we prove that $US^tU = S$ so $U \in \text{SpO}_N(\mathbb{R})$.

Since $U$ is a symplectic matrix, its determinant is 1, so $U \in SO_{2N}(\mathbb{R})$.

(5) Let us use the previous point to write that there exists $R, U$ such that $U^tMM^tU = R^2$, since $R^2$ has the same shape as $R$ with $t_i$ replaced by $2t_i$. If we set $K = M^tUR^{-1}$, we see that $K \in G$, in particular it has determinant 1. Moreover, $M = KRU$ and $^tKK = R^{-1}U^tMM^tUR^{-1} = 1$.
(4) We first prove that $^tMM$ has an orthonormal basis of eigenvectors which can be written

$$ (v_1, \dots, v_d, Jv_1, \dots, Jv_d, Sv_1, \dots, Sv_d, JSv_1, \dots, JSv_d) $$

when $N = 2d$ and

$$ (v_1, \dots, v_d, v_{d+1}, Jv_1, \dots, Jv_d, Jv_{d+1}Sv_1, \dots, Sv_d, JSv_1, \dots, JSv_d) $$

when $N = 2d + 1$. To this purpose, we split the space $\mathbb{R}^{2N}$ into 2 subspaces: $V_1$ is the eigenspace of $^tMM$ associated with the eigenvalue 1 and $V_0 = V_1^\perp$.

Let us first consider $V_0$: we prove by induction that its dimension is a multiple of 4 and that it has a basis of the desired form. If the dimension is 0, there is nothing to prove. Otherwise, we take an eigenvector $v_1$ of $^tMM$, associated with its largest eigenvalue $e^{t_1} > 1$ which has norm 1. We see by Point (2) that $JSv$ is an eigenvector associated with the same eigenvalue and, by Point (3), it is orthogonal to $v_1$. It has norm 1 as well. Since $^tMM$ is symmetric, $Jv_1$ and $Sv_1$ are by Point (2) eigenvectors of it as well, associated with the eigenvalue $e^{-t_1} \neq e^{t_1}$. They both have norm 1, are orthogonal to each other (since $Jv_1 = JSSv_1$), and orthogonal to $v_1$ and $JSv_1$ since the eigenspaces associated with different eigenvalues are orthogonal. Then, the space spanned by the 4 vectors $v_1, Jv_1, Sv_1$ and $JSv_1$ has dimension 4 and has an orthonormal basis of the desired form. We can now consider the orthogonal of the space spanned by these 4 eigenvectors to apply the induction hypothesis.

We now consider the space $V_1$. Note that this space is stable under $S$. Therefore, $V_1 = V_1^+ \oplus V_1^-$, where the spaces $V_1^\pm := \{w \in V_1, Sw = \pm w\}$, are orthogonal. These 2 spaces are moreover isomorphic since, if $w \in V_1^+$, then $Jw \in V_1^-$ and vice versa. Let us assume that $\dim(V_1) \geq 4$, which implies that $\dim(V_1^+) \geq 2$. We can consider two orthogonal vectors $w_1, w_2 \in V_1^+$, both with norm 1. If we define $v_{r+1} = (w_1+Jw_2)/\sqrt{2}$, we can see that the vectors $v_{r+1}, Jv_{r+1} = (Jw_1-w_2)/\sqrt{2}$, $Sv_{r+1} = (w_1-Jw_2)/\sqrt{2}$ and $JSv_{r+1} = (Jw_1+w_2)/\sqrt{2}$ are orthonormal. We can complete the basis by applying the same process to the orthogonal of the space spanned by these 4 vectors in $V_1$. If $N$ is even, the dimension of $V_1$ is a multiple of 4 so we can construct the whole basis in this way. If $N$ is odd, we are left with a 2-dimensional subspace of $V_1$, of which we can take an orthonormal basis of the form $(v_{d+1}, Jv_{d+1})$.

We now construct the matrix $U$ in the following way. For $i = 1, \dots, d$, we define $Uv_i = 1/\sqrt{2}(e_{2i} + e_{2i-1})$, $UJv_i = 1/\sqrt{2}(e_{2i+N} + e_{2i-1+N})$, $USv_i = 1/\sqrt{2}(e_{2i} - e_{2i-1})$ and $UJSv_i = 1/\sqrt{2}(e_{2i+N} - e_{2i-1+N})$. If $N$ is odd, we define too $Uv_{d+1} = e_N$ and $UJv_{d+1} = e_{2N}$. We easily see that such a $U$ is unitary. Moreover, by construction, $U^tMM^tUe_{2i-1} = \cosh t_i e_{2i-1} + \sinh t_i e_{2i}$, $U^tMM^tUe_{2i} = \cosh t_i e_{2i} + \sinh t_i e_{2i-1}$, $U^tMM^tUe_{2i-1+N} = \cosh t_i e_{2i-1+N} - \sinh t_i e_{2i+N}$ and $U^tMM^tUe_{2i+N} = \cosh t_i e_{2i+N} - \sinh t_i e_{2i-1+N}$, so $U^tMM^tU = R$. We can explicitly compute that $UJ^tUe_i = e_{N+i}$ and $UJ^tUe_{N+i} = -e_i$ so $UJ^tU = J$. Similarly, we prove that $US^tU = S$ so $U \in \text{SpO}_N(\mathbb{R})$.

Since $U$ is a symplectic matrix, its determinant is 1, so $U \in SO_{2N}(\mathbb{R})$.

(5) Let us use the previous point to write that there exists $R, U$ such that $U^tMM^tU = R^2$, since $R^2$ has the same shape as $R$ with $t_i$ replaced by $2t_i$. If we set $K = M^tUR^{-1}$, we see that $K \in G$, in particular it has determinant 1. Moreover, $M = KRU$ and $^tKK = R^{-1}U^tMM^tUR^{-1} = 1$.
(6) Let us prove that the group $SO_{2N}(\mathbb{R}) \cap SpO_N(\mathbb{R})$ is pathwise connected. This, together with the decomposition given by point (5), will imply that $SpO_N(\mathbb{R})$ is pathwise connected. To this purpose, we introduce the matrices

$$ S' = \begin{pmatrix} I_k & 0 \\ 0 & -I_{k'} \end{pmatrix} \quad \text{and} \quad J' = \begin{pmatrix} 0 & -I_{k/2} & 0 & 0 \\ I_{k/2} & 0 & 0 & 0 \\ 0 & 0 & 0 & -I_{k'/2} \\ 0 & 0 & I_{k'/2} & 0 \end{pmatrix}, $$

where $k = k' = N$ when $N$ is even and $k = N + 1$ and $k' = N - 1$ when $N$ is odd. We see that $SO_{2N}(\mathbb{R}) \cap SpO_N(\mathbb{R})$ is homeomorphic to $SO_{2N}(\mathbb{R}) \cap G_5'$, where we have

$$ G_5' = \{ M \in \mathcal{M}_{2N}(\mathbb{R}), {}^t M J' M = J', {}^t M S' M = S' \}. $$

Indeed, the matrices $J$ and $J'$ (resp. $S$ and $S'$) are unitarily equivalent since $J'$ is obtained from $J$ by permutating rows and columns. Therefore, $SpO_N(\mathbb{R})$ and $SpO_N(\mathbb{R})'$ are unitarily equivalent hence homeomorphic.

We take a matrix $M = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in SO_{2N}(\mathbb{R}) \cap G_5'$, where the blocks have respective size $k$ and $k'$. Since ${}^t M M = I_{2N}$,

$$ (A.3) \qquad \left\{ \begin{array}{l} {}^t A B + {}^t C D = 0 \\ {}^t A A + {}^t C C = I_k \\ {}^t B B + {}^t D D = I_{k'} \end{array} \right. $$

Similarly, since ${}^t M S' M = S'$,

$$ \left\{ \begin{array}{l} {}^t A B - {}^t C D = 0 \\ {}^t A A - {}^t C C = I_k \\ {}^t B B - {}^t D D = -I_{k'} \end{array} \right. $$

These two series of equations imply that $B = C = 0$. We have then that $A$ and $D$ are orthogonal matrices. Finally, the fact that ${}^t M J' M = J'$ implies that the two matrices $A$ and $D$ are symplectic matrices. As a consequence, all matrix $M \in SpO_N(\mathbb{R})' \cap SO_{2N}(\mathbb{R})$ can be written $\begin{pmatrix} A & 0 \\ 0 & D \end{pmatrix}$, where $A, D \in Sp_{k^*/2}(\mathbb{R}) \cap SO_{k^*}(\mathbb{R})$, $k^*$ being $k$ or $k'$.

One easily checks that all such matrices are in $SpO_N(\mathbb{R})' \cap SO_{2N}(\mathbb{R})$. But, for all $k^*$, $Sp_{k^*/2}(\mathbb{R}) \cap SO_{k^*}(\mathbb{R})$ is pathwise connected as an intersection of 2 pathwise connected Lie groups so $SpO_N(\mathbb{R})' \cap SO_{2N}(\mathbb{R})$ is itself pathwise connected. For the pathwise connectedness of $SO_{k^*}(\mathbb{R})$, see [2]. For the pathwise connectedness of the symplectic group, one uses the fact that this group is generated by the symplectic transvections (see [19], Lemma 1, p. 392) to construct a continuous path between any symplectic matrix and the identity matrix.

(7) By differentiating both relations ${}^t M J M = J$ and ${}^t M S M = S$ one gets that the Lie algebra of $SpO_N(\mathbb{R})$ is the set

$$ \{ M \in \mathcal{M}_{2N}(\mathbb{R}) \mid {}^t M J + J M = 0 \text{ and } {}^t M S + S M = 0 \} \subset \mathfrak{sp}_N(\mathbb{R}). $$

If $M = (A_B)_D \in \mathcal{M}_{2N}(\mathbb{R})$ satisfies ${}^t M J + J M = 0$ one already gets that $M = (A_B^{-t} A'_C)$ with $A \in \mathcal{M}_N(\mathbb{R})$, $B = B$ and $C = C$. For such a matrix $M$, we write the relation
(6) Let us prove that the group $SO_{2N}(\mathbb{R}) \cap \text{SpO}_N(\mathbb{R})$ is pathwise connected. This, together with the decomposition given by point (5), will imply that $\text{SpO}_N(\mathbb{R})$ is pathwise connected. To this purpose, we introduce the matrices

$$S' = \begin{pmatrix} I_k & 0 \\ 0 & -I_{k'} \end{pmatrix} \quad \text{and} \quad J' = \begin{pmatrix} 0 & -I_{k/2} & 0 & 0 \\ I_{k/2} & 0 & 0 & 0 \\ 0 & 0 & 0 & -I_{k'/2} \\ 0 & 0 & I_{k'/2} & 0 \end{pmatrix},$$

where $k = k' = N$ when $N$ is even and $k = N + 1$ and $k' = N - 1$ when $N$ is odd. We see that $SO_{2N}(\mathbb{R}) \cap \text{SpO}_N(\mathbb{R})$ is homeomorphic to $SO_{2N}(\mathbb{R}) \cap G'_5$, where we have

$$G'_5 = \{ M \in \mathcal{M}_{2N}(\mathbb{R}), {}^t M J' M = J', {}^t M S' M = S' \}.$$

Indeed, the matrices $J$ and $J'$ (resp. $S$ and $S'$) are unitarily equivalent since $J'$ is obtained from $J$ by permutating rows and columns. Therefore, $\text{SpO}_N(\mathbb{R})$ and $\text{SpO}_N(\mathbb{R})'$ are unitarily equivalent hence homeomorphic.

We take a matrix $M = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in \text{SO}_{2N}(\mathbb{R}) \cap G'_5$, where the blocks have respective size $k$ and $k'$. Since ${}^t M M = I_{2N}$,

$$(A.3) \qquad \left\{ \begin{array}{l} {}^t A B + {}^t C D = 0 \\ {}^t A A + {}^t C C = I_k \\ {}^t B B + {}^t D D = I_{k'} \end{array} \right.$$

Similarly, since ${}^t M S' M = S'$,

$$\left\{ \begin{array}{l} {}^t A B - {}^t C D = 0 \\ {}^t A A - {}^t C C = I_k \\ {}^t B B - {}^t D D = -I_{k'} \end{array} \right.$$

These two series of equations imply that $B = C = 0$. We have then that $A$ and $D$ are orthogonal matrices. Finally, the fact that ${}^t M J' M = J'$ implies that the two matrices $A$ and $D$ are symplectic matrices. As a consequence, all matrix $M \in \text{SpO}_N(\mathbb{R})' \cap \text{SO}_{2N}(\mathbb{R})$ can be written $\begin{pmatrix} A & 0 \\ 0 & D \end{pmatrix}$, where $A, D \in \text{Sp}_{k^*/2}(\mathbb{R}) \cap \text{SO}_{k^*}(\mathbb{R})$, $k^*$ being $k$ or $k'$.

One easily checks that all such matrices are in $\text{SpO}_N(\mathbb{R})' \cap \text{SO}_{2N}(\mathbb{R})$. But, for all $k^*$, $\text{Sp}_{k^*/2}(\mathbb{R}) \cap \text{SO}_{k^*}(\mathbb{R})$ is pathwise connected as an intersection of 2 pathwise connected Lie groups so $\text{SpO}_N(\mathbb{R})' \cap \text{SO}_{2N}(\mathbb{R})$ is itself pathwise connected. For the pathwise connectedness of $\text{SO}_{k^*}(\mathbb{R})$, see [2]. For the pathwise connectedness of the symplectic group, one uses the fact that this group is generated by the symplectic transvections (see [19], Lemma 1, p. 392) to construct a continuous path between any symplectic matrix and the identity matrix.

(7) By differentiating both relations ${}^t M J M = J$ and ${}^t M S M = S$ one gets that the Lie algebra of $\text{SpO}_N(\mathbb{R})$ is the set

$$\{ M \in \mathcal{M}_{2N}(\mathbb{R}) \mid {}^t M J + J M = 0 \text{ and } {}^t M S + S M = 0 \} \subset \mathfrak{sp}_N(\mathbb{R}).$$

If $M = (A \ B \ D) \in \mathcal{M}_{2N}(\mathbb{R})$ satisfies ${}^t M J + J M = 0$ one already gets that $M = (A \ B \ -{}^t A)$ with $A \in \mathcal{M}_N(\mathbb{R})$, $B = B$ and $C = C$. For such a matrix $M$, we write the relation
$^tMS + SM = 0$:

$$ \begin{pmatrix} {}^t A & C \\ B & -A \end{pmatrix} \begin{pmatrix} K & 0 \\ 0 & K \end{pmatrix} + \begin{pmatrix} K & 0 \\ 0 & K \end{pmatrix} \begin{pmatrix} A & B \\ C & -{}^t A \end{pmatrix} = 0. $$

Since $B, C$ and $K$ are symmetric matrices, it implies that ${}^tAK + KA = 0$ and $BK + KC = 0$. This is equivalent, by definitions of the matrix $K$ and of the operator $^s$, to $A = {}^sA$ and $C = {}^sB$.

(8) First, we remark that since the Lie algebra of the group $SpO_N(\mathbb{R})$ is a sub Lie algebra of the Lie algebra of the symplectic group, its Killing form is the restriction of the Killing form of the Lie algebra of the symplectic group to the Lie algebra of $SpO_N(\mathbb{R})$ given by $(X, Y) \mapsto 2(N+1)\text{Tr}(XY)$ on $\text{Lie}(SpO_N(\mathbb{R})) \times \text{Lie}(SpO_N(\mathbb{R}))$ (see [31]). Then, a direct computation shows that this Killing form is non-degenerate, hence the Lie algebra of $SpO_N(\mathbb{R})$ is semisimple. Moreover, since the relationships defining $SpO_N(\mathbb{R})$ are polynomial in the matrix coefficients of its elements, the group $SpO_N(\mathbb{R})$ is an algebraic group (actually it is an algebraic subgroup of the symplectic group). Hence using for example [31, Proposition 27.2.2], the semisimplicity of the Lie algebra of $SpO_N(\mathbb{R})$ imply the semisimplicity of $SpO_N(\mathbb{R})$ as an algebraic group : the derived group of $SpO_N(\mathbb{R})$ is equal to $SpO_N(\mathbb{R})$. $\square$

*Remark A.2.* The semisimplicity of $SpO_N(\mathbb{R})$ implies that it is topologically perfect hence we can apply to $SpO_N(\mathbb{R})$ the result of Breuillard and Gelander, Theorem 3.5, as in the case of the symplectic group.

## REFERENCES

[1] L. Arnold, *Random Dynamical Systems*, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.

[2] Y. C. Wong, Y. H. Au-Yeung, An elementary and simple proof of the connectedness of the classical groups, *Am. Math. Mon.* 74, 964–966, 1967.

[3] J.-M. Barbaroux, H.D. Cornean, and S. Zalczer. Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices. *Doc. Math.*, 24:65–93, 2019.

[4] W. Borrelli, P. Briet, D. Krejčířík, and T. Ourmières-Bonafos. Spectral Properties of Relativistic Quantum Waveguides. *Ann. Henri Poincaré* 23, 4069–4114, 2022.

[5] Ph. Bougerol and J. Lacroix, *Products of Random Matrices with Applications to Schrödinger Operators*, Progress in probability and statistics, Birkhäuser, Boston, 1985.

[6] H. Boumaza, *Exposants de Lyapounov et Densité d'États Intégrée pour des opérateurs de Schrödinger continus à valeurs matricielles*, PhD thesis, Université Paris Diderot – Paris 7, 2007.

[7] H. Boumaza, Hölder continuity of the integrated density of states for matrix-valued Anderson models, *Rev. Math. Phys.* 20(7), 873–900, 2008.

[8] H. Boumaza, Localization for a matrix-valued Anderson model, *Math. Phys. Anal. Geom.* 12(3), 255–286, 2009.

[9] H. Boumaza, Localization for an Anderson-Bernoulli model with generic interaction potential, *Tohoku Math. J.* 65(1), 57–74, 2013.

[10] H. Boumaza. Localization for random quasi-one-dimensional models, *J. Math. Phys.* 64, No. 9, Article ID 091101, 2023.

[11] E. Breuillard and T. Gelander, On dense free subgroups of Lie groups, *J. Algebra* 261:448–467, 2003.

[12] R. Carmona, A. Klein, and F. Martinelli. Anderson localization for Bernoulli and other singular potentials. *Comm. Math. Phys.*, 108:41–66, 1987.
$^tMS + SM = 0$:

$$ \begin{pmatrix} {}^t A & C \\ B & -A \end{pmatrix} \begin{pmatrix} K & 0 \\ 0 & K \end{pmatrix} + \begin{pmatrix} K & 0 \\ 0 & K \end{pmatrix} \begin{pmatrix} A & B \\ C & -{}^t A \end{pmatrix} = 0. $$

Since $B, C$ and $K$ are symmetric matrices, it implies that ${}^tAK + KA = 0$ and $BK + KC = 0$. This is equivalent, by definitions of the matrix $K$ and of the operator $^s$, to $A = {}^sA$ and $C = {}^sB$.

(8) First, we remark that since the Lie algebra of the group $SpO_N(\mathbb{R})$ is a sub Lie algebra of the Lie algebra of the symplectic group, its Killing form is the restriction of the Killing form of the Lie algebra of the symplectic group to the Lie algebra of $SpO_N(\mathbb{R})$ given by $(X, Y) \mapsto 2(N+1)\text{Tr}(XY)$ on $\text{Lie}(SpO_N(\mathbb{R})) \times \text{Lie}(SpO_N(\mathbb{R}))$ (see [31]). Then, a direct computation shows that this Killing form is non-degenerate, hence the Lie algebra of $SpO_N(\mathbb{R})$ is semisimple. Moreover, since the relationships defining $SpO_N(\mathbb{R})$ are polynomial in the matrix coefficients of its elements, the group $SpO_N(\mathbb{R})$ is an algebraic group (actually it is an algebraic subgroup of the symplectic group). Hence using for example [31, Proposition 27.2.2], the semisimplicity of the Lie algebra of $SpO_N(\mathbb{R})$ imply the semisimplicity of $SpO_N(\mathbb{R})$ as an algebraic group : the derived group of $SpO_N(\mathbb{R})$ is equal to $SpO_N(\mathbb{R})$. $\square$

*Remark A.2.* The semisimplicity of $SpO_N(\mathbb{R})$ implies that it is topologically perfect hence we can apply to $SpO_N(\mathbb{R})$ the result of Breuillard and Gelander, Theorem 3.5, as in the case of the symplectic group.

## REFERENCES

[1] L. Arnold, *Random Dynamical Systems*, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.

[2] Y. C. Wong, Y. H. Au-Yeung, An elementary and simple proof of the connectedness of the classical groups, *Am. Math. Mon.* 74, 964–966, 1967.

[3] J.-M. Barbaroux, H.D. Cornean, and S. Zalczer. Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices. *Doc. Math.*, 24:65–93, 2019.

[4] W. Borrelli, P. Briet, D. Krejčířík, and T. Ourmières-Bonafos. Spectral Properties of Relativistic Quantum Waveguides. *Ann. Henri Poincaré* 23, 4069–4114, 2022.

[5] Ph. Bougerol and J. Lacroix, *Products of Random Matrices with Applications to Schrödinger Operators*, Progress in probability and statistics, Birkhäuser, Boston, 1985.

[6] H. Boumaza, *Exposants de Lyapounov et Densité d'États Intégrée pour des opérateurs de Schrödinger continus à valeurs matricielles*, PhD thesis, Université Paris Diderot – Paris 7, 2007.

[7] H. Boumaza, Hölder continuity of the integrated density of states for matrix-valued Anderson models, *Rev. Math. Phys.* 20(7), 873–900, 2008.

[8] H. Boumaza, Localization for a matrix-valued Anderson model, *Math. Phys. Anal. Geom.* 12(3), 255–286, 2009.

[9] H. Boumaza, Localization for an Anderson-Bernoulli model with generic interaction potential, *Tohoku Math. J.* 65(1), 57–74, 2013.

[10] H. Boumaza. Localization for random quasi-one-dimensional models, *J. Math. Phys.* 64, No. 9, Article ID 091101, 2023.

[11] E. Breuillard and T. Gelander, On dense free subgroups of Lie groups, *J. Algebra* 261:448–467, 2003.

[12] R. Carmona, A. Klein, and F. Martinelli. Anderson localization for Bernoulli and other singular potentials. *Comm. Math. Phys.*, 108:41–66, 1987.
[13] R. Carmona and J. Lacroix, *Spectral Theory of Random Schrödinger Operators*, Probability and Its Applications, Birkhäuser, Boston, 1990.

[14] D. Damanik and P. Stollmann, Multi-scale analysis implies strong dynamical localization, *Geom. Funct. Anal.* 11.1:11–29, 2001.

[15] M. Evaldsson, I. V. Zozoulenko, Hengyi Xu, and T. Heinzel. Edge-disorder-induced Anderson localization and conduction gap in graphene nanoribbons. *Phys. Rev. B* 78:161407, 2008.

[16] F. Germinet and A. Klein. Bootstrap multiscale analysis and localization in random media. *Comm. Math. Phys.*, 222:415–448, 2001.

[17] I.Ya. Goldsheid and G.A. Margulis, Lyapunov indices of a product of random matrices, *Russian Math. Surveys* 44:11–71,1989.

[18] R. A. Horn and C. R. Johnson. *Topics in Matrix Analysis*. Cambridge University Press, Cambridge, 1991.

[19] N. Jacobson, *Basic algebra I. 2nd ed.*, New York, W. H. Freeman and Company, 1985.

[20] W. Kirsch. On a class of random Schrödinger operators. *Adv. in Appl. Math.*, 6:177-187, 1985.

[21] W. Kirsch. An Invitation to Random Schrödinger operators. In *Random Schrödinger Operators*, volume 25 of *Panoramas et Synthèses*. Société Mathématique de France, 2008.

[22] A. Klein. Multiscale analysis and localization of random operators. In *Random Schrödinger Operators*, volume 25 of *Panoramas et Synthèses*. Société Mathématique de France, 2008.

[23] A. Klein, J. Lacroix and A.Speis. Localization for the Anderson model on a strip with singular potentials, *Journal of Functional Analysis*, 94, 135–155, 1990.

[24] D. Macera and S. Sodin. Anderson Localisation for Quasi-One-Dimensional Random Operators, *Ann. Henri Poincaré* 23, No. 12, 4227–4247, 2022.

[25] C.D. Nunez, P.A. Orellana and L. Rosales. Electron localization due to side-attached molecules on graphene nanoribbons, *J. Appl. Phys.* 120 (16):164310, 2016.

[26] R. A. Prado and C. R. de Oliveira. Dynamical lower bounds for 1D Dirac operators. *Math. Z.*, 259:45–60, 2008.

[27] R. A. Prado, C. R. de Oliveira, and S. L. Carvalho. Dynamical localization for discrete Anderson Dirac operators. *J. Stat. Phys.*, 167:260–296, 2017.

[28] R.A. Prado, C. de Oliveira, and E. de Oliveira. Density of states and Lifshitz tails for discrete 1D random Dirac operators. *Math. Phys. Anal. Geom.*, 24:30, 2021.

[29] C. Sadel and H. Schulz-Baldes. Random Dirac operators with time reversal symmetry, *Comm. Math. Phys.*, 295:209-242, 2010.

[30] J. R. da Silva, A. L.R. Barbosa and L. F. C. Pereira. Localization effects in graphene nanoribbons with quasiperiodic hopping modulation, *Micro Nanostructures*, 168:207295, 2022.

[31] P. Tauvel, R. W. T. Yu, *Lie algebras and algebraic groups*, Springer Monographs in Mathematics, Springer Berlin (2005).

[32] B. Thaller. *The Dirac Equation*. Springer-Verlag, Berlin, 1992.

[33] S.-J. Xiong and Y. Xiong. Anderson localization of electron states in graphene in different types of disorder. *Phys. Rev. B* 76 (21):214204, 2007.

[34] S. Zalczer. Regularity of the density of states for random Dirac operators. *J. Math. Phys.*, 64 (6): 062103, 2023.

[35] S. Zalczer. Localization for One-Dimensional Anderson-Dirac Models, *Ann. Henri Poincaré* 24, 37-72, 2023.

LAGA, UNIVERSITÉ SORBONNE PARIS NORD, 99 AVENUE J.B. CLÉMENT, F-93430 VILLETANEUSE
*Email address: boumaza@math.univ-paris13.fr*

KIT, ENGLERSTRASSE 2, 76131, KARLSRUHE, GERMANY
*Email address: sylvain.zalczer@kit.edu*
[13] R. Carmona and J. Lacroix, *Spectral Theory of Random Schrödinger Operators*, Probability and Its Applications, Birkhäuser, Boston, 1990.

[14] D. Damanik and P. Stollmann, Multi-scale analysis implies strong dynamical localization, *Geom. Funct. Anal.* 11.1:11–29, 2001.

[15] M. Evaldsson, I. V. Zozoulenko, Hengyi Xu, and T. Heinzel. Edge-disorder-induced Anderson localization and conduction gap in graphene nanoribbons. *Phys. Rev. B* 78:161407, 2008.

[16] F. Germinet and A. Klein. Bootstrap multiscale analysis and localization in random media. *Comm. Math. Phys.*, 222:415–448, 2001.

[17] I.Ya. Goldsheid and G.A. Margulis, Lyapunov indices of a product of random matrices, *Russian Math. Surveys* 44:11–71,1989.

[18] R. A. Horn and C. R. Johnson. *Topics in Matrix Analysis*. Cambridge University Press, Cambridge, 1991.

[19] N. Jacobson, *Basic algebra I. 2nd ed.*, New York, W. H. Freeman and Company, 1985.

[20] W. Kirsch. On a class of random Schrödinger operators. *Adv. in Appl. Math.*, 6:177-187, 1985.

[21] W. Kirsch. An Invitation to Random Schrödinger operators. In *Random Schrödinger Operators*, volume 25 of *Panoramas et Synthèses*. Société Mathématique de France, 2008.

[22] A. Klein. Multiscale analysis and localization of random operators. In *Random Schrödinger Operators*, volume 25 of *Panoramas et Synthèses*. Société Mathématique de France, 2008.

[23] A. Klein, J. Lacroix and A.Speis. Localization for the Anderson model on a strip with singular potentials, *Journal of Functional Analysis*, 94, 135–155, 1990.

[24] D. Macera and S. Sodin. Anderson Localisation for Quasi-One-Dimensional Random Operators, *Ann. Henri Poincaré* 23, No. 12, 4227–4247, 2022.

[25] C.D. Nunez, P.A. Orellana and L. Rosales. Electron localization due to side-attached molecules on graphene nanoribbons, *J. Appl. Phys.* 120 (16):164310, 2016.

[26] R. A. Prado and C. R. de Oliveira. Dynamical lower bounds for 1D Dirac operators. *Math. Z.*, 259:45–60, 2008.

[27] R. A. Prado, C. R. de Oliveira, and S. L. Carvalho. Dynamical localization for discrete Anderson Dirac operators. *J. Stat. Phys.*, 167:260–296, 2017.

[28] R.A. Prado, C. de Oliveira, and E. de Oliveira. Density of states and Lifshitz tails for discrete 1D random Dirac operators. *Math. Phys. Anal. Geom.*, 24:30, 2021.

[29] C. Sadel and H. Schulz-Baldes. Random Dirac operators with time reversal symmetry, *Comm. Math. Phys.*, 295:209-242, 2010.

[30] J. R. da Silva, A. L.R. Barbosa and L. F. C. Pereira. Localization effects in graphene nanoribbons with quasiperiodic hopping modulation, *Micro Nanostructures*, 168:207295, 2022.

[31] P. Tauvel, R. W. T. Yu, *Lie algebras and algebraic groups*, Springer Monographs in Mathematics, Springer Berlin (2005).

[32] B. Thaller. *The Dirac Equation*. Springer-Verlag, Berlin, 1992.

[33] S.-J. Xiong and Y. Xiong. Anderson localization of electron states in graphene in different types of disorder. *Phys. Rev. B* 76 (21):214204, 2007.

[34] S. Zalczer. Regularity of the density of states for random Dirac operators. *J. Math. Phys.*, 64 (6): 062103, 2023.

[35] S. Zalczer. Localization for One-Dimensional Anderson-Dirac Models, *Ann. Henri Poincaré* 24, 37-72, 2023.

LAGA, UNIVERSITÉ SORBONNE PARIS NORD, 99 AVENUE J.B. CLÉMENT, F-93430 VILLETANEUSE
*Email address: boumaza@math.univ-paris13.fr*

KIT, ENGLERSTRASSE 2, 76131, KARLSRUHE, GERMANY
*Email address: sylvain.zalczer@kit.edu*