Abstract
In 1990, Klein, Lacroix, and Speis proved (spectral) Anderson localisation for the Anderson model on the strip of width \(W \geqslant 1\), allowing for singular distribution of the potential. Their proof employs multiscale analysis, in addition to arguments from the theory of random matrix products (the case of regular distributions was handled earlier in the works of Goldsheid and Lacroix by other means). We give a proof of their result avoiding multiscale analysis and also extend it to the general quasionedimensional model, allowing, in particular, random hopping. Furthermore, we prove a sharp bound on the eigenfunction correlator of the model, which implies exponential dynamical localisation and exponential decay of the Fermi projection. The method is also applicable to operators on the halfline with arbitrary (deterministic, selfadjoint) boundary condition. Our work generalises and complements the singlescale proofs of localisation in pure one dimension (\(W=1\)), recently found by Bucaj–Damanik–Fillman–Gerbuz–VandenBoom–Wang–Zhang, Jitomirskaya–Zhu, Gorodetski–Kleptsyn, and Rangamani.
Introduction
The Operator, Transfer Matrices and Lyapunov Exponents
Let \(W \geqslant 1\). Let \(\{L_x\}_{x \in {\mathbb {Z}}}\) be a sequence of identically distributed \(W \times W\) random matrices in \({\text {GL}}(W, {\mathbb {R}})\), and let \(\{ V_x \}_{x \in {\mathbb {Z}}}\) be a sequence of identically distributed \(W \times W\) real symmetric matrices, so that \(\{L_x\}_{x \in {\mathbb {Z}}}, \{V_x\}_{x \in {\mathbb {Z}}}\) are jointly independent. Denote by \({\mathcal {L}}\) the support of \(L_0\) and by \({\mathcal {V}}\)—the support of \(V_0\). Throughout this paper, we assume that

(A)
there exists \(\eta > 0\) such that
$$\begin{aligned} {\mathbb {E}} (\Vert V_0 \Vert ^\eta + \Vert L_0\Vert ^\eta + \Vert L_0^{1}\Vert ^\eta ) < \infty ~; \end{aligned}$$ 
(B)
the Zariski closure of the group generated by \({\mathcal {L}} \, {\mathcal {L}}^{1}\) in \({\text {GL}}(W, {\mathbb {R}})\) intersects \({\mathcal {L}}\) (this holds, for example, when \(\mathbbm {1} \in {\mathcal {L}}\));

(C)
\({\mathcal {V}}\) is irreducible (i.e. has no common invariant subspaces except for \(\{0\}\) and \({\mathbb {R}}^W\)), and \({\mathcal {V}}  {\mathcal {V}}\) contains a matrix of rank one.
We are concerned with the spectral properties of the random operator H acting on (a dense subspace of) \(\ell _2 ({\mathbb {Z}} \rightarrow {\mathbb {C}}^W)\) via
This model, often referred to as a quasionedimensional random operator, is the general Hamiltonian describing a quantum particle with W internal degrees of freedom in random potential and with nearestneighbour random hopping. The special case \(L_x \equiv \mathbbm {1}\) is known as the block Anderson model; it is in turn a generalisation of the Anderson model on the strip \({\mathbb {Z}} \times \{1, \ldots , W\}\), and, more generally, on \({\mathbb {Z}} \times \Gamma \), where \(\Gamma \) is any connected finite graph (the assumption that \(\Gamma \) is connected ensures that \({\mathcal {V}}\) is irreducible). Another known special case of (1) is the Wegner orbital model.
Fix \(E \in {\mathbb {R}}\). If \(\psi : {\mathbb {Z}} \rightarrow {\mathbb {C}}^W\) is a formal solution of the equation
then
where the onestep transfer matrix \(T_x \in {\text {GL}}(2W, {\mathbb {R}})\) is given by
The multistep transfer matrices \(\Phi _{x, y} \in {\text {GL}}(2W, {\mathbb {R}})\), \(x, y \in {\mathbb {Z}}\), are defined by
so that
In particular, \(T_x = \Phi _{x+1,x}\). We abbreviate \(\Phi _{N} = \Phi _{N,0}\). The Lyapunov exponents \(\gamma _j(E)\), \(1 \leqslant j \leqslant 2W\), are defined as
where \(s_j\) stands for the jth singular value. It is known [11] that (for fixed E) this limit in expectation is also an almost sure limit. The cocycle \(\{\Phi _{x,y}\}\) is conjugate to a symplectic one (see Sect. 3.1), and hence
Further, as we shall see in Sect. 3.2, using the work of Goldsheid [15] to verify the conditions of the Goldsheid–Margulis theorem [16] on the simplicity of the Lyapunov spectrum, that
We also mention that the Lyapunov exponents \(\gamma _j(E)\) are continuous functions of E. This was proved by Furstenberg and Kifer in [12]; it can also be deduced from the large deviation estimate (27)—see Duarte and Klein [8].
The Main Results
Theorem 1
Assume (A)–(C). Then, the spectrum of H is almost surely pure point. Moreover, if
is the collection of eigenpairs of H, then
i.e. each eigenfunction decays exponentially, with the rate lowerbounded by the slowest Lyapunov exponent.
Remark 1.1
It is believed that the lower bound is sharp, i.e. the rate of decay cannot be faster than the slowest Lyapunov exponent:
We refer to [18] for a discussion and partial results in this direction. For \(W=1\), (7) was proved by Craig and Simon in [7].
The property of having pure point spectrum with exponentially decaying eigenfunctions is a manifestation of Anderson localisation of the random operator H. The mathematical work on Anderson localisation in one dimension was initiated by Goldsheid, Molchanov and Pastur [17], who considered the case \(W=1\), \(L_x \equiv 1\) and established the pure point nature of the spectrum under the assumption that the distribution of \(V_x\) is regular enough (absolutely continuous with bounded density). A different proof of the result of [17] was found by Kunz and Souillard [25]. Under the same assumptions, the exponential decay of the eigenfunctions was established by Molchanov [29]. The case of singular distributions was treated by Carmona, Klein, and Martinelli [6].
The case \(W >1\) was first considered by Goldsheid [14], who established the pure point nature of the spectrum for the case of the Schrödinger operator on the strip, i.e. when \(L_x \equiv \mathbbm {1}\), \(V_x\) is tridiagonal with the offdiagonal entries equal to 1 and the diagonal ones independent and identically distributed, under the assumption that the distribution of the diagonal entries of \(V_x\) is regular. In the same setting, Lacroix [26,27,28] proved that the eigenfunctions decay exponentially. The case of the Anderson model on a strip with general (possibly, singular) distributions was settled by Klein–Lacroix–Speis [23], who established localisation in the strong form (6).
Unlike the earlier, more direct arguments treating regular distributions, the works [6, 23] allowing singular distributions involve a multiscale argument (as developed in the work of Fröhlich–Spencer [10] on localisation in higher dimension); the theory of random matrix products is used to verify the initial hypothesis of multiscale analysis. Recently, proofs of the result of [6] avoiding multiscale analysis were found by Bucaj et al. [5], Jitomirskaya and Zhu [22], and Gorodetski and Kleptsyn [20]; the general onedimensional case (allowing for random hopping) was settled by Rangamani [30]. Our Theorem 1 can be seen as a generalisation of these works, and especially of [22, 30], to which our arguments are closest in spirit: we give a relatively short and singlescale proof of localisation which applies to arbitrary \(W \geqslant 1\), and allows for rather general distributions of \(V_0\) and \(L_0\) (under no regularity assumptions on the distribution of the potential). In particular, we recover and generalise the result of [23].
In fact, we prove a stronger result pertaining to the eigenfunction correlators, introduced by Aizenman [1] (see further the monograph of Aizenman–Warzel [3]). If \(\Lambda \subset {\mathbb {Z}}\) is a finite set, denote by \(H_\Lambda \) the restriction of H to \(\ell _2(\Lambda \rightarrow {\mathbb {C}}^W)\), i.e.
where \(P_\Lambda : \ell _2({\mathbb {Z}} \rightarrow {\mathbb {C}}^W) \rightarrow \ell _2(\Lambda \rightarrow {\mathbb {C}}^W)\) is the coordinate projection. If \(I \subset {\mathbb {R}}\) is a compact interval, denote
Here \(\Vert f(H_{\Lambda })_{x,y}\Vert \) is the operator norm of the (x, y) block of \(f(H_{\Lambda })\), and the functions f in the supremum are assumed to be, say, Borel measurable.
Theorem 2
Assume (A)–(C). For any compact interval \(I \subset {\mathbb {R}}\),
It is known (see [3]) that Theorem 2 implies Theorem 1. By plugging in various choices of f, it also implies (almost sure) dynamical localisation with the sharp rate of exponential decay, the exponential decay of the Fermi projection, et cet. (see e.g. [2] and [3]). We chose to state Theorem 1 as a separate result rather than a corollary of Theorem 2 since its direct proof is somewhat shorter than that of the latter.
We refer to Bucaj et al. [5], Jitomirskaya–Zhu [22], and Ge–Zhao [13] for earlier results on dynamical localisation for \(W = 1\).
Main Ingredients of the Proof
Similarly to many of the previous works, including [6, 23] and also the recent works [5, 20, 22], the two main ingredients of the proof of localisation are a large deviation estimate and a Wegnertype estimate. We state these in the generality required here. Let \(I \subset {\mathbb {R}}\) be a compact interval, and let \(F \subset {\mathbb {R}}^{2W}\) be a Lagrangian subspace (see Sect. 3). Denote by \(\pi _F: {\mathbb {R}}^{2W} \rightarrow F\) the orthogonal projection onto F.
Proposition 1.2
Assume (A)–(C). For any \(\epsilon > 0\) there exist \(C, c> 0\) such that for any \(E \in I\) and any Lagrangian subspace \(F \subset {\mathbb {R}}^{2W}\)
The proof is essentially given in [23]; we outline the necessary reductions in Sect. 3.1. The second proposition could be also proved along the lines of the special case considered in [23]; we present an alternative (arguably, simpler) argument in Sect. 3.3.
For an operator H and E in the resolvent set of H, we denote by \(G_E[H] = (HE)^{1}\) the resolvent of H and by \(G_E[H](\cdot , \cdot )\) its matrix elements. If E lies in the spectrum of H, we set \(G_E[H](\cdot , \cdot ) \equiv \infty \).
Proposition 1.3
Assume (A)–(C). For any \(\epsilon > 0\) there exist \(C, c> 0\) such that for any \(E \in I\) and \(N \geqslant 1\)
Remark 1.4
The arguments which we present can be applied to deduce the following strengthening of (10):
We content ourselves with (10) which suffices for the proof of the main theorems.
Klein, Lacroix and Speis [23] use (special cases of) Propositions 1.2 and 1.3 to verify the assumptions required for multiscale analysis. We deduce Theorems 1 and 2 directly from these propositions. In this aspect, our general strategy is similar to the cited works [5, 20, 22]. However, several of the arguments employed in these works rely on the special features of the model for \(W=1\); therefore, our implementation of the strategy differs in several crucial aspects.
Proof of the Main Theorems
Resonant Sites; The Main Technical Proposition
Let \(\tau > 0\) be a (small) number. We say that \(x\in {\mathbb {Z}}\) is \((\tau , E, N)\)nonresonant (\(x \notin {\text {Res}}(\tau , E,N)\)) if
and \((\tau , E,N)\)resonant (\(x \in {\text {Res}}(\tau , E,N)\)) otherwise. The following proposition is the key step towards the proof of Theorems 1 and 2.
Proposition 2.1
Assume (A)–(C). Let \(I \subset {\mathbb {R}}\) be a compact interval, and let \(\tau > 0\). There exist \(C, c> 0\) such that for any \(N \geqslant 1\)
The remainder of this section is organised as follows. In Sect. 2.2, we express the Green function in terms of the transfer matrices. Using this expression and Propositions 1.2 and 1.3, we show that the probability that \(x \in {\text {Res}}(\tau , E, N)\) (for a fixed \(E \in {\mathbb {R}}\)) is exponentially small. In Sect. 2.3, we rely on this estimate to prove Proposition 2.1. Then, we use this proposition to prove Theorem 1 (Sect. 2.4) and Theorem 2 (Sect. 2.5).
Reduction to Transfer Matrices
Fix \(N \geqslant 1\). Consider the \(W \times W\) matrices
The Green function of \(H_{[N,N]}\) can be expressed in terms of these matrices using the following claim, which holds deterministically for any H of the form (1). A similar expression has been employed already in [14].
Claim 2.2
If \(E \notin \sigma (H_{[N,N]})\), then:

1.
$$\begin{aligned} \left( {\begin{array}{c}\Psi _{\pm 1}^\pm \\ \Psi _{0}^\pm \end{array}}\right) G_E[H_{[N,N]}](0, \pm N) = \left( {\begin{array}{c}G_E[H_{[N,N]}](0, \pm 1)\\ G_E[H_{[N,N]}](0,0)\end{array}}\right) ;\end{aligned}$$

2.
for any \(i,j \in [N, N]\),
$$\begin{aligned} G_E[H_{[N, N]}](i, i) = {\left\{ \begin{array}{ll} \Psi _j^{} (\Psi _i^)^{1} \left( \Psi _{i+1}^+ (\Psi _i^+)^{1}  \Psi _{i+1}^ (\Psi _i^)^{1}\right) ^{1} L_i^{1}, &{}i \geqslant j\\ \Psi _j^{+} (\Psi _i^+)^{1} \left( \Psi _{i+1}^+ (\Psi _i^+)^{1}  \Psi _{i+1}^ (\Psi _i^)^{1}\right) ^{1} L_i^{1}, &{}i \leqslant j. \end{array}\right. } \end{aligned}$$
Proof
Abbreviate \(G_E = G_E[H_{[N,N]}]\), and set \(G_E(i, j) = 0\) for \(j \notin [N, N]\). The matrices \(G_E(i, j)\), \(N \leqslant j \leqslant N\), are uniquely determined by the system of equations
We look for a solution of the form
where
The first equation ensures that (14) defines \(G_E(i,i)\) consistently, while the second one guarantees that (13) holds for \(j=i\). For the other values of j, (13) follows from the construction of the matrices \(\Psi ^{\pm }_j\).
The solution to (15)–(16) is explicitly found by elimination:
This implies the second part of the claim. For the first part, note that for \(j \geqslant i\)
Observing that \(\Psi _N^+ = \mathbbm {1}\), we conclude that
as claimed. Similarly,
\(\square \)
Proof of Proposition 2.1
Fix a small \(\tau > 0\). Without loss of generality, I is short enough to ensure that
(this property is valid for short intervals due to the continuity of \(\gamma _W\); the statement for larger intervals I follows by compactness). Fix such I (which will be suppressed from the notation), and let
For \(x \in {\mathbb {Z}}\), let
where \(\Vert A\Vert _{1,\infty } = \max _{1 \leqslant \alpha , \beta \leqslant W} A_{\alpha ,\beta }\). For N large enough (\(N \geqslant N_0(\tau )\)),
By (A) and the Chebyshev inequality
Hence, the proposition boils down to the following statement:
The proof of (17) rests on two claims. The first one is deterministic:
Claim 2.3
\({\text {Res}}^*(\tau , x, N)\) is the union of at most \(C_W N\) disjoint closed intervals.
Proof
By Cramer’s rule, for each \(\alpha ,\beta \in \{1, \ldots , W\}\) and ± the function
is the ratio of two polynomials of degree \(\leqslant W(2N+1)\). Hence, the level set
is of cardinality \(\leqslant W(2N+1)\) (note that the \(\leqslant W(2N+1)\) discontinuity points of \(g_{\alpha ,\beta }^\pm \) are poles; hence, they cannot serve as the endpoints of the superlevel sets of this function). Hence, our set
is the union of at most \(\leqslant W(2N+1)/2\) closed intervals, and \({\text {Res}}^*(\tau , x, N)\) is the union of at most
closed intervals. \(\square \)
Claim 2.4
Assume (A)–(C). For any compact interval \(I \subset {\mathbb {R}}\) there exist \(C, c> 0\) such that for any \(N \geqslant 1\) and any \(E \in I\),
Proof
According to Claim 2.2,
hence
By Propositions 1.2 and 1.3, both terms decay exponentially in N, locally uniformly in E. \(\square \)
Now we can prove (17). By Claim 2.3 both \({\text {Res}}^*(\tau , x, N)\) and \({\text {Res}}^*(\tau , y, N)\) are unions of at most \(C_W N\) closed intervals. If these two sets intersect, then either one of the edges of the intervals composing the first one lies in the second one, or vice versa. The operators \(H_{[xN, x+N]}\) and \(H_{[yN, y+N]}\) are independent due to the assumption \(xy> 2N\), hence by Claim 2.4
This concludes the proof of (17) and of Proposition 2.1. \(\square \)
Spectral Localisation: Proof of Theorem 1
The proof of localisation is based on Schnol’s lemma, which we now recall (see [21] for a version applicable in the current setting). A function \(\psi : {\mathbb {Z}} \rightarrow {\mathbb {C}}^W\) is called a generalised eigenfunction corresponding to a generalised eigenvalue \(E \in {\mathbb {R}}\) if
Schnol’s lemma asserts that any spectral measure of H is supported on the set of generalised eigenvalues. Thus, we need to show that (with full probability) any generalised eigenpair \((E, \psi )\) satisfies
Fix a compact interval \(I \subset {\mathbb {R}}\), and \(\tau > 0\). Consider the events
By Proposition 2.1 and the Borel–Cantelli lemma,
We shall prove that on any \({\mathcal {G}}_M(I, \tau )\) every generalised eigenpair \((E, \psi )\) with \(E \in I\) satisfies
From (18), we have for any x
If \(x \notin {\text {Res}}(\tau , E, N)\), this implies
whence \(f_\tau (x) \overset{\text {def}}{=} e^{\tau x} \Vert \psi (x)\Vert \) satisfies
The function \(f_\tau \) is bounded due to (19); hence, it achieves a maximum at some \(x_\psi \in {\mathbb {Z}}\). For
(22) cannot hold at \(x = x_\psi \), thus on \({\mathcal {G}}_M(I, \tau )\) for all
we have:
Thus, (22) holds whenever x, N are such that \(3N < x \leqslant N^2\) and \(N \geqslant N_0\).
For each \(x \in {\mathbb {Z}}\), let N(x) be such that \(N^2/10 \leqslant x \leqslant N^2 / 5\). If x is large enough, \(N(x) \geqslant N_0\). Applying (22) \(\lfloor x/(N+1) \rfloor  4\) times, we obtain
which implies (21). \(\square \)
Eigenfunction Correlator: Proof of Theorem 2
Fix a compact interval \(I \subset {\mathbb {R}}\), and let \(\gamma = \min _{E \in I} \gamma _W(E)\). The proof of (8) relies on the following fact from [9, Lemma 4.1], based on an idea from [3]:
Our goal is to bound on this quantity uniformly in the interval \(\Lambda \supset \{x,y\}\). Without loss of generality we can assume that \(x = 0\). Choose N such that \(N^2/10 \leqslant y \leqslant N^2/5\). By Proposition 2.1, for any \(\tau \in (0, \gamma )\)
We show that on the event
we have
Expand the Green function \(G_E[H_\Lambda ](0, y)\) as follows. First, iterate the resolvent identity
starting from \(x = 0\) at most y/N times, or until the first argument of \(G_E[H_\Lambda ]\) reaches the set \({\text {Res}}(\tau , E, N)\). Then, apply the identity
starting from \(u = y\) at most y/N times, or until the second argument of \(G_E[H_\Lambda ]\) reaches the set \({\text {Res}}(\tau , E, N)\). The resulting expansion has \(\leqslant 2^{2y/N}\) addends, each of which has the form
where \(x_0 = 0\), \(x_{j+1} = x_j \pm N\), \(y_0 = y\), \(y_{j+1} = y_j\pm N\), and (by the construction of the event (24)) \(k + \ell \geqslant y/N4\). All the terms in the first and third line of (26) are bounded in norm by \(e^{(\gamma \tau )N}\), hence
Now we raise this estimate to the power \(1 \epsilon \) and integrate over \(E \in I\):
It remains to let \(\epsilon \rightarrow + 0\) while making use of the two inequalities in (23). \(\square \)
Properties of Transfer Matrices
Preliminaries
Denote
A matrix \(Q \in {\text {GL}}(2W, {\mathbb {R}})\) is called symplectic, \(Q \in {\text {Sp}}(2W, {\mathbb {R}})\), if \(Q^\intercal J Q = J\).
The matrices \(T_x\) are, generally speaking, not symplectic. However, the cocycle \(\{\Phi _{x,y}\}_{x,y,\in {\mathbb {Z}}}\) is conjugate to a symplectic one. Indeed, observe that
Claim 3.1
If \(L \in {\text {GL}}(W, {\mathbb {R}})\) and Z is \(W \times W\) real symmetric, then \(Q(L, Z) = \left( \begin{array}{cc} L^{1} Z &{} L^{1} \\ L^\intercal &{} 0 \end{array} \right) \) is symplectic.
Denote \(D_x =\left( \begin{array}{cc} \mathbbm {1} &{} 0 \\ 0 &{} L_x^\intercal \end{array} \right) \), then
Thus, also
Simplicity of the Lyapunov Spectrum and Large Deviations
Goldsheid and Margulis showed [16] that if \(g_j\) are independent, identically distributed random matrices in \({\text {Sp}}(2W, {\mathbb {R}})\), and the group generated by the support of \(g_1\) is Zariski dense in \({\text {Sp}}(2W, {\mathbb {R}})\), then the Lyapunov spectrum of a random matrix product \(\{ g_N \ldots g_1 \}\) is simple, i.e.
Goldsheid showed [15] that if \({\mathcal {V}}\) is irreducible and \({\mathcal {V}}  {\mathcal {V}}\) contains a rankone matrix, then for any \(E \in {\mathbb {R}}\) the group generated by \(Q(\mathbbm {1}, E \mathbbm {1}  V)\), \(V \in {\mathcal {V}}\), is Zariski dense in \({\text {Sp}}(2W, {\mathbb {R}})\).
Corollary 3.2
Assume (A)–(C). Then, for any \(E \in {\mathbb {R}}\)
Proof
Observe that
whence
If the Zariski closure of the group generated by \({\mathcal {L}} {\mathcal {L}}^{1}\) intersects \({\mathcal {L}}\), then the Zariski closure of the group generated by \(\{ Q(L, E \mathbbm {1}  V)\}_{L \in {\mathcal {L}}, V \in {\mathcal {V}}}\) contains that of the group generated by \(\{ Q(\mathbbm {1}, E \mathbbm {1}  V)\}_{V \in {\mathcal {V}}}\). \(\square \)
Having the corollary at hand, we deduce from [23, Proposition 2.7] applied to the matrices \({{\widetilde{\Phi }}}_N(E)\):
Proposition 3.3
Assume (A)–(C). For any \(\epsilon > 0\) there exist \(C, c> 0\) such that for any \(E \in I\) and \(1 \leqslant j \leqslant W\)
Further, for any Lagrangian subspace \(F \subset {\mathbb {R}}^{2W}\)
Proof
The estimate (28) is a restatement of [23, Proposition 2.7], whereas (27) follows from (28) applied to a \(\delta \)net on the manifold of Lagrangian subspaces of \({\mathbb {R}}^{2W}\) (the Lagrangian Grassmannian). We note that (27) is also proved directly in [8]. \(\square \)
Note that Proposition 1.2 follows from (28).
Now fix \(\epsilon \) and a Lagrangian subspace F, and let
According to Proposition 3.3,
where the constants are locally uniform in E. Let
be the singular value decomposition of \({\widetilde{\Phi }}_N(E)\). Assume that the singular values on the diagonal of \(\Sigma _N(E)\) are arranged in nonincreasing order; the choice of the additional degrees of freedom is not essential for the current discussion. Denote
Claim 3.4
Let \(F \subset {\mathbb {R}}^{2W}\) be a Lagrangian subspace. For N large enough (depending on \(\epsilon \)), one has (deterministically) on the event \(\Omega _\epsilon ^F[{\widetilde{\Phi }}_N(E)]\) defined in (29)
Remark 3.5
For future reference, we also record the dual version of the claim: on \(\Omega _\epsilon ^F[{\widetilde{\Phi }}_N(E)^\intercal ]\)
Proof
We abbreviate \(\Sigma = \Sigma _N(E)\), \(V = V_N(E)\), and \(\gamma _j = \gamma _j(E)\). On the other hand, the constants with \(\epsilon \) not explicitly present in the notation will be uniform in \(\epsilon \rightarrow + 0\).
Clearly, \(s_j(\pi _{F_+ }V^\intercal \pi _F^*) \leqslant \Vert \pi _{F_+ }V^\intercal \pi _F^* \Vert \leqslant 1\). Hence, it will suffice to show that on \(\Omega _\epsilon ^F[{\widetilde{\Phi }}_N(E)]\)
Let \(\Sigma ^+\) be the diagonal matrix obtained by setting the (k, k) matrix entries of \(\Sigma \) to zero for \(k > W\). Then, on \(\Omega _\epsilon ^F[{\widetilde{\Phi }}_N(E)]\) we have
(with \(c>0\) uniform in \(\epsilon \rightarrow +0\)). Thus, \(s_j({\widetilde{\Phi }}_N \pi _F^*) = s_j(\Sigma V^\intercal \pi _F^*)\) satisfies
Observing that \(s_j(\Sigma ^{+} V^\intercal \pi _F^*) = s_j({\widehat{\Sigma }}^{+}\pi _{F_+} V^\intercal \pi _F^*)\), where \({\widehat{\Sigma }}^+ = \pi _{F_+} \Sigma ^+ \pi _{F_+}^*\), and that
on \(\Omega _\epsilon ^F\), we get (for sufficiently large N):
On the other hand, using the submultiplicativity of the operator norm and the equality between the norm of the jth wedge power of a matrix and the product of its j top singular values, we have
whence
thus concluding the proof of (31) and of the claim. \(\square \)
WegnerType Estimate: Proof of Proposition 1.3
Let us first show that for any \(i \in [N, N]\)
By Claim 2.2,
where
Hence,
where
and the subscripts 11 and 21 represent extracting the corresponding \(W\times W\) blocks from a \(2W \times 2W\) matrix (i.e. \(Y_{11} = \pi _{F_+} Y \pi _{F_+}^*\), \(Y_{21} = \pi _{F_} Y \pi _{F_+}^*\), in the notation of (30)). Both matrices \(X^\pm \) are Hermitian, as follows from the symplectic property of the transfer matrices.
Without loss of generality, we can assume that \(i \geqslant 0\). We shall prove that
To this end, denote
In the notation of Claim 3.4, consider the transfer matrix \({\widetilde{\Phi }}_{i+1,N}\), and let
(note that \({\widetilde{\Phi }}_{i+1,N}\) is independent of \(X^+\) and thus also of F). It suffices to show that on \(\Omega _\epsilon \)
Let us write the singular value decomposition of \({\widetilde{\Phi }} = {\widetilde{\Phi }}_{i+1,N}\) in block form:
whence on \(\Omega _\epsilon \)
Further, by Claim 3.4 we have on \(\Omega _\epsilon \):
Let us show that
To this end, start with the relation
In view of the bound
we can set \(E_2' = E_2 (U_{21} {\widehat{\Sigma }}^+ V_{11}^\intercal )^{1}\) and rewrite
which implies (35).
Now, the matrix \(X^+\) is symmetric, therefore \(x  X^+ y = 0\) for \(\left( {\begin{array}{c}x\\ y\end{array}}\right) \in F^\perp \), whence for any \(\left( {\begin{array}{c}x\\ y\end{array}}\right) \in {\mathbb {R}}^{2W}\)
(where the first term is a \(1 \times 2\) block matrix). Therefore, we have, by another application of Claim 3.4:
This, together with (35) and (34), concludes the proof of (33), and of (32).
Now we consider the elements \(G_E[H_{[N,N]}](i, i\pm 1)\). We have:
The norm of \( G_E[H_{[N,N]}](i, i)\) is controlled by (32), whereas \(\Psi _{i+1}^\pm (\Psi _i^\pm )^{1} = L_i^{1} X^\pm \) are controlled using (35) and Claim 3.4. \(\square \)
On Generalisations
Other distributions The assumptions (A)–(C) in Theorems 1 and 2 can probably be relaxed. Instead of a finite fractional moment in (A), it should be sufficient to assume the existence of a sufficiently high logarithmic moment:
for a sufficiently large \(A > 1\). To carry out the proof under this assumption in place of (A), one would need appropriate versions of large deviation estimates for random matrix products.
As we saw in the previous section, the rôle of the assumptions (B)–(C) is to ensure that the conditions of the Goldsheid–Margulis theorem [16] are satisfied. That is, our argument yields the following:
Theorem 3
Let \(I \subset {\mathbb {R}}\) be a compact interval. Assume (A) and that for any \(E \in I\) the group generated by
is Zariskidense in \({\text {Sp}}(2W, {\mathbb {R}})\). Then:

1.
The spectrum of H in I is almost surely pure point, and
$$\begin{aligned} {\mathbb {P}} \left\{ \forall (E, \psi ) \in {\mathcal {E}}[H] \,\,\, E \in I \Longrightarrow \limsup _{x \rightarrow \pm \infty } \frac{1}{x} \log \Vert \psi (x)\Vert \leqslant  \gamma _W(E)\right\} =1;\nonumber \\ \end{aligned}$$(36) 
2.
for any compact subinterval \(I' \subset I\) (possibly equal to I) one has:
$$\begin{aligned} {\mathbb {P}}\left\{ \limsup _{x \rightarrow \pm \infty } \frac{1}{x} \log Q_I(x, y) \leqslant  \inf _{E \in I} \gamma _W(E) \right\} = 1. \end{aligned}$$(37)
As we saw in the previous section, the second condition of this theorem is implied by our assumptions (B)–(C). Most probably, weaker assumptions should suffice, and, in fact, we believe that the conclusions of Theorems 1 and 2 hold as stated without the assumption (B). A proof would require an appropriate generalisation of the results of Goldsheid [15].
Another interesting class of models appears when \(V_x \equiv 0\). The complex counterpart of this class, along with a generalisation in which the distribution of \(L_x\) depends on the parity of x, has recently been considered by Shapiro [31], in view of applications to topological insulators. An interesting feature of such models is that the slowest Lyapunov exponent \(\gamma _W(E)\) may vanish at \(E=0\). This circle of questions (in particular, the positivity of the smallest Lyapunov exponent and Anderson localisation) is studied in [31] under the assumption that the distribution of \(L_0\) in \({\text {GL}}(W, {\mathbb {C}})\) is regular. In order to extend the results of [31] (for matrices complex entries) to singular distributions, one would first need an extension of [16] to the Hermitian symplectic group.
Returning to the (real) setting of the current paper, assume that (B)–(C) are replaced with
 (B\('\)):

the group generated by \({\mathcal {L}}\) is Zariskidense in \({\text {GL}}(W, {\mathbb {R}})\);
 (C\('\)):

\(V_x \equiv 0\).
Along the arguments of [31], one can check that the conditions of [16] hold for any \(E \ne 0\). From Theorem 3, one deduces that the conclusion of Theorem 1 holds under the assumptions (A), (B\('\)), (C\('\)), whereas the conclusion (37) of Theorem 2 holds for compact intervals I not containing 0. If \(\gamma _W(0)= 0\), (37) is vacuous for \(I \ni 0\). If \(\gamma _W(0) > 0\), (37) is meaningful and probably true for such intervals, however, additional arguments are required to establish the large deviation estimates required for the proof.
Finally, we note that Theorem 3 remains valid if the independence assumption is relaxed as follows: \(\{(V_x, L_x)\}_{x \in {\mathbb {Z}}}\) are jointly independent (i.e. we can allow dependence between \(V_x\) and the corresponding \(L_x\)).
The halfline Similar results can be established for random operators on the halfline. For simplicity, we focus on the case \(L_x \equiv \mathbbm {1}\). Fix a Lagrangian subspace \(F \subset {\mathbb {R}}^{2W}\), and consider the space \({\mathcal {H}}_F\) of squaresummable sequences \(\psi : {\mathbb {Z}}_+ \rightarrow {\mathbb {C}}^W\) such that \(\left( {\begin{array}{c}\psi (1)\\ \psi (0)\end{array}}\right) \in F\). Define an operator \(H_F\) acting on \({\mathcal {H}}_F\) so that
(see e.g. [4] for details).
Theorem 4
Fix a Lagrangian subspace \(F \subset {\mathbb {R}}^{2W}\). Under the assumptions (A) and (C) with \(L_x \equiv 1\), the spectrum of \(H_F\) in any compact interval I is almost surely pure point, and
Remark 4.1

1.
For general \(L_x\), the boundary condition has to be prescribed in a different way. However, in the Dirichlet case \(F = F_+\) the result holds as stated for general \(L_x\) satisfying (A)–(B).

2.
Combining the proof of Theorem 2 with the additional argument described below, one can also prove dynamical localisation.

3.
For \(W=1\), a result of Kotani [24] implies that the operator \(H_F\) has pure point spectrum for almost every boundary condition F; a similar statement is valid for \(W>1\). As to fixed (deterministic) boundary conditions, the only published reference known to us is the work of Gorodetski–Kleptsyn [20], treating Schrödinger operators in \(W=1\) with Dirichlet boundary conditions.

4.
The event of full probability provided by Theorem 4 depends on the boundary condition F. And indeed, a result of Gordon [19] implies that (almost surely) there exists a residual set of initial conditions F for which the spectrum of \(H_F\) is not pure point (and in fact has only isolated eigenvalues).
Sketch of proof of Theorem 4
We indicate the necessary modifications with respect to the proof of Theorem 1. First, we modify the definition (11) of \({\text {Res}}(\tau , E, N)\) as follows: \(x \geqslant N+1\) is said to be \((\tau ,E,N)\)nonresonant (\(x \notin {\text {Res}}(\tau , E,N)\)) under the same condition
while \(x \in \{1, \ldots , 2 N\}\) is said to be \((\tau ,E,N)\)nonresonant if
(this condition does not depend on x and only depends on the restriction of the operator to [1, N]). We claim that Proposition 2.1 is still valid:
To prove this estimate, it suffices to show that for any \(1 \leqslant x < y \leqslant N^2\) with \(yx>2N\) one has
The case \(x, y > 2N\) is covered by the current Proposition 2.1. If \(x \leqslant 2N\) and \(y > 2N\) the events \(x \in {\text {Res}}(\tau , E, N)\) and \(y \in {\text {Res}}(\tau , E, N)\) are independent; the probability that \(x \in {\text {Res}}(\tau , E, N)\) is exponentially small due to the large deviation estimate (9), and the collection of E violating (40) is the union of \(\leqslant N^{2W}\) intervals. From this point the proof of (42) mimics the argument in the proof of Proposition 2.1.
As in the proof of Theorem 1, let \(\psi \) be a generalised solution at energy \(E \in I\) given by Schnol’s lemma, \(x^{1} \log \Vert \psi (x) \Vert \rightarrow 0\). Letting \(u_x = \left( {\begin{array}{c}\psi (x)\\ \psi (x1)\end{array}}\right) \), we have
hence for sufficiently large N one has
On the other hand, on an event of full probability one has for all \(E \in I\) and all sufficiently large N
due to a version of the Craig–Simon theorem [7] (cf. [18, Lemma 2.2]). This implies
which contradicts (40) when \(\tau > 0\) is small enough. Thus, for N large enough
and thus \(\psi \) decays exponentially as in the proof of Theorem 1. \(\square \)
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Acknowledgements
We are grateful to Ilya Goldsheid for helpful discussions and to Alexander Elgart for spotting a number of lapses in a preliminary version of this paper.
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Macera, D., Sodin, S. Anderson Localisation for QuasiOneDimensional Random Operators. Ann. Henri Poincaré (2022). https://doi.org/10.1007/s0002302201191z
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DOI: https://doi.org/10.1007/s0002302201191z