Anderson localisation for quasi-one-dimensional random operators

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 multi-scale 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 multi-scale analysis, and also extend it to the general quasi-one-dimensional 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. Our work generalises and complements the single-scale 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.


The operator, transfer matrices and Lyapunov exponents
Let ⩾ 1. Let { } ∈ℤ be a sequence of identically distributed × random matrices in GL( , ℝ), and let { } ∈ℤ be a sequence of identically distributed × real symmetric matrices, so that { } ∈ℤ , { } ∈ℤ are jointly independent. Denote by  the support of 0 and by  -the support of 0 . Throughout this paper we assume that (A) there exists > 0 such that 1 Introduction 2 (C)  is irreducible (i.e. has no common invariant subspaces except for {0} and ℝ ), and  −  contains a matrix of rank one.
We are concerned with the spectral properties of the random operator acting on (a dense subspace of) 2 This model, often referred to as a quasi-one-dimensional random operator, is the general Hamiltonian describing a quantum particle with internal degrees of freedom in random potential and with nearest-neighbour random hopping. The special case ≡ ½ is known as the block Anderson model; it is in turn a generalisation of the Anderson model on the strip ℤ×{1, ⋯ , }, and, more generally, on ℤ × Γ, where Γ is any connected finite graph (the assumption that Γ is connected ensures that  is irreducible). Another known special case of (1) is the Wegner orbital model.
Fix ∈ ℝ. If ∶ ℤ → ℂ is a formal solution of the equation where the one-step transfer matrix ∈ GL(2 , ℝ) is given by The multi-step transfer matrices Φ , ∈ GL(2 , ℝ), , ∈ ℤ, are defined by so that In particular, = Φ +1, . We abbreviate Φ = Φ ,0 . The Lyapunov exponents ( ), 1 ⩽ ⩽ 2 , are defined as ( ) = lim Further, as we shall see in Section 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 ( ) are continuous functions of . 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
is the collection of eigenpairs of , then i.e. each eigenfunction decays exponentially, with the rate lower-bounded by the slowest Lyapunov exponent.
We refer to [18] for a discussion and partial results in this direction. For = 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 . The mathematical work on Anderson localisation in one dimension was initiated by Goldsheid, Molchanov and Pastur [17], who considered the case = 1, ≡ 1 and established the pure point nature of the spectrum under the assumption that the distribution of 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 > 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 ≡ ½, is tridiagonal with the off-diagonal entries equal to 1 and the diagonal ones independent and identically distributed, under the assumption that the distribution of the diagonal etries of 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 multi-scale 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 multi-scale analysis. Recently, proofs of the result of [6] avoiding multi-scale analysis were found by Bucaj et al. [5], Jitomirskaya and Zhu [22], and Gorodetski and Kleptsyn [20]; the general one-dimensional 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 single-scale proof of localisation which applies to arbitrary ⩾ 1, and allows for rather general distributions of 0 and 0 (under no regularity assumptions on the distribution of the potential). In particular, we recover and generalise the result of [23].
Here ‖ ( Λ ) , ‖ is the operator norm of the ( , ) block of ( Λ ), and the functions in the supremum are assumed to be, say, Borel measurable.

Theorem 2. Assume (A)-(C). For any compact interval
It is known (see [3]) that Theorem 2 implies Theorem 1. By plugging in various choices of , 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.

Main ingredients of the proof
Similarly to many of the previous works, including [6,23] and also the recent works [5,22,20], 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 ⊂ ℝ be a compact interval, and let ⊂ ℝ 2 be a Lagrangian subspace (see Section 3). Denote by ∶ ℝ 2 → the orthogonal projection onto .

Proposition 1.2. Assume (A)-(C). For any > 0 there exist , > 0 such that for any ∈ and any Lagrangian subspace
The proof is essentially given in [23]; we outline the necessary reductions in Section 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 Section 3.3.
For an operator and in the resolvent set of , we denote by [ ] = ( − ) −1 the resolvent of and by [ ](⋅, ⋅) its matrix elements. If lies in the spectrum of , we set [ ](⋅, ⋅) ≡ ∞.
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 multi-scale analysis. We deduce Theorems 1 and 2 directly from these propositions. In this aspect, our general strategy is similar to the cited works [5,22,20]. However, several of the arguments employed in these works rely on the special features of the model for = 1; therefore our implementation of the strategy differs in several crucial aspects.
2 Proof of the main theorems

Resonant sites; the main technical proposition
and ( , , )-resonant ( ∈ Res( , , )) otherwise. The following proposition is the key step towards the proof of Theorems 1 and 2.
Proposition 2.1. Assume (A)-(C). Let ⊂ ℝ be a compact interval, and let > 0. There exist , > 0 such that for any ⩾ 1 The remainder of this section is organised as follows. In Section 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 ∈ Res( , , ) (for a fixed ∈ ℝ) is exponentially small. In Section 2.3, we rely on this estimate to prove Proposition 2.1. Then we use this proposition to prove Theorem 1 (Section 2.4) and Theorem 2 (Section 2.5).

Reduction to transfer matrices
The Green function of [− , ] can be expressed in terms of these matrices using the following claim, which holds deterministically for any of the form (1). A similar expression has been employed already in [14].
The matrices ( , ), − ⩽ ⩽ , are uniquely determined by the system of equations We look for a solution of the form where The first equation ensures that (14) defines ( , ) consistently, while the second one guarantees that (13) holds for = . For the other values of , (13) follows from the construction of the matrices Ψ ± .

Proof of Proposition 2.1
Fix a small > 0. Without loss of generality is short enough to ensure that (this property is valid for short intervals due to the continuity of ; the statement for larger intervals follows by compactness). Fix such (which will be suppressed from the notation), and let

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: is the ratio of two polynomials of degree ⩽ (2 + 1). Hence the level set is of cardinality ⩽ (2 +1) (note that the ⩽ (2 +1) discontinuity points of ± , are poles, hence they can not serve as the endpoints of the superlevel sets of this function). Hence our set is the union of at most ⩽ (2 + 1)∕2 closed intervals, and Res * ( , , ) is the union of at most 2 ( + 1) 2 (2 + 1) 2 ⩽ closed intervals.

Claim 2.4. Assume (A)-(C)
. For any compact interval ⊂ ℝ there exist , > 0 such that for any ⩾ 1 and any ∈ , Proof. According to Claim 2.2, By Propositions 1.2 and 1.3, both terms decay exponentially in , locally uniformly in . This concludes the proof of (17) and of Proposition 2.1.

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 ∶ ℤ → ℂ is called a generalised eigenfunction corresponding to a generalised eigenvalue ∈ ℝ if Schnol's lemma asserts that any spectral measure of is supported on the set of generalised eigenvalues. Thus we need to show that (with full probability) any generalised eigenpair ( , ) satisfies lim sup Fix a compact interval ⊂ ℝ, and > 0. Consider the events By Proposition 2.1 and the Borel-Cantelli lemma, We shall prove that on any  ( , ) every generalised eigenpair ( , ) with ∈ satisfies From (18) The function is bounded due to (19), hence it achieves a maximum at some ∈ ℤ. For Thus (22) holds whenever , are such that 3 < | | ⩽ 2 and ⩾ 0 . For each ∈ ℤ, let ( ) be such that 2 ∕10 ⩽ | | ⩽ 2 ∕5. If | | is large enough, ( ) ⩾ 0 . Applying (22) ⌊| |∕( + 1)⌋ − 4 times, we obtain which implies (21).

Eigenfunction correlator: proof of Theorem 2
Fix a compact interval ⊂ ℝ, and let = min ∈ ( ). 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 Λ ⊃ { , }. Without loss of generality we can assume that = 0. Choose such that 2 ∕10 ⩽ | | ⩽ 2 ∕5. By Proposition 2.1, for any ∈ (0, ) We show that on the event we have Expand the Green function where 0 = 0, +1 = ± , 0 = , +1 = ± , and (by the construction of the event (24)) + ⩾ | |∕ − 4. All the terms in the first and third line of (26)

are bounded in norm by
Now we raise this estimate to the power 1 − and integrate over ∈ : It remains to let → +0 while making use of the two inequalities in (23).

Preliminaries
The matrices are, generally speaking, not symplectic. However, the cocycle {Φ , } , ,∈ℤ is conjugate to a symplectic one. Indeed, observe that Thus alsõ

Simplicity of the Lyapunov spectrum and large deviations
Goldsheid and Margulis showed [16] that if are independent, identically distributed random matrices in Sp(2 , ℝ), and the group generated by the support of 1 is Zariski dense in Sp(2 , ℝ), then the Lyapunov spectrum of a random matrix product { ⋯ 1 } is simple, i.e.

Corollary 3.2. Assume (A)-(C). Then for any
Proof. Observe that If the Zariski closure of the group generated by  −1 intersects , then the Zariski closure On the other hand, the constants with not explicitly present in the notation will be uniform in → +0.
Clearly, ( Let Σ + be the diagonal matrix obtained by setting the ( , ) matrix entries of Σ to zero for > . Then on Ω [Φ ( )] we have Observing that (Σ + ⊺ * ) = (Σ + + ⊺ * ), whereΣ + = + Σ + * + , and that on Ω , we get (for sufficiently large ): On the other hand, using the submultiplicativity of the operator norm and the equalitiy between the norm of the -th wedge power of a matrix and the product of its top singular values, we have thus concluding the proof of (31) and of the claim.
Without loss of generality we can assume that ⩾ 0. We shall prove that To this end, denote = ∈ ℝ 2 ∶ = − + .

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: (log + ‖ 0 ‖ + log + ‖ 0 ‖ + log + ‖ −1 0 ‖) < ∞ for a sufficiently large > 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 ⊂ ℝ be a compact interval. Assume (A) and that for any ∈ the group generated by is Zariski-dense in Sp(2 , ℝ). Then: 1. The spectrum of in is almost surely pure point, and 2. for any compact subinterval ′ ⊂ (possibly equal to ) one has: 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 ≡ 0. The complex counterpart of this class, along with a generalisation in which the distribution of depends on the parity of , 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 ( ) may vanish at = 0. This circle of questions (in partiular, the positivity of the smallest Lyapunov exponent and Anderson localisation) is studied in [31] under the assumption that the distribution of 0 in GL( , 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  is Zariski-dense in GL( , ℝ); Along the arguments of [31], one can check that the conditions of [16] hold for any ≠ 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 not containing 0. If (0) = 0, (37) is vacuous for ∋ 0. If (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: {( , )} ∈ℤ are jointly independent (i.e. we can allow dependence between and the corresponding ).
The case , > 2 is covered by the current Proposition 2.1. If ⩽ 2 and > 2 the events ∈ Res( , , ) and ∈ Res( , , ) are independent; the probability that ∈ Res( , , ) is exponentially small due to the large deviation estimate (9), and the collection of violating (40) is the union of ⩽ 2 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 be a generalised solution at energy ∈ given by Schnol's lemma, On the other hand, on an event of full probability one has for all ∈ and all sufficiently large