Thouless formula for random nonHermitian Jacobi matrices
Abstract
Random nonHermitian Jacobi matrices of increasing dimension are considered. We prove that the normalized eigenvalue counting measure of converges weakly to a limiting measure as . We also extend to the nonHermitian case the Thouless formula relating and the Lyapunov exponent of the secondorder difference equation associated with the sequence . The measure is shown to be logHölder continuous.
1 Introduction
Let , , and are three given sequences of complex numbers. Consider the secondorder difference equation for
(1.1) 
This equation can be also written as
(1.2) 
Denote by the solution of (1.1) satisfying the initial condition , . In terms of the transfer matrix ,
(1.3) 
Obviously, is a polynomial in of degree ,
(1.4) 
Its roots are the eigenvalues of the tridiagonal (Jacobi) matrix
(1.5) 
In this paper we are concerned with the limiting distribution of the eigenvalues of as for random , , and .
If all are real and for all the matrices are Hermitian. The eigenvalue distribution of such matrices was studied extensively in the past in the context of the Anderson model, see e.g. [18, 3]. In this case the eigenvalues are always real and there are several ways to prove that the normalized eigenvalue counting measure of converges to a limiting measure as . None of these proofs works in the nonHermitian case and little is known about the limiting eigenvalue distribution of random nonHermitian Jacobi matrices, however, see [5, 11].
Our interest to such matrices is partly motivated by nonHermitian quantum mechanics of Hatano and Nelson [9, 10] which, in one dimension, leads to equation (1.1) with the coefficients , , and chosen randomly from the special class defined by the restrictions
(1.6) 
In this class the Liouville substitution^{1}^{1}1, where and for . reduces equation (1.1) to the symmetric equation
(1.7) 
where . However, the situation here is much richer than in the Hermitian case as the choice of boundary conditions to accompany equation (1.1) has a profound effect on the spectrum of the associated Jacobi matrix. If the Dirichlet boundary conditions, and , are chosen then the corresponding Jacobi matrix is (1.5). As the Dirichlet boundary conditions are preserved by the Liouville transformation, the spectrum of is real provided the coefficients belong to the HatanoNelson class (1.6). On the other hand, if one imposes the periodic boundary conditions, and , then the spectrum of the corresponding Jacobi matrix turns out to be complex. This is not surprising of course as the Liouville substitution transforms the periodic boundary conditions for into highly asymmetric boundary conditions for . What is surprising however is that in the limit the complex eigenvalues lie on analytic curves [15] and are regularly spaced even if the coefficients in equation (1.1) are chosen randomly [16]. These effects are specific to the HatanoNelson class and the proofs and analysis of the limiting eigenvalue distribution given in [15, 16] exploit the relation between equations (1.1) and (1.7). Of course, in the general case of arbitrary coefficients no such relation exists and one requires a different approach in order to investigate the eigenvalue distribution of . We develop such an approach in the present paper.
Throughout this paper we assume that:

is a sequence i.i.d. random vectors.

For some .

The support of the probability distribution of the random vector contains at least two different points and .
If all mass of the probability distribution of is concentrated at one point then of course we have a tridiagonal matrix with constant diagonals. This is a particular case of Töplitz matrices. Eigenvalue distribution of nonHermitian Töplitz matrices was extensively studied in the past, see e.g. survey [21].
Our main result expresses the limiting distribution of the eigenvalues of in terms of the (upper) Lyapunov exponent
of equation (1.1). It is well known that (for every complex ) the above limit exists with probability one and is nonrandom. This follows from Oseledec’s multiplicative ergodic theorem [17]. A more subtle fact is that in our case can be calculated using the well known Furstenberg formula [7], and moreover
(1.8) 
The function is subharmonic in the entire complex plane [4] and bounded from below,
(1.9) 
This inequality easily follows from . The subharmonicity implies that , where is the distributional Laplacian in variables and , defines a measure on , see e.g. [12]. Our main result is as follows.
Theorem 1.1
Let be the normalized eigenvalue counting measure of , i.e. , where are the eigenvalues of . Then:

With probability one, converges weakly to as .

(Thouless formula) For every
(1.10) 
The limiting eigenvalue counting measure is logHölder continuous. More precisely, for any , ,
(1.11) where as .
We deduce Theorem 1.1 from Theorem 1.2 which is of independent interest in the context of second order difference equations.
Theorem 1.2
With probability one
(1.12) 
for almost all with respect to the Lebesgue measure on .
To prove Theorem 1.2, we use the theory of products of random matrices. Of course this is unnecessary in the Hermitian case. In this case (1.12) and the Thouless formula (1.10) follow directly from the fact that converges weakly to a limiting measure [1, 6] and the latter can be established independently and by more elementary means. We would like to emphasize that in the nonHermitian case we follow the opposite direction route: the weak convergence of and the Thouless formula are deduced from (1.12). To this end we make use of the relation
(1.13) 
where the equality is to be understood in the sense of distribution theory. Relation (1.13) is well known in the function theory. It holds for arbitrary polynomial of degree and can be easily derived with the help of the GaussGreen formula. In this general setup it was shown by Widom [20, 21] that if the measures for all are supported inside a bounded region and in the limit the function converges to a limiting function almost everywhere in the complex plane then converges weakly to . We shall need the following simple extension of this result to the case when the supports of are not necessarily bounded.
Let be a (deterministic) sequence of square matrices of increasing dimension , and
where is identity matrix and is the normalized eigenvalue counting measure of . Define
(1.14) 
Proposition 1.3
Assume that there is a function such that as almost everywhere in . If then it follows that is locally integrable, is a unit mass measure,
(1.15) 
and the sequence of measures converges weakly to as . If, in addition, then we also have that
(1.16) 
Remark. In view of (1.15), the integral on the RHS in (1.16) is a locally integrable function of taking values in .
For the sake of completeness, we give a proof of this Proposition in Appendix A.
In order to estimate the tails of eigenvalue distributions as required in the above Proposition 1.3 we use the following inequalities^{2}^{2}2Note that .:
(1.17) 
and for any and
(1.18) 
These inequalities can be derived with the help of Weyl’s Majorant Theorem, for details of derivation see Appendix B.
Let us now return to the random Jacobi matrices . Straightforward but tedious calculations show^{3}^{3}3For any Hermitian matrix we have with , . Therefore if is a nondecreasing function then, by the CourantFisher minimax principle, . that
for some independent of ’s and . Therefore if the random sequence is stationary and
(1.19) 
then the Ergodic Theorem asserts that with probability one the limits in (1.17) (1.18) are finite which implies and , as required in Proposition 1.3. The assumptions of stationarity and (1.19) are less restrictive than assumptions A1A3. However we are only able to prove Theorem 1.2 (which is the main ingredient to our proof of Theorem 1.1) under these more restrictive assumptions.
2 Products of random matrices
Our proof of Theorem 1.2 makes use of several facts from the theory of products of random matrices. We list these facts below (Propositions 2.1  2.3).
Let be a probability distribution on the group of invertible complex matrices and be an infinite sequence of independent samples from this distribution.
As before for By we denote the projective space on which every nondegenerate matrix acts in a natural way. Let be a probability measure on . We say that preserves if for any Borel set (here is the result of the action of on ). By we denote the closure of the subgroup of generated by all matrices belonging to the support of . We say that preserves if is preserved by every .
Proposition 2.1
Let be the singular values of . If
and are both finite  (2.1) 
then with probability one the following limits
(2.2) 
exist and are nonrandom.
The limiting values and are called the Lyapunov exponents of the sequence .
Proposition 2.2
If in addition to condition (2.1), no measure is preserved by then the Lyapunov exponents of the sequence are distinct, i.e. .
Proposition 2.3
If condition (2.1) is satisfied and no measure is preserved by then

For any unit vector the probability is one that
(2.3) 
If in addition for some then for any positive there is a constant such that uniformly in
(2.4)
Remarks. 1. As all norms in are equivalent, the choice of norm in (2.3) and (2.4) is not important. However it is convenient to deal with the standard Euclidian norm.
2. Propositions 2.1  2.3 are well known in the classical case of the real matrices, see e.g. [17, 2] for proofs of Propositions 2.1 and 2.3 and [7, 19] for proofs of Proposition 2.2. For complex matrices, Propositions 2.1 and 2.3 are proved in the same way as in [17, 2]. However, the proof of Proposition 2.2 is somewhat different from that given in [7, 19]. We shall now discuss the necessary changes which would allow the interested reader to reconstruct the proof in question simply by examining the one in [19]. Namely, the main ingredient of this proof is the fact that the mapping , where
defines a unitary representation of the group in Hilbert space with being the natural Lebesgue measure on the unit sphere . (Obviously, we are interested in the case when .)
In the case of the complex space the representation is defined by
in Hilbert space with being again the natural Lebesgue measure on the unit sphere . After that the proof proceeds in the way suggested in [19].
3 Proofs of Theorems 1.1 and 1.2
In order to be able to apply Propositions 2.1 – 2.3 we have to verify that under assumptions A1–A3 our matrices defined in (1.2) satisfy the conditions of these Propositions.
Is is apparent that assumption A2 guarantees that condition (2.1) is satisfied and . It remains to check that assumption A3 implies that no measure is preserved by (here is the measure induced on the group of matrices by the distribution of ). To this end we note that if
then
It remains to check that for almost all that the group generated by the matrices is rich enough in the sense that no measure is preserved by all matrices of this group. The main idea is as follows. For a ”typical” we construct two matrices, say and , from such that the eigenvalues of are of different moduli. It is easy to see then that the only measure preserved by all matrices of the form is the one supported by the lines in generated by the eigenvectors of . The matrix is then chosen so that its action on does not preserve these lines which means that the measure in question does not exist. We would like to emphasize that the presence of the parameter plays a crucial role in this situation.
More precisely, if is such that
then the matrix has eigenvalues with different moduli. In other words the moduli are different if does not belong to a certain half line. The plays then the role of (once again when lies outside of certain curves). This statement can be checked by direct calculation and is sufficient for our purposes.
However, in some important cases much more precise statements can be made. In particular if then each of triangular matrices and is nontrivial for all but may be two values of and a similar idea applies, see [2] page 213.
Now we are in a position to apply Propositions 2.1  2.3. For any two nonzero vectors and define
where is the scalar product in . The function is the natural angular distance between and on the projective space .
The following Lemma is the key element in the proof of Theorem 1.2. (In this Lemma and thereafter the abbreviation a.s. refers to the probability measure, i.e. any equality with the letters a.s. above it holds with probability one)
Lemma 3.1
Proof. For any one can always find two orthogonal unit vectors and such that and . In view of Proposition 2.1,
Obviously the sequence satisfies condition (3.1) and we first prove the large deviation estimate (3.2) for this sequence.
Let be a fixed unit vector. Then for every , and, since and , we have that
Therefore if then
and hence with probability one,
It follows now from Proposition 2.3 that
(3.3) 
for some and all where depends on the matrices , and also on and .
Now, let be an arbitrary sequence of random unit vectors satisfying condition (3.1), and let be a sequence of unit vectors orthogonal to , i.e. for all . Obviously, and, since , we have that with probability one
It is then apparent that is also exponentially small for large and therefore the large deviation estimate (3.2) for follows from (3.3).
Proof of Theorem 1.2. Let
Then
and therefore
(3.4) 
In view of (1.3) and Proposition 2.3(i),
(3.5) 
where is the upper Lyapunov exponent of the sequence of transfer matrices . On the other hand, , and therefore
It follows now from Lemma 3.1 (applied to the matrices and the vectors and ^{4}^{4}4If and are the Lyapunov exponents of a sequence then the sequence has the Lyapunov exponents and .) and the BorelCantelli Lemma that
Therefore, in view of (3.4) and (3.5), for any fixed the probability is one that
(3.6) 
But then the probability is one that (3.6) holds almost everywhere in the complex plane. This follows from the Fubini Theorem. Our proof of Theorem 1.2 is complete.
Proof of Theorem 1.1. As explained in introduction, under assumptions A1 – A3, the probability is one that for some nonrandom and . Therefore parts (a) and (b) of Theorem 1.1 follow immediately from Theorem 1.2 by the way of Proposition 1.3.
The logHölder continuity of is a corollary of the Thouless formula and the fact that the Lyapunov exponent is bounded from below. This is very much in the same way as in the Hermitian case, see [4].
To prove (1.11), we first note that the integral converges absolutely for every . Indeed, it follows from (1.15) that
and this inequality together with the Thouless formula and the lower bound (1.9) imply that
as well. Therefore,
(3.7) 
Obviously, for ,
and part (c) of Theorem 1.1 follows. Our proof of Theorem 1.1 is now complete.
Appendix A Appendix
Proof of Proposition 1.3. The local integrability of and the condition imply that the functions are uniformly integrable on bounded sets in . It follows from this that is locally integrable and as in , the space of Schwartz distributions in . Since is continuous on distributions, we also have that in . Obviously , hence is defined by a measure, see e.g. [13]. As any sequence of measures converging as distributions must converge weakly we conclude that weakly as measures.
For any ,
Therefore the inequality implies that the sequence of measures is tight, and hence cannot lose mass. As each of has unit mass, so has the limiting measure .
It follows from the weak convergence of to and (1.14) that
for any . This implies (1.15). Similarly, if then
(A.1) 
It remains to prove relation (1.16). It will suffice to show that
(A.2) 
when . Let be a continuous function with bounded support. Then
with
The function is continuous and when . Assume now that . Then
and
because of (A.1). It now follows from the weak convergence of to that
Therefore
and (A.2) follows.
Appendix B Appendix
Derivation of inequalities (1.17) and (1.18). Let and be respectively the eigenvalues and singular values of labeled so that and . Weyl’s Majorant Theorem, see [14], page 39, asserts that
for any function such that is convex on . Obviously the function satisfies this requirement for , and therefore
where is the number of eigenvalues of such that . Obviously,
and therefore
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