Products of Random Matrices with Applications to Schrödinger Operators

In this chapter we define the upper Lyapunov exponent γ which gives the exponential rate of growth of the norm of products of independent identically distributed (i.i.d.) random matrices. In order to prove the analogue of the law of large numbers we develop some basic results on G-spaces which will often be used in the sequel.

We present here the main arguments we shall use for studying products of random matrices of arbitrary order, but in the case of 2 × 2 matrices. Our main interest is not the theorems in themselves and actually shorter proofs are available when dealing with 2 × 2 matrices. We rather intend to explicit in this simple situation the general approach, valid for matrices of any order. This chapter is thus introductive. Although all the results are particular cases of more general statements to be proved later, we give them in full detail for the convenience of the reader who is only interested in matrices of order 2.

We now consider i.i.d. invertible random matrices Y₁, Y₂, ... of arbitrary order, say d . This chapter is devoted to the study of the basic almost sure properties of the products S_n = Y_n...Y₁. We shall derive their salient feature, which is the contracting action of S_n on the set of directions. In particular we shall give (see 3.4, 4.3 and 6.1), following Guivarc’h and Raugi [34], a condition ensuring that

In the preceding chapter we have given a criterion ensuring that the two upper Lyapunov exponents are distinct. It will give us all we need for the study of limit theorems. But a sharp study of the behaviour at infinity of the random products S requires a precise knowledge of the relations between all the exponents. For instance they provide all the limit values of \( \frac{1}{n}\;Log\;\left\| {{S_{n}}\left( \omega \right)x} \right\| \) , when ω is kept fixed and x runs through ℝ^d (Osseledec’s theorem) and determine the possible boundaries towards which S_n converges.

In 1970 Kaijser showed in some particular but typical cases that the contractive action of the random products S_n = Y_n...Y₁ on P(ℝ^d) implies that Log ∥S_nx∥ suitably normalized converges in distribution to a gaussian law (see [38], [39], [40]). This idea was later fully developed by Le Page in [49] where he proved that, loosely speaking, Log ∥S_nx∥ behaves like a sum of i.i.d. real random variables and satisfies analogues of the main classical limit theorems. We shall give here a detailed introduction to the important work of Le Page and present, with some improvements, his main results (namely, the central limit theorem with speed of convergence and an estimate of the large deviations of Log ∥S_nx∥). We also study the tightness of {S_nx, n ≧ 1} without moment assumption. This chapter ends with an application to linear stochastic differential equations.

The main result to be proved in this chapter is that under suitable hypotheses, the μ-invariant distribution v on P(ℝ^d) satisfies for some α > 0 .

Let L be the linear space of complex sequences ψ = (ψ_n) where n runs through the set of integers ZZ. The operator H is associated to two given real sequences a and b with b_n ≠ 0 ∀n ε ZZ, and acts on L by the formula: . For a complex number X every solution of the difference equation Hψ = λψ lies in a two dimensional subspace of ob spanned by the solutions p(λ) and q(λ) constructed from the initial values p (X) = q_-1(λ) = 1, p_-1 (λ) = q_o (λ) = 0, such that: . From now in order to avoid too complicated notations we don’t write the variable X in the solutions of the difference equation. A solution ψ of the difference equation is constructed from initial values ψ_o and ψ_-1. by a product of “transfer matrices” Y_n defined by: . The transfer matrices Y_n and therefore the products S_n belong to the group SL(2,c) of two by two matrices with complex entries and of determinant one. If A is real then Y and S belong to the subgroup SL(2,ℝ) with real entries. The construction of the solutions of the difference equation by such products of matrices is the essential link between the two parts of this book.

We now suppose that (a_n,b_n), n ε ZZ, is a stationary random process This means that the real random variables (a_n (ω), b_n(ω)) are defined on some complete probability space (Ω,a,ℙ) and that there exists an invertible measurable transformation θ of Ω, leaving ℙ invariant and such that a_n+1=a_nºθ, b_n+1, ºθ. In general we don’t write the variable ω and when a property depends only upon the common law of the sequence (a_n, b_n) we omit the index n and speak of the variables (a) and (b). We say that the family H(ω) of associated operators on H is ergodic if a θ invariant measurable subset of Ω is of zero or one ℙ measure. It’s easily seen that H° θ = U^-1 H U where U is the shift measure. It’s easily seen that H o θ = U^-1 H U where U is the shift operator on ZZ, (Uψ)_n = ψ_n-1.

We now prove the strongest singularity property for the spectrum of Schrödinger operators, that is the pure point spectrum property. Until the end of the year 1984 the only known proofs were very close from the original one given by the Russian school (see the introduction). In most of these works it was not clear what was really necessary in order to obtain the pure point spectrum property, since they dealed essentially with the independent or Markov cases under strong assumptions on these processes. A very clarifying idea of Kotani introduced in II.7 and applied to the independent case in the first section gives very easily the expected result, at least in the classical Schrödinger case, that is when b_n = 1, ∀n ε ZZ. It’s not clear that such a procedure can be easily applied to the Helmotz case. Furthermore Kotani’s theory does not use any approximation of the spectral properties of H by means of “thermodynamic limits” along sequences of boxes going to ZZ. Thus if we want to obtain some information about the speed of convergence of these spectral measures computed in boxes, we have to use an other way. In this purpose the second section is devoted to the study of the Laplace transform on SL(2,ℝ). The corresponding operators were already introduced in A.V. In section 4, using the spectral properties of these operators, we prove again the pure point spectrum property in the independent case, thank’s to the “good” approximations of the spectral measures of H obtained in I. These results are also used in section 5, in order to obtain a representation formula for the density of states.

Most of the results obtained for the one dimensional Schrödinger operator in the preceding chapters can be adapted to the case of the strip. Deterministic properties are easily generalized, replacing at time, the Poincare half plane by the generalized Siegel half plane. For random operators, the theory of Lyapunov exponents is more involved and we refer heavily to part A chapter IV for related results. In contrast with the one dimensional case, positivity of some exponents is not easily checked and needs more assumptions on the “disorder”. As in chapter III we give two proofs of the pure point spectrum property. The first proof is a straightforward adaptation of Kotani’s criterion. In order to treat the Helmotz case and to obtain some information about “thermodynamic limits” we need some Laplace analysis on symplectic groups. Such results, easily obtained on SL(2,ℝ), require here much more work since we have to deal with the Poisson kernels associated to the boundaries of symplectic groups.