Random Products of Unimodular Matrices

	It is the nature of all greatness not to be exact.
	EDMUND BURKE

Using measure theoretical methods, FURSTENBERG has derived a theorem that is useful for proving ANDERSON localization on one-dimensional lattices as, for example, in chapter 5 of this study. The theorem deals with a specific class of random matrices $X_m$, $m\in\mathbb{N}$, and states that generically the norms of the vectors $\, X_m X_{m-1}\cdots X_1\vec{u}_0 \,$ tend to infinity at an exponential rate with $m\to\infty$, provided that certain -- not very restrictive -- conditions are met. The theorem is applied in chapter 5, where the role of the $X_m$ is taken by the transfer matrices ${\cal{T}}_m$.

Consider the classical group $\mbox{SL}(N,\mathbb{R})$ of unimodular real $(N,N)$ matrices with the usual matrix product. A subgroup $G$ of $\mbox{SL}(N,\mathbb{R})$ is called irreducible if the only subspaces of $\mathbb{R}^N$ left fixed by the matrices of $G$ are $\mathbb{R}^N$ and $\{(0,\dots,0)^t\}$; otherwise $G$ is called reducible. Let $\mu$ be a measure on $\mbox{SL}(N,\mathbb{R})$, induced by the distribution of the elements of a given set of unimodular matrices $\mbox{$\big\{X_m, \; m\in\mathbb{N}\big\}$} \subseteq \mbox{SL}(N,\mathbb{R})$, and let $G$ be the smallest closed subgroup of $\mbox{SL}(N,\mathbb{R})$ that contains the support of $\mu$.

FURSTENBERG's theorem discloses some details of the asymptotic behaviour of the norms $\big\Vert\vec{u}_m\big\Vert$ of the vectors

$$\vec{u}_m \; := \; X_m X_{m-1}\cdots X_1\vec{u}_0, \quad m\in\mathbb{N}$$ (B.1)

for $\vec{u}_0\in\mathbb{R}^N$. The vector norm $\Vert\cdot\Vert$ used here is the conventional Euclidean 2-norm and not to be confused with the matrix norm $\Vert\cdot\Vert _{\rm m}$ used below.

Theorem. If $G$ is irreducible and $\mu$ satisfies

$$\int\limits _{G} {\mbox{d}}\mu(M) \, \log(\Vert M\Vert _{\rm m}) \; < \; \infty ,$$ (B.2)

then for random matrices $\{X_m\}$ which are independently distributed according to $\mu$, with probability one the limit

$$\lim_{m\to\infty} \frac{ \log \left\Vert % \X_m\X_{m-1}\... ...1\vec{u} \vec{u}_m \right\Vert }{m} \; =: \; \gamma_\mu$$ (B.3)

exists for all nonzero vectors $\vec{u}_0\in\mathbb{R}^N$.
Moreover, if $G$ is noncompact, then $\gamma_\mu$ is strictly positive.

The theorem holds with probability one only, due to the random nature of the $X_m$. Almost all such sequences of matrices -- and thus almost all distributions of matrices obtained in this way -- give the desired result, but there are some that do not. The probability of coming by such an exceptional case is zero, though, such that for all practical applications the theorem can be considered to be true. The (rather technical) proof of the theorem is given in FURSTENBERG's paper [Fur63].

In a typical application one is given a set $\big\{X_m\big\}$ of unimodular matrices with a distribution $\mu$ on $\mbox{SL}(N,\mathbb{R})$. First one has to identify the corresponding matrix group $G$; then irreducibility and noncompactness of $G$ need to be confirmed. This, by FURSTENBERG's theorem, establishes the result that for almost all such sets $\big\{X_m\big\}$ with a sufficiently well-behaved measure $\mu$, the norms of the vectors $\vec{u}_m$ grow exponentially for large $m$,

$$\left\Vert \vec{u}_m \right\Vert \; \sim \; e^{\gamma_\mu m} \quad \mbox{with} \quad \gamma_\mu>0,$$ (B.4)

provided that the initial vector $\vec{u}_0$ is nonzero. The rate of growth is given by the ``LIAPUNOV exponent $\gamma_\mu$ of the set $\big\{X_m\big\}$'' and does not depend on $\vec{u}_0$. For checking the requirement (B.2) on $\mu$, any matrix norm $\Vert\cdot\Vert _{\rm m}$ can be used; often the maximum norm $\Vert M\Vert _{\rm m}=\max\limits_{i,j}\vert M_{ij}\vert$
is most convenient to work with.

For $N=2$, the most important example is $\mbox{SL}(2,\mathbb{R})$ itself:

$$G_1 \; := \; % \left\{ \big\{ % M \! = \! (M_{ij}), ... ... \det(M) \! = \! 1 % M_{11}M_{22}-M_{21}M_{12}=1 \big\}$$ (B.5)

defines the appropriate group containing all the transfer matrices (5.61) that model the tight binding equation of the kicked rotor. $G_1$ also plays the same role with respect to the kicked harmonic oscillator and its transfer matrices (5.114, 5.117, 5.120). Clearly, $G_1$ is irreducible and noncompact and thus for sufficiently well-behaved $\mu$ generically gives rise to exponentially growing $\big\Vert X_m X_{m-1}\cdots X_1\vec{u}_0 \big\Vert$ for $X_m\in G_1$. This observation is used, for example, in subsection 5.1.3. There, for the ANDERSON-LLOYD model, a measure $\mu$ is used that is generated by the Lorentzian $p(\epsilon_m)$ (5.70) with respect to the transfer matrices $\{{\cal{T}}_m\}$ of equation (5.61), such that

$$\int\limits _{G_1} {\mbox{d}}\mu(M) \, \log(\Vert M\Vert _{\rm m}) = \int\limits _{-\infty}^\infty {\mbox{d}}\epsilon_m \; p(\epsilon_... ...\left( \max\left( \left\vert\frac{\epsilon_m}{W_1}\right\vert,1 \right) \right)$$

$$= \frac{2}{\pi} \int\limits _{\vert W_1\vert}^\infty {\mbox{d}}\eps... ...g\displaystyle \frac{\epsilon_m}{\vert W_1\vert}}{\displaystyle 1+\epsilon_m^2}$$ (B.6)

$$= -\log\vert W_1\vert +\frac{i}{\pi}\Big( \mbox{Li}_2(-i\vert W_1\vert)-\mbox{Li}_2(i\vert W_1\vert) \Big)$$

is obtained, with the dilogarithm function $\mbox{Li}_2$ [AS72]. The expression (B.6) is finite for any given nonzero $W_1$, such that the integrability condition (B.2) is satisfied. Analogous calculations for the measures corresponding to the two-dimensional transfer matrix models defined by equations (5.114, 5.117, 5.120) with respect to the kicked harmonic oscillator show that -- in the cases considered in subsection 5.3.2 -- the theorem applies to these models as well, because either the matrix elements used there are Lorentzian distributed, or at least their distribution is sufficiently localized for allowing condition (B.2) to hold.

The group given by

$$G_2 \; := \; \left\{ \left( \begin{array}{@{}c@{\hspac... ... \end{array} \right), \; M_{12} \in\mathbb{R} \right\},$$ (B.7)

on the other hand, may serve as a counterexample for $N=2$. It is easy to verify that $G_2$ indeed defines a subgroup of $\mbox{SL}(2,\mathbb{R})$. But $G_2$ is reducible, since $\mbox{span}\left\{\vec{e}_1\right\} \subset\mathbb{R}^2$ is mapped onto itself by the matrices of $G_2$. Therefore, the norms $\big\Vert X_m X_{m-1}\cdots X_1\vec{u}_0 \big\Vert, \; X_m\in G_2$ should not be expected to grow exponentially.

For $N=3$, the theorem is used in subsection 5.3.3 with respect to $\mbox{SL}(3,\mathbb{R})$ itself: the transfer matrices (5.123a) and (5.124a) belong to the group defined by

$$G_3 \; := \; % \left\{ \big\{ % M \! = \! M \! = \! ... ...\in\mathbb{R}, \;\; \det(M) \! = \! 1 % \right\} \big\}$$ (B.8)

Again, irreducibility and noncompactness of $G_3$ are obvious, and FURSTENBERG's theorem applies, once the integrability condition (B.2) has been verified -- as in subsection 5.3.3.