Random Products of Unimodular Matrices

It is the nature of all greatness not to be exact. |
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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 , , and states that generically the norms of the vectors tend to infinity at an exponential rate with , provided that certain -- not very restrictive -- conditions are met. The theorem is applied in chapter 5, where the role of the is taken by the transfer matrices .

Consider the classical group of unimodular real matrices with the usual matrix product. A subgroup of is called irreducible if the only subspaces of left fixed by the matrices of are and ; otherwise is called reducible. Let be a measure on , induced by the distribution of the elements of a given set of unimodular matrices , and let be the smallest closed subgroup of that contains the support of .

FURSTENBERG's theorem discloses some details of the asymptotic behaviour
of the norms
of the vectors

(B.1) |

**Theorem.**
If is irreducible and satisfies

then for random matrices which are independently distributed according to , with probability one the limit

(B.3) |

Moreover, if is noncompact, then is strictly positive.

The theorem holds with probability one only, due to the random nature of the . 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
of unimodular matrices with a
distribution on
. First one has to
identify the corresponding matrix group ; then irreducibility and
noncompactness of need to be confirmed.
This, by FURSTENBERG's theorem,
establishes the result
that for almost all such sets
with a
sufficiently well-behaved
measure ,
the norms of the vectors
grow exponentially
for large ,

(B.4) |

is most convenient to work with.

For , the most important example is
itself:

(B.5) |

is obtained, with the dilogarithm function [AS72]. The expression (B.6) is finite for any given nonzero , 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

(B.7) |

For , the theorem is used in
subsection 5.3.3
with respect to
itself:
the transfer matrices
(5.123a) and (5.124a)
belong to the group defined by

(B.8) |