Next: A Picture Book of
Up: m_engel_diss_2ol_l2h
Previous: Typical Applications
Contents
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
,
,
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) |
for
.
The vector norm
used here
is the
conventional
Euclidean 2-norm
and not to be confused with the matrix norm
used
below.
Theorem.
If is irreducible and satisfies
|
(B.2) |
then for random matrices which are independently
distributed according to , with probability one the limit
|
(B.3) |
exists for all
nonzero
vectors
.
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) |
provided
that
the initial vector is
nonzero.
The
rate
of growth is given by the ``LIAPUNOV exponent
of the set
''
and does not depend on .
For checking the requirement (B.2) on ,
any matrix norm
can be used; often the maximum norm
is most convenient to work with.
For , the most important example is
itself:
|
(B.5) |
defines
the appropriate group
containing all the transfer matrices
(5.61)
that model the tight binding equation of the kicked rotor.
also plays the same role with respect to the kicked harmonic
oscillator and its transfer matrices
(5.114, 5.117, 5.120).
Clearly,
is irreducible and noncompact and
thus
for
sufficiently
well-behaved
generically gives rise to exponentially growing
for
.
This observation is used, for example, in
subsection 5.1.3.
There,
for the ANDERSON-LLOYD model,
a measure is used that is generated by the Lorentzian
(5.70) with respect to the transfer matrices
of equation (5.61),
such that
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) |
on the other hand, may serve as a counterexample for .
It is easy to verify that indeed defines a
subgroup
of
.
But is reducible, since
is mapped onto itself by the matrices of . Therefore, the norms
should not
be expected to grow exponentially.
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) |
Again, irreducibility and noncompactness of are obvious, and
FURSTENBERG's theorem applies, once the integrability condition
(B.2) has been verified
-- as in subsection 5.3.3.
Next: A Picture Book of
Up: m_engel_diss_2ol_l2h
Previous: Typical Applications
Contents
Martin Engel 2004-01-01