Journal of Physics A: Mathematical and General
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Rigorous bounds and the replica method for
products of random matrices
To cite this article: J P Bouchard et al 1986 J. Phys. A: Math. Gen. 19 L1145
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J. Phys. A: Math. Gen. 19 (1986) L1145-Ll152. Printed in Great Britain
LElTER TO THE EDITOR
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Rigorous bounds and the replica method for products of
random matrices
J P Bouchaudi, A Georgest, D Hansel§, P Le Doussalt and J M
Maillard /I
t Laboratoire de Spectroscopie Hertzienne de I’Ecole Normale Suptrieure, CNRS, 24 rue
Lhomond, 75231 Paris Cedex, France
$ Laboratoire de Physique Thtorique de I’Ecole Normale Suptrieure, CNRS, associt a
I’Universitt de Paris-Sud, 24 rue Lhomond, 75231 Paris Cedex, France
5 Centre de Physique Thtorique de I’Ecole Polytechnique, Groupe de Recherche du CNRS
no 48, Route de Saclay, 91128 Palaiseau Cedex, France
)I Laboratoire de Physique Thtorique et Hautes Energies, associt au CNRS et I’UniversitC
Pierre et Marie Curie, 4 place Jussieu, 75230 Paris Cedex, France
Received 16 January 1986
Abstract. We first develop a method to obtain rigorous bounds for the Lyapounov exponent
of products of a random matrix. When applied to a class of I D problems, including
localisation, it reproduces the correct scaling behaviour at the band edge and gives very
good approximations of the prefactors. We then study analytically the successive moments
of the distribution law for the trace of the random matrix product within the whole energy
band. The band centre anomaly is found to affect the whole statistics of the problem and
the exact anomalous value of the Lyapounov exponent is recovered through the replica trick.
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A direct approach to random matrix products through the calculation of the successive
moments of their trace allows to obtain three different kinds of information.
( i ) The second moment gives an upper bound on the Lyapounov exponent ( L E )
of the problem.
(ii) Analytical continuation of the moments towards q +. 0, when possible, gives the
value of this L E .
(iii) Finally, the whole set of moments allows us to discuss the statistics of this trace
through its distribution function.
Point (i) (corresponding to an annealed approximation in disordered spin problems)
has been initiated in the review paper of Ishii (1973) but has not been extensively used
since. Moreover, the ‘replica trick’ (point (ii)) has very seldom been used in actual
calculations in the context of random matrices (see Kirkman and Pendry 1984).
The first aim of this letter is to develop point (i) for the study of some products
of random matrices providing both rigorous upper bounds and approximations of the
Lyapounov exponents ( L E ) . As a test of the efficiency of these methods, we have
considered the one-dimensional discretised localisation problem (Halperin 1965, Thouless 1972, 1974, Ishii 1973, Derrida and Gardner 1984). For this problem, we give an
upper bound on the LE which reproduces the exactly known U ’ / ’ scaling behaviour at
band edge.
More generally, we have considered the n x n random matrices associated with the
discretisation of the I D differential equation
where i+h(“) is the nth derivative of I,!I and
0305-4470/86/181145 +08$02.50
E(X)
a centred white noise of variance
@ 1986 The Institute of Physics
U.
L1145
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L1146
Letter to the Editor
We obtain the following bound on the largest
LE
when
(+ +
0:
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This 1/(2n - 1 ) scaling behaviour can also be predicted for instance using a renormalisation argument (Bouchaud and Le Doussal 1986), while the amplitude is unknown
(except for n = 2). A numerical analysis furthermore indicates that the upper bound
(2) reproduces the exact amplitude in the limit n -$CO.
We finally turn to points (ii) and (iii) for the one-dimensional localisation problem
and develop a simple method to calculate the replica moments for low disorder,
allowing us in particular to apply the 'replica trick' even at the anomalous band centre
(Kappus and Wegner 1981).
Let us consider n x n matrices M ( E ) depending on a random parameter E of mean
value p and variance cr. One is interested in the real part of the Lyapounov exponent
of the product n M(EO
N
N+x
2N
M(E,)
n M(E,)
(3)
where the angle bracket denotes an average over the realisations of the noise (the
are independent).
For the example of the discretised I D Schrodinger equation in a random potential,
one has ( n = 2)
Consider now the linear mapping operating in the space of n x n matrices:
(5)
q,:u+q,(u)= M(E,)UM+(E,).
This transformation can be represented by the n2 x n2 matrix q E ,= M ( E , )0M ( E , ) where
0 denotes the usual tensorial product. A can be rewritten in term of q E , :
Using the convexity of the logarithm, one can give an upper bound on A :
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where A,,, is the largest eigenvalue of the averaged n 2 x n 2 matrix ( c p c , ) =
( M ( E ) O M ( E ) ) .(The last equality in ( 7 ) holds provided U,,> has a component in the
eigenspace corresponding to A,,, , which is the generic case.)
The bound (7) thus amounts to replacing the original disordered problem by an
effective pure one: in this sense, it can be called an annealed method. Note however
that the dimension of the corresponding matrices has been increased from n x n to
n 2x n2,
Letter to the Editor
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L1147
For the localisation problem (4), one has to compute the largest eigenvalue of the
following 4 x 4 matrix:
-
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(2-p)2+c
2-P
2-CL
1
p-2
p-2
0
-1
-1
1
0
0
0
0
0-
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0
whose characteristic polynomial is
x[-x’+
( ( r + p 2 - 4 p ) x 2 + ( 3 e + p 2- 4 p ) x + 2 a ] = 0
(9)
where A = 1 + x.
From (9), we obtain the following results.
(i) At band edge ( p = 0 or 4) the LE A ( p = 0, e) is bounded from above by
2-2/3 e1 / 3 - - 0 . 6 3 ~ ” ~to
, be compared with the exact result 0.289.. . e1l3(Derrida and
Gardner 1984).
given by (9)
(ii) In the vicinity of the band edge, the upper bound on
depends, for small U, on the scaling variable
as is the case for the exact result.
This bound is plotted in figure 1 together with the result of Derrida and Gardner. As
expected, these two curves are very close far from the critical value = 0.
(iii) Away from the band edge, (9) leads to
which (see equation (14) below) is just twice the exact value known from weak disorder
expansions (forgetting the values of p where an anomaly arises (Kappus and Wegner
1981): p = 2( 1 +cos 7~ n l m ) ) . (In figure 1, we also give numerical results concerning
the imaginary noise case, for which the bound is surprisingly nearly exact in a wide
range of p / e 2 ” . )
-2
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-1
0
1
- p 102”
Figure 1. The ratio A / u ” ~ as a function of the scaling variable P / u * ’ ~ ,near the band
edge for small U. Full curve: upper bound; broken curve: imaginary noise, dotted curve:
real noise.
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L1148
letter to the Editor
Let us also mention that these bounds can be improved using a method detailed
in another paper on the example of Z D spin systems (Georges er a1 1986). It is based
on the following trick: let f(E ) be an arbitrary positive function of E ; one has
(In Trcp,)S(In l / f ) + I n ( f T r cp,).
One can now optimise the bound by minimising functionally the right-hand side with
respect to f:
At the band edge ( p -f 0) and for U + 0, it can be seen that the minimum is obtained
for
b = 0.665
which leads to the upper bound A G 0.435~"~.
This can also be systematically done in the vicinity of the band edge, leading to a
function of p / a 2 I 3lying halfway between the two curves of figure 1.
Let us now turn to problem described by (1) for arbitrary n, with a zero mean value
for E ( x ) . The discretised version of (1) is (up to irrelevant terms in the weak noise
limit)
VI,,
= M,V,
where M , is the n x n matrix
and VI = (A"-'$,, . . . , A'$,); A' is the finite difference operator and ( E , & , ) = U&,.
The average mapping (cp,,) is now represented by the n 2 x n 2 matrix (M,@ M,)
whose characteristic polynomial can be evaluated using the algebra of the following
n x n matrices:
(Ak)t,
= 8rj-k
1 s i, j , k s n.
The rules of this algebra are given by
&Ap = A k + p
=O
ifk+pGn
if not.
This allows us to compute the matrix ln(l,~-A(cp)) and the characteristic polynomial
is obtained by using the formula det M = exp Tr In M. Finally, it turns out that the
largest root A,,, of this polynomial of degree n2 satisfies the following equation:
The highest pole in (1 - A ) and its residue can easily be extracted from this series,
and this leads to the bound (2) when a+O. The 1 / ( 2 n - l ) exponent is the correct
one, as can be intuitively understood by noting that the tensorial product in (6) acts
as a first step of a block renormalisation analysis.
We have numerically studied the relevance of this bound: the ratio of the simulated
LE on the 'annealed bound' is plotted in figure 2 as a function of n.
Letter to the Editor
L1149
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2
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6
4
0
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Dimension n of t h e matrices
Figure 2. The ratio of the numerically computed LE on the annealed bound (2) as a function
of the size of the matrix n.
This suggests that (2) might be the exact analytical expression of A ( a + O ) in the
would require one to solve a partial
limit n + w . The exact prefactor of
differential equation of n - 1 variables generalising the simple differential equation
(32) of Derrida and Gardner (1984).
The above bound can in fact be seen as the second replica moment of Tr rI M ( .si).
More generally, in order to obtain information on the entire probability distribution
of Tr rI M ( . s i ) ,it is possible to study
N
N
one has to estimate the highest eigenvalues Aq of a 2' x 2' matrix (AI:'), which can
in fact be reduced to a ( q + 1) x ( q + 1) matrix using symmetry properties (Pendry 1982).
Actual calculations can be performed in the weak disorder limit within the
framework of standard perturbation theory. This method has been applied by Kirkman
and Pendry (1984) to study analytically the moments of an analogous quantity, namely
the transmission coefficient, away from the band centre ( p = 2) where the perturbation
theory becomes degenerate and where only a semi-numerical treatment of the diagonalisation problem was performed.
Let us notice that for the moments of Tr M, one has a nice interpretation in terms
of tensorial products of q f spins, which greatly simplifies actual calculations and
allows a complete analytical treatment of the degenerate case (band centre). Denoting
by S the total spin operator, one has, in every eigenspace of S,,
for2cos 8 = 2 - p # 0
and
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L1150
Letter to the Editor
For p # 2, L(q, N ) can be evaluated for any finite N and analytically continued to
any real q. This allows us to discuss the influence of the lattice in the metallic (finite-size)
regime and will be addressed in a subsequent work.
In the thermodynamic limit N + 00, one obtains
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U
L ( q , a)=
max
{ ( S ( S + 1)) - 3 m 2 }= A( q + ; q 2 )
p ( 4 - p ) S = q / 2 , q / 2 - 1 . ...
for q even
(14)
/ml=S
allowing one to analytically continue for any real q the expression of the average
modulus of the trace (In MIq),showing that the distribution of this quantity is purely
log-normal with LE A = u / 2 p ( 4 - p ) . Analogous results have first been obtained by
Melnikov (1981a) in the context of the random phase model (Anderson et a1 1980)
and also by Kirkman and Pendry (1984).
In the interesting band centre case ( p = 2 ) , the spectrum of operator (13) for an
integer q is given by the roots of two sets of polynomials P",X) and P i ( X ) ,corresponding to the representation of total spin S (running from 0 (or f ) to q / 2 , the maximally
symmetrical representation). These P$b obey the same recursion relations with different
initial conditions:
Q F ( X ) = (as,,, - X ) Q F - ' ( X ) - C i + m C ; - m + 2 Q Y 4 ( X )
with
asm= S(S
(15a)
+ 1) - 3 m 2
and, for integer S,
Pg = QZ
QiS-'(x) = 1
Q i S ( X )= as.-s - x
p i = QZ-'
Q i S - ' ( X )= 1
QSS"(X) = uS,-S+~ - X
(156)
while for half-integer S,
pg
=p
i
=
QZ-'
Q S S - ' ( X )= 1
Q i s ( X )= uS,-S - X .
(15~)
The recursion relation naturally closes for q integer and the analytic continuation
simply corresponds to the iteration of ( 1 5 a ) ad injnitum. This introduces spurious
roots which turn out to be always smaller than the natural one.
Taking the thermodynamic limit amounts to selecting the largest root which is then
easy to follow for real q, as seen in figure 3 which represents L(q, a)/qat the band
centre. This figure calls for three remarks.
(a) The q+O limit gives the exact Kappus-Wegner value of the LE as will be
explained below. Thus no difficulty arises when taking the thermodynamic limit before
the q = 0 limit. This is true for all energies within the band, and is due to the fact that
eigenvalues do not accumulate in this limit (see figure 3).
(b) Melnikov (1981b), by studying a simplified version of the model, has obtained
the correct asymptotic ( q + 00) slope of L( q ) / q.
(c) For negative q, L ( q ) / q does not seem to exhibit any peculiarities, such as the
annulation of L ( - 2 ) , as suggested by Melnikov (1981b).
One can obtain the exact value for the L E A by developing the recursion (15a) for
small q. The A term turns out to be the limiting ratio of two sequences satisfying
U,+:! = -12n2u,+, - n ( 2 n - 1 ) 2 ( n - l)u,
(16~)
0.5
.
P
P
0
-0.5
z
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Letter to the Editor
L1151
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2
0
4
6
Figure 3. L ( q ) / q for the band centre case. Plain lines: the roots of polynomials Po and
Pb as functions of q. The value of L ( q ) / q is the largest root for a given q (note the
occurrence of spurious roots). Chain curve: Melnikov's estimation; dotted line: continuation to the case p = 2 of the result valid only for p # 2 (without anomaly) for the modulus
of the trace.
This ratio converges extremely rapidly towards the exact value A = 0,1142(Derrida
and Gardner 1984). Furthermore, a generating function treatment of (16a)allows us
to exactly obtain
A = -111-R
8 12-R
with
and
I(a, b, e ) =
Id
zyx
-;)-" dt.
tb-'(l - f)'-.-'( 1
The complete distribution law for the band centre, which is no longer log-normal,
can, in principle, be reconstructed from the knowledge of L( q ) in the thermodynamic
limit.
In the case p = 0 (band edge), no perturbation theory has yet been applied. We
have nonetheless shown that for q integer 2 2 , the u1I3behaviour always holds for
weak disorder. The q = 2 and 4 replica moments yield
A - 1 2'13 421/3
-
4=-
qa'/'
2 ' 4
and a linear extrapolation gives A = 0 . 3 9 ~ ~ while
' / ~ the exact value is 0 . 2 8 9 ~ ' / ~If.
however one includes the third moment ( q = 3) (Tr(II:, M i ) 3 ) ,one obtains the intermediate value (12)'13/3 and obtains A =0.2830'/~, again very close to the true value.
L1152
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Letter to the Editor
Let us summarise the results contained in this letter. They include
(i) a systematic way of obtaining rigorous upper bounds on the LE of products of
random matrices,
(ii) an upper bound on the LE of the first problem (i), leading to a conjecture on
its exact behaviour in the limit n +a,and
(iii) a study of the moments of Tr II M in the infinite-size limit showing that the
statistics at the band centre is far more complicated than the log-normal distribution
holding within the band.
Similar considerations have led us to improvements of the annealed approaches
for ZD disordered spin systems (Georges et a1 1986), which provide approximations
of the critical lines in good agreement with numerical results. Let us emphasise that
these methods apply beyond the weak disorder limit and can also be used in the case
of complex random matrices.
The evolution of the statistics with respect to the size of the sample will be the
object of a subsequent publication.
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We thank B Derrida and R Rammal for their encouragment.
References
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Anderson P W, Thouless D J, Abrahams E and Fisher D 1980 Phys. Rev. B 22 350
Bouchaud J P and Le Doussal P 1986 J. Phys. A: Marh. Gen. 19 797
Derrida B and Gardner E 1984 J. Physique 45 1283
Georges A, Hansel D, Le Doussal P, Maillard J M and Bouchard J P 1986 J. Physique 47 947
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