variety of systems, and a very few of them will deviate much from a properly chosen
ensemble average.
One of the surprising aspects of this "new kind of statistical mechanics" is that its
utility is far reaching beyond its origination. Since its introduction to the many-body
complex nuclei systems, RMT has been applied to a wide variety of systems in diverse
areas, including many-body atoms and molecules, quantum chaos, mesoscopic
disordered conductor, 2-D quantum gravity, conformal field theory, chiral phase
transitions as well as zeros of Riemann zeta function, scale-free networks, biological
networks, communication systems and financial markets [6-18]. This broad range of
applicability of RMT in seemingly unrelated areas highlights the universal features of
the correlations of the eigenvalues in RMT. Within the classical or Gaussian model
pioneered by Wigner, these correlations are known as the Wigner-Dyson (WD) statistics
of the Gaussian ensembles, which are qualitatively different from the statistical features
of completely uncorrelated eigenvalues given by the Poisson statistics.
1.2 Complex Systems with Power-law Distributions
In the past decades, there have been growing interest of RMT applications to
generic complex systems that appear in the study of natural, social, economical
and biological systems. In these systems, the microscopic interactions among the
constituents are so complicated that the statistical hypothesis made in RMT seems
relevant. Namely, we are forced to "renounce exact knowledge not of the state of the
system but of the system itself." Thus, it appears to be appropriate to apply the idea of
RMT to understand certain statistical features of these systems.
One such application of RMT is found in the analysis of the empirical covariance
matrices in a multivariate setting such as financial assets and climate systems [11, 12,
19] based on classical RM models. In these works, the application aims at decomposing
genuine correlations, present in the eigenvalues and eigenvectors of the empirical
covariance matrices from pure noise components, the RMT predictions. For example,
005 I
Gausslan ---
X=I 0 -
0045 9 09 .-
045' *X=08 ---
X=07 ----
004
0 035
003
^ 0025
002
0015 /
001 i
0005
0
1 15 2 25 3 35 4 45 5
s
Figure 3-10. Normal component of the cluster function for A-ensembles for A = 0.9, 0.7,
0.5 as well as 1.0 (7 = 0.50).
breaking and thus, indicative of the non-trivial character of the eigenvector correlations,
namely the multi-fractal dimensionality as can be seen immediately from the Eq. 3-32
and the A-dependent sum-rule deficiency. In this regard, the study of the dimensional
dependence of the critical statistics will be important to further understand the role of
the parameter A since the multi-fractal dimension of the eigenvector correlations at the
critical states is dependent on the spacial dimension [63].
3.2.3.2 Two-level Kernel of A-ensembles
It is known from rigorous results that for the soft-confinment potential of the critical
ensembles, the translational-invariance is broken and the density does not depend
on N. Thus the unfolding procedure is non-trivial [54, 64, 65]. In fact, the deformation
from the Gaussian universality, the 'sine kernel', is the result of the non-trivial unfolding
[54, 64, 65] In the semi-classical regime (7 < 27 for the critical ensembles), the kernel
for an arbitrary weight function can be written as [65-67]
sin(w(u v))
x(u, ) () () (3-34)
x(u) y(v)
and two-level cluster function
T2(x, y) = -R2(x, y) + R(x, x) R(y, y) (2-31)
= KN(X, y)2 (2-32)
N -1 x ( ) (2 3 3 )
2.2.4 Unfolding
Within the invariant class ensembles, the one-level correlation function (eigenvalue
density) aN(x) = KN(x, x) is not universal and it is dependent on the specific form of the
probability measure dP(H). So in the limit of large matrix size, e.g., N oo, we need to
introduce a proper scaling variable to study the universality of correlation function such
that in the scaled variable the mean spacing of eigenvalues become unity. It is achieved
by defining the unfolding variable
du lim KNv(x,x)dx. (2-34)
N->oo
The unfolded two-level kernel can be written in terms of the unfolding variables
((u, v) = lim KN() (2-35)
N- VKAN(X,x)KN(y, y)
where u and v are the unfolding variables. Now all the correlation functions and the
cluster functions can be written in terms of this unfolded kernel. For example, unfolded
one-level cluster functions Y(u) = 1 and the two level cluster function
Y2(u, v) = K(u, v)2 (2-36)
Thus, the study of correlation functions requires understanding of two level kernel in the
properly scaled variable which requires the knowledge of asymptotic behavior of the
orthogonal polynomials.
y=0.5 A
y=1.0 o
1.5 -
0.5
0
1-]
-0.5
-1 I I I I
0 0.5 1 1.5 2
Figure 3-2. The exponent a(A) as a function of A for two different values of 7 In(l/q).
A = 1 corresponds to the q-polynomials describing the critical ensembles.
classical orthogonal polynomials whose weight function is given by w(x) = e-x' with
m > 0 where Rn oc n.
3.2.2 Eigenvalue Density
The density of the eigenvalues p(x) = KN(x, x) can now be obtained for different
values of A from Eq. 2-20 by summing the products numerically. The results are shown
in Figure 3-3. Earlier it was understood in Ref. [55] that the density of the A-ensembles
is given by a pure power-law, e.g., p(x) = -i For A = 1, 0 = 0 and for A > 1 and A < 1,
0 > 0 and 0 < 0.
However, careful investigation shows that the earlier interpretation is only approximate
and more accurate form of the eigenvalue density is given by
[In x]^-1
p(x) oc I for x > A (3-20)
X
where the lower cutoff A is dependent on the regularization of the confining potential
V(x). i.e., for the choice of V(x) [In x]2, the singularity, albeit unphysical, can be
extended in the vicinity of x 0 limit [60]. For our choice Eq. 3-1, such singularity
the probability measure can be written down as
P(H)dH oc e-TrV(H)dH. (2-9)
It is also possible to consider RMEs with non-inavarant form of probability measure
under transformation, which is the so-called "Wigner class" ensembles [35]. In this class,
each matrix element of the ensemble of N x N matrix are randomly taken from i.i.d.
probability distributions so that the generic form of the probability measure of Wigner
ensembles can be given by
P(M)dM oc f(Mi) H dMi (2-10)
ij ij
where f(x) is the probability distribution of each individual matrix element and the
product n[i is performed for all the degrees of the freedom of the ensemble under
consideration. Some of the examples are:
* Wigner Guassian ensembles : Hermitian matrices whose entries are given by i.i.d.
Gaussian distributions.
Levy matrices (Wigner Levy matrices) : Real symmetric matrices whose entries
are given by i.i.d. Levy distributions.
Banded matrices : Symmetric matrices whose entries m, are non-zero for i-jl < r
and zero otherwise where r determines band width.
Adjacency or Laplacian matrices of random graphs or scale-free networks.
In the following, we will mainly focus on the study of the invariant class ensembles
and its universality.
2.2 Orthogonal Polynomial Method
The orthogonal polynomial method has been a fruitful tool in the development of
RMT. The merit of the orthogonal polynomial method is that it allows to write down
all the correlation functions of the eigenvalues in terms of orthogonal polynomials
corresponding to the probability measure characterizing a RME. In particular, the
two-level correlation function written in a properly scaled variable in the asymptotic large
CHAPTER 1
MOTIVATION
1.1 Random Matrix Theory and Complex Systems
Random matrix theory (RMT) deals with eigenvalue and eigenvector correlations
of random matrix ensembles drawn in a stochastic manner. The earlier application
of RMT [1, 2] in physics dates back to the 1950s when Wigner introduced RMT
to explain the fluctuation properties of heavy nuclei energy spectrum. In a slow
nuclear reaction, incident nucleon interacts with the constituents of target nucleus in
a complicated manner such that extra energy carried in by the an incident nucleon is
shared non-trivially with the nucleons of the target nucleus while forming the excited
states. However, the fluctuation properties of the excitation energy levels of the
compound nucleus are not very well understood [3]. To explain the statistical nature
of the energy spectrum of the compound nucleus, Wigner suggested to consider
ensembles of Hamiltonians whose entries are randomly drawn from the Gaussian
distribution. It turns out that the statistical properties of the eigenvalue correlations of
such random Hamiltonian ensembles, e.g. the Gaussian ensembles, showed a good
agreement with the statistical behaviors of the eigenvalue spectrum obtained from
various heavy nuclei reaction experiments [3, 4]. The underlying idea of RMT is well
explained by Dyson [5]:
What is required is a new kind of statistical mechanics, in which we renounce
exact knowledge not of the state of the system but of the system itself. We picture a
complex nucleus as a black box in which a large number of particles are interacting
according to unknown laws. As in the orthodox statistical mechanics we shall consider
an ensemble of Hamiltonians, each of which could describe a different nucleus. There is
a strong logical expectation, though no rigorous mathematical proof, that an ensemble
average will correctly describe the behavior of one particular system which is under
observation. The expectation is strong, because the system might be one of a huge
To my family
CHAPTER 2
RANDOM MATRIX THEORY
2.1 Random Matrix Ensembles
The first step to construct a RME is to consider a set of N x N Hermitian matrices H
with the following probability measure
PN(H)dH oc e-TrV(H)dH, (2-1)
where V(H) is a suitably increasing function such that the probability measure is
normalizable. Tr is the matrix trace and dH the invariant measure. Especially, the
TrV(H) ensures the rotational invariance under orthogonal, unitary and symplectic
transformations. More explicitly, it means that under a transformation R,
H -- M = RtHR, RtR = 1 (2-2)
the probability measure remains the same
PN (H)dH = PN(M)dM. (2-3)
Thus, by construction, all orientations of the eigenbasis are equally likely or in other
words, there is no preferential basis. Each of the symmetries determines the structure of
the hamiltonian systems, e.g.,
* Orthogonal symmetry (time-reversal invariant systems with rotational symmetry)
Real symmetric matrices
Unitary symmetry (systems in which time-reversal symmetry is broken)
Complex Hermitian matrices
Symplectic symmetry (time-reversal invariant systems with half-integer spin and
broken rotational symmetry) Self-dual quaternion matrices.
In order to find a specific realization of a RME, one needs to specify the probability
measure V(H) of a RME. One way to find V(H) comes from the maximization of the
entropy or the minimization of information content of the RME. Suppose that nothing is
0 045 / =1 J (Tit)
!i\,, 1= 1 2 (fit)
', 1=1 1 (fit)
004 1=1 3
A=I 2 -----
=X 1 -
0 035
003 -
P 0025
002 '
0015 \
0 01 -
0005 ...
1 15 2 25 3 35 4 45 5
s
Figure 3-15. Fitting results for normal component of the cluster function for A-ensembles
for A = 1.1, 1.2, 1.3 as well as 1.0 ( = 1.50).
0.99 -
0.98 -
0.97 -
0.96 -
0.95
S 0.94
X=1.0 -
0.93 k =0.9 (fit)
X=0.8 (fit)
0.92 =0.7 (fit)
X=0.9 ...............
0.91 X=0.8 ------
X=0.7 ---.---
0.9 I I I 1
-9 -8.5 -8 -7.5 -7 -6.5 -6 -5.5 -5
v
Figure 3-16. Fitting results for anomalous component of the cluster function for
A-ensembles for A =0.9, 0.8, 07 as well as 1.0 (7 = 0.50).
10
=0.6 ----
k=1.0
S=, X=1.8 -
0.1
0.01 "
.o 0.001 -
0.0001
le-05
1e-06
1 10 100 1000 10000 100000 le+06
log x
Figure 3-3. Density of eigenvalues for different values of A.
doesn't exist. In general, it is expected that the regularization behavior (the cut-off A) is
different for the choice of V(x) sharing the same asymptotic logarithmic behavior.
The following facts convince us that the spectral density should be given by Eq
3-20. First of all, it can be shown [61] that for A > 1, the density should be of the form
3-20 based on the mean-field approach, which reproduces the exact inverse power-law
density as well as the sinh kernel of the critical ensembles (A = 1.) Second, the validity
of this form for all A > 0 can be checked by considering the normalization condition of
the spectral density
2 p(x)dx= N. (3-21)
Here the factor 2 comes from the fact that p(x) is symmetric around origin. The upper
bound DN is given by the largest zero of the orthogonal polynomials of order N, namely
DN oc vRN. As pointed out, R oc exp[nl^]. We notice that Ref. [62] studied the largest
zeros of the orthogonal polynomials to the weight function of exp[-c(Inx)"] for c > 0
and m a positive even integer and reported that it is of order exp(nm1), which is the
exact same behavior as the coefficient Rn of recurrence relation of the generalized
3-17 Fitting results for anomalous component of the cluster function for A-ensembles
for A =1.1, 1.2, 1.3 as well as 1.0 (7 = 1.50). .... 59
3-18 Number variance for A-ensembles for A =0.5, 0.8, 1,0, 1.5, 2.0 (7 = 0.5). 61
It is expected that the asymptotic tail of the spacing distribution is also given by a similar
exponential form.
Lastly, we show that the number variance is linear in L for large L. The slope of
number variance or the level compressibility is dependent on the parameter A. As A
decreases, the slope increases, which is consistent with that fact that anomaly or the
deficiency of the sum rule increases as A decrease.
These results seem to imply that the A-ensembles in general are relevant to
the description of the critical states of the localization-delocalization transition in the
disordered systems. Since the critical level statistics are universal, depending only on
the critical exponent and the dimensionality of space for a given symmetry class, it is
conceivable that the parameter A can be associated with these parameters. In particular,
the study of the dimensional dependence of the critical statistics is interesting in this
respect. At the same time, it is also interesting to see if A-ensembles are also applicable
in the study of quantum chaos. We note that recently the critical statistics has been
found relevant in some cases of quantum chaos as well [68, 69]. It turns out that the two
level kernel of chaotic systems with logarithmic singularity [70] have the exact same form
as that of the critical ensembles.
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In terms of the difference variable s = u v, we define the two-level correlation function
for the A-ensembles as
R (s) 6(s) YA(s) (3-57)
For s < a, RX(s) ~ s2 for all values of A, which is an expected feature due to the unitary
symmetry of the ensembles. For large s > a, the results are simply obtained from
Eq. 3-53
R'(s, a) oc e-s for A 0 (3-58)
oc e- forA= 1 (3-59)
oc s-2 for A -oo (3-60)
We can see that the asymptotic behavior of the two-level correlation function interpolates
that of Gaussian ensembles for A oo limit, that of critical ensembles A = 1 and the free
Levy matrices for A -- 0 limit.
3.2.4 Number Variance
As an example of the level statistics obtained from the proposed form of the
two-level kernel Eq. 3-50, we show the evaluation of the number variance E(L) within
a range L shown in Figure 3-18. It clearly shows that the number variance is linear in L
the for all values of A with a fixed 7 = 0.5. As A become smaller, the slope of the number
variance increases. This is consistent with the fact that deficiency of sum rule rule X(A)
increases as A decreases since the slope of the number variance is directly related to
the deficiency of the sum rule by X(A) = dZ))
dKL)
3.3 Discussion
In this work, we further study an invariant-class of random matrix ensembles
characterized by logarithmic soft-confinement potentials introduced in Ref. ??
which we refer to as the A-ensembles. As a first step, we carefully reinvestigate the
spectral density of the A-ensembles and show that the spectral density is given more
2010 Jinmyung Choi
After substituting f(b) and making a change of variable, 2= 2 -dependence can
be taken out from the integral. Thus we have,
8 a2N2 A p (B-8)
6Y2(p, X) lim 22 /) ) (B-10)
N-oo a2 p2X2 F(A) aN
where
(; p, = d e- 2 in [ os- (1 2)] sin [ cos-1 E 1 2 )]
(B-11)
In the limit N oo, ~ N, with other variables order of 1, the argument of exponential
in Eq. B- 1 can be made very small. So the leading order of the integral Eq. B- 1 goes
as 1. With this in mind, putting A =1 into Eq. B-10, we can see that 6Y2(p, X) ~ N.
Next, we calculate the Yo in a similar way. After a little calculation, we get
Y2O(p, a) lim 22 2 (N;pp, E) (B-1A2)
IM _) ( P /2 (B-1 2)
N-poo a2 p2 X F(A) aN2
where
,1 2-le_ 2 Sin[( )]2 [sin N(u )]2
/2( p, A) <= L .2) 2)1/2 N X(p-x)2 (B-13)
Here u and v are defined as
u= -. cos-1 (B-14)
22 X X
and
v= V- 2-. cos-1 (B-15)
When N(D <) < 1, Eq. B-13 becomes N-independent so Yo goes as N with A =
and p N-1 otherwise, Yo goes to zero.
From above, it is shown that unfolded two-level cluster function carries non-trivial
N dependence although one-level correlation function can be made to have power-law
asymptotic behavior.
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0.99 / \ .
0.98
0.97
U 0.96
0.95
0.94
X=1.0
0.93 -=1. 1 .........
X=1.3 -------
k=1.3 --------
0.92 I I I I
-8 -7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4
v
Figure 3-13. Anomalous component of the cluster function for A-ensembles for A =1.1,
1.2, 1.3 as well as 1.0 (7 = 1.50).
Thus we expect that in the same limit the difference between the A-ensembles and the
critical ensembles also arises from the difference in unfolding. To implement this, we
define the unfolding variable for the critical ensembles in the following form,
X c
u' u- U -dt = Inx/xo (3-35)
In the similar manner, we can define the unfolding variable for A-ensembles for A / 1,
u'= u U dt = ([In x]l [In xo]A) (3-36)
t
where u0 = jox p(x)dx and xo >> A, the cut off of the density that arises as a result of
the regularization close to the origin. The constant c oc p(x = 0). Note that p(x = 0) is a
function of the parameters that do not depend on N. For the critical ensembles, p(0) = 1
and for the A-ensembles, p(O) = f(A, 7) where f is some function of A and 7. Rewriting
the original variable x in terms of the unfolding variable, we obtain for both ensembles
x = exp [7*(u + a)l' ] (3-37)
Note that if the kernel satisfy Eq. 2-23 or in order words, the kernel is reproducing,
the unfolded two-level kernel satisfies the normalization of sum rule.
I dvY(u, v)
K K(x, y)K
lim j KN(y, y) K(x,y)2 dy
N-rn KN(x, x)KN(y, y)
lim Kx,y)Ky,x)dyy
N-oo KN (x, x)
n KN (x, x)
limN =K 1.
N-oo KN(x, x)
2.3 Gaussian Ensembles
2.3.1 Semi-Circle law
For the Gaussian ensembles V(x)
well-known semi-circle law.
o-(x) = KN(x, x)=
x2, the eigenvalue density is given by the
i(x) (x) M /2N x2
(2-40)
where the wave function ,',,(x) = hn(x)e-x2/2 satisfies
0o
,',(x), (y)e- 2dx = vF22nn!nm
(2-41)
where hn(x) is the Hermite polynomial of degree n. The derivation of semi-circle law can
be given in the following way. First of all, we recognize that Eq. 2-19, the determinant
form of the JPDF, can be considered as the probability density of the ground state of the
many-body wave function of non-interacting fermions, e.g.,
P(xi, ..., XN) ox I| (x1, ..., xN) 2
where
bo (X2)
b l(Xl)
(2-42)
Y" (X1, ..., X )
(2-37)
(2-38)
(2-39)
bo (xi)
bi(Xi)
... ^o(XV)
... bl (Xl)
(2-43)
the Hamiltonian H({xi}) [53] in the following form.
1
P({x,}) = Z exp[-PH({x,})], (2-63)
where
H({x}) = In Ix, -x + V(xi). (2-64)
In this view point, the eigenvalues x, can be considered fictitious particles interacting
with each other through logarithmic repulsion at temperature 1/3 while they are confined
by the external potential V(x). In particular, one can assume that the particle density
p(x) = "N 6(x x) reaches a continuum in the large asymptotic system size N oo
limit. In this limit, the Hamiltonian H({xi}) can be read as an functional of the density.
H[p] = I j dxdyp(x)p(y) In x y + dxp(x) V(x) (2-65)
where J denotes the support of the density. Using the saddle point approximation, we
can obtain an integral equation for the average particle (mean-field) density of (p(x)) 2
I dx(p(x)) In |y x = V(y) + const. (2-67)
where the additional constant term can be determined by the normalization condition of
the density fp(x)dx = N. The two level correlation function can be written in terms of
the functional derivative of (p(x)) with respect to V(x),
2
(p(x)e- V(y)Dp
(p(x)) = fx)e- Dp (2-66)
fJ e-1H[p] Dp
where 7* (')1/ and the a [Inxo]^. In the limit A = 1, it reduces to
x = xoe .
(3-38)
Thus for the critical ensembles we find the expected form of the kernel by using Eq.
3-34
1
x y -sinh[ (u v)].
2c
(3-39)
For A # 1, the same procedure results in
x y sinh[ [(u' + a) (v' a)' ]].
2
(3-40)
Thus if we redefine the unfolding variables by u u = u' + a and u v = v'+ a, we
obtain a suggested form of the two-level kernel for the A-ensembles
r sin[7(D v)]
Kn(,, 2 ) = sin- )] (3-41)
2x sinh[ -(ul/^ ,1/,)]'
where F is introduced to ensure that the kernel satisfy the condition o(u) =
lim, (u =) = 1. To identify the necessary form of F, we consider the kernel in
the limit of v u
lim K (u, v)
v->
F -v
7* l/1A 1/A
from which we obtain
F = F(u, v) 7y*g(D, v)
(3-43)
Here, we introduce a function g(b, v)
D -/A I 1/X
g(u, v)
u-v
(3-44)
Then, we finally obtain the proper form of the regular component of the two-level kernel
(uv > 0) for the A-ensembles as
(3-45)
(3-42)
S(uD, sin[7r(u i)]
Sn( 2 --
27 sinh[l ( t)]
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
RANDOM MATRIX ENSEMBLES WITH SOFT-CONFINEMENT POTENTIAL
By
Jinmyung Choi
August 2010
Chair: Khandker Muttalib
Major: Physics
In this work, we study invariant-class of random matrix ensembles characterized
by the asymptotic logarithmic soft-confinement potential V(H) ~ [In H](1+') (A > 0),
named "A-ensembles". The suggestion is inspired by the existing random matrix
models such as the critical ensembles (A=1), the free Levy matrices (A 0 limit)
and the Gaussian ensembles (A oo limit) in an effort to investigate the novel
universality associated with the fat-tail random matrix ensembles as well as the
logarithmic soft-confinement potential within the framework of rotationally invariant
random matrix theory. First of all, we show that the orthogonal polynomials with respect
to the weight function exp[-(In x)1+'] belong to a novel orthogonal polynomial system,
named "A-generalization of q-polynomials". Second, we show that based on numerical
construction of the "A-generalization of q-polynomials", we can study the one-level and
the two-level correlation functions as well as the level statistics of the A-ensembles.
Third, we show that the one-level correlation (eigenvalue density) has a power-law
form p(x) oc [In x]-I1/x and the unfolded two-level correlation function possesses
the normal/anomalous structure, characteristic of the critical ensembles. We further
show that the anomalous part, so-called "ghost-correlation peak" is controlled by
the parameter A; decreasing A increases the anomaly. Third, we also identify the
two-level kernel of the A-ensembles in the semi-classical regime, which can be written
in a sinh-kernel form with more general argument that reduces to that of the critical
LIST OF FIGURES
Figure
3-1 Log Rn as a function of n for different values of A. Solid line corresponds to
the critical ensemble A = 1 ........................
3-2 The exponent a(A) as a function of A for two different values of 7 In(1/q).
A = 1 corresponds to the q-polynomials describing the critical ensembles. .
3-3 Density of eigenvalues for different values of A. ..............
3-4 Eigenvalue density (x
= 0.50), 0.5 (7 = 0.25)
3-5 Eigenvalue density (x
= 0.50), 0.5 (7 = 0.25)
3-6 Eigenvalue density (x
= 2.00), 1.5 (7 = 4.00)
3-7 Eigenvalue density (x
= 2.00), 1.5 (7 = 4.00)
3-8 Eigenvalue density for
(7 = 0.25) as well as 1
3-9 Eigenvalue density for
= 0 104) for A-ensembles for A
as well as 1 (7 = 0.75). .....
= 0 40) for A-ensembles for A
as well as 1 (7 = 0.75) .....
= 0 104) for A-ensembles for A
as well as 1 (7 = 0.75). .....
= 0 40) for A-ensembles for A
as well as 1 (7 = 0.75).
A-ensembles for A = 0
(7 = 0.75). ........
A-ensembles for A = 1
(7 = 4.00) as well as 1 (7 = 0.75) .. .....
3-10 Normal component of the cluster function for
0.5 as well as 1.0 (7 = 0.50) .......
3-11 Normal component of the cluster function for
1.3 as well as 1.0 (7 = 1.50) .......
3-12 Anomalous component of the cluster function
0.8, 07 as well as 1.0 (7 = 0.50) .....
0.9 (7 = 0.75), 0.7 (7
0.9 (7 = 0.75), 0.7 (7
1.1 (7 = 0.75), 1.3 (7
1.1 (7 = 0.75), 1.3 (7
.9 (7 = 0.75), 0.7 ( =
.1 (7 = 0.75), 1.3 ( =
A-ensembles for A = C
A-ensembles for A = 1
for A-ensembles for A
3-13 Anomalous component of the cluster function for A-ensembles for A
0.50), 0.5
2.00), 1.5
).9, 0.7,
.1, 1.2,
=0.9,
=1.1,
1.2, 1.3 as well as 1.0 (7 = 1.50). ...........................
3-14 Fitting results for normal component of the cluster function for A-ensembles
for A = 0.9, 0.7, 0.5 as well as 1.0 (7 = 0.50) ................
3-15 Fitting results for normal component of the cluster function for A-ensembles
for A = 1.1, 1.2, 1.3 as well as 1.0 (7 = 1.50) ................
3-16 Fitting results for anomalous component of the cluster function for A-ensembles
for A =0.9, 0.8, 07 as well as 1.0 (7 = 0.50). ................... .
= (
: t
RANDOM MATRIX ENSEMBLES WITH SOFT-CONFINEMENT POTENTIAL
By
JINMYUNG CHOI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2010
Then, the Rn can be determined in terms of Gn by [58]
GnGon-2
Gn Gn, 2(3-11)
'n-1
An alternative view of the above procedure is adopted in Ref. [59] to obtain the
polynomials recursively. Following the Ref. [59], we define
Qn,m xmbn(x)w(x)dx (3-12)
--00
Using the fact that x" = Qn(x) + j aj@(x), we can find
Qn, n= hn, (3-13)
n-1
Qn,n+l = hn S,, (3-14)
J=0
and
Qn,m = Qn-l,m+l Sn-lQn-l,m Rn-lQn-2,m. (3-15)
In this case, the determination of Rn and Sn in order to calculate the polynomials of
degree n < N 1 requires only the knowledge of the 2N 1+ integrals of Qo,m (or the
moments)
Qo,m = xmw(x)dx (3-16)
--00
for m = 0,.... 2N.
3.2 Results
3.2.1 A-generalization of q-polynomials
For the study of the A-ensembles, we adopted the latter approach to construct the
orthogonal polynomials. So first, we calculated Qo,m with Mathmatica
Qom xme-v(x)dx, (3-17)
--00
with the V(x) shown in the Eq. 3-1. Since we choose V(x) to be symmetric around
the origin, e.g., V(-x) = V(x), the Sn = 0 for all n. Rn alone defines the corresponding
of the A-ensembles. For our study, we particularly choose the following form of the
potential,
1
V(x) = [sinh- x] +; A >0; q < 1. (3-1)
In(1/q)
For simplicity, we will introduce 7 In(1/q) so that the model has two parameter A
and 7. The merit of choosing the Eq. 3-1 is that in the limit A = 1, it coincides with
the one possible form of the weight function of the q-RMEs so that we can compare
our results with those of q-RMEs. For the q-RMEs, the mathematical properties of the
corresponding orthogonal polynomials, known as "the Ismail-Masson q-polynomials"
[52], are well established and accordingly, the two-level kernel (sinh kernel) and the all
the spectral properties (the critical statistics) are very well known [48, 49].
In the following sections, we will show that for A 4 1, we can construct the
corresponding orthogonal polynomials, "A-generalization of q-polynomials" in a rigorous
numerical method and thus we can study the one-level (spectral density) and the
two-level kernel that is central to the test of the universality. We will also show that
the spectral density and the two-level correlation function of the A ensembles exhibit
novel feature; the spectral density is given by a power-law form and two-level kernel
possess the anomalous component which is considered one of the characteristics of
the critical ensembles. In addition, we will suggest a novel form of two-level kernel of the
A-ensembles based on the unfolding analysis and discuss the details of its behavior as
well as the properties of the level statistics that can be deduced from the kernel. Finally,
we will discuss the implications and applications of the our results.
3.1 Method
The main difficulty of studying A-ensembles is that the orthogonal polynomials
corresponding to the weight function w(x) = e-v(x) for the arbitrary A values (expect
A = 1) are not known. Thus, the first task is to obtain the orthogonal polynomials
corresponding to the weight function of the A-ensembles. In the following, we will
N limit is central to establishing the universality of RMEs. Thus, in the following, we
will review the orthogonal polynomial method and the results obtained for classical and
Gaussian RMEs and the critical ensembles.
2.2.1 Joint Probability Distribution Function (JPDF)
In order to study the eigenvalue correlations, the probability measure P(H)dH
needs to be transformed from the matrix element basis to the eigenvalue/eigenvector
basis,
H RtXR with RtR = 1. (2-11)
Here, the matrix X is the diagonal matrix containing eigenvalues ({x,}, i = 1 ..., N)
and the matrix R is the rotational transformation containing eigenvectors. After the
transformation is considered, the invariant measure dH in the matrix element basis can
be rewritten in the eigenvalue/eigenvector basis, e.g.,
dH = J(X, R)dXdR; i = 1... N (2-12)
where J(X, R) is the Jacobian of the transform. The dX and dR refer to the measure
associated with the eigenvalues and parameters of the rotational transformations
respectively. Here, the Jacobian factor can be further calculated for a given symmetry
class and it can be shown that [4]
J(X,R) = ]x xj,- f(R) (2-13)
i
where the 3 is the symmetry parameter of the rotational transformations such as
Orthogonal (3 = 1), Unitary (3 = 2) and Symplectic (3 = 4) symmetries. The f(R) is
the function of the parameters associated with the rotational transformations. Once the
parameters associated with transformation is integrated out, we obtain the JPDF of N
eigenvalues in the following form.
PN({x,})dX oc I Ix,- X w({x,})dx ... dxN (2-14)
i
known about the detailed dynamics of a system other than the fundamental symmetries,
which gives no knowledge about the matrix elements of the hamiltonian systems other
than the global symmetries. The information (I) contained in the N x N Hamiltonian
matrix H can be defined as
I dHP(H)InP(H), (2-4)
with the condition that the probability is normalized to 1,
dHP(H) = 1. (2-5)
requiring the variance of the matrix elements be finite
I dHTr(H)P(H) < o. (2-6)
The information in Eq. 2-4 can be minimized subject to the constraints in Eq. 2-5 and
Eq. 2-6 by using Lagrange multiplier method. For an arbitrary variation 6P(H) of P(H),
61 = dHP(H) { In P(H) ATr(H2)P(H)} = 0 (2-7)
Thus,
P(H)dH oc e-T'(H2)dH. (2-8)
This is the probability measure of the well-known Gaussian ensembles. The Gaussian
ensembles can also be obtained based on the two assumptions; invariance under
transformation (Orthogonal, Unitary, Symplectic) and statistical independence of matrix
elements [4]. The three different symmetries define three different classes of Gaussian
ensembles: Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE),
and Gaussian Symplectic Ensemble (GSE). If we only require the rotational invariance
of P(H)dH, dropping the assumption of the statistical independence of the matrix
elements, then P(H) may be any function of the traces of powers of H. Thus, in general
005
Gausslan --
X= 1 0 -
0045 / X =O 9 (fit)
X=O 8 (fit)
X=O 7 (fit)
004 X1= 9 -
)=O 8 ------
S=O 7 --- -
0035
003
S0025
002
0015
001 / \
0005 \ ,
1 15 2 25 3 35 4 45
s
Figure 3-14. Fitting results for normal component of the cluster function for A-ensembles
for A = 0.9, 0.7, 0.5 as well as 1.0 (y = 0.50).
be regularized near the origin. We also show that in the similar range of the parameters,
the anomalous components of the kernel fits well with the proposed form of the kernel
Eq. 3-46 which are shown in the figures 3-16 and 3-17.
3.2.3.3 Universality of A-ensembles
In the following, we will discuss novel asymptotic behavior of the two-level kernel of
the A-ensembles by further examining Eq. 3-50. First, we note that
F(s, a) oc sl- for s > a (3-51)
leads to
K,(s, a) oc s 1-'e-S'/ for s > a. (3-52)
Looking at the A-dependence, we observe that
Kn(s,a) oc e-s for A O
oc e- for A = 1
o s-1 for A -oo. (3-53)
Refs. [11, 12] study the empirical cross-correlation matrices of the stock price returns
of S&P 500 and the largest 1000 US companies for certain period time such that the
elements of such matrices represent the correlations among the companies in a given
period, e.g.,
S- (r(t)r(t)). (1-1)
Here < ... > denotes time average over the period studied and r,(t) represents a
normalized time series of stock price return of a company i defined as
( R(t) (R,(t)) (1-2)
r,(t) (1-2)
where the volatility (R,(t)2) (Ri(t))(R(t)) and R,(t) represents the logarithm of
the price change,
S,(t A t) S,(t)
Ri(t) Int A) t + t) InS(t) (1-3)
si(t)
Here, Si(t) is the time series of the stock price. Thus, the covariance matrix C, is by
construction a real symmetric matrix. The statistical properties of the covariance matrix
C. can be compared to the well known results of appropriate classical RM models, e.g.
Gaussian orthogonal ensembles (GOE) or Wishart matrices (Laguerre ensembles).
The study shows that most of the eigenvalues except some outliers are well within the
RMT expectations in terms of the eigenvalue density, the nearest spacing distribution,
the number variance and the inverse participation ratio, implying that the empirical
covariance matrices are dominated by random Gaussian noise for the most part. The
small fraction of the outliers carries some meaningful informations useful for the risk
management.
This explanation, however, is not fully satisfactory because the a priori assumption
that cross-correlations present in the stock price returns are dominated by the Gaussian
noise is not in accordance with the observation that the distribution of stock price returns
follows an asymptotic fat-tail distribution, the so called inverse-cubic law [20]. To be
Since the wave function in the determinant is the solution of one-dimensional harmonic
oscillator, it satisfies the Schrodinger equation Thus,
( d2 1
-h -dX2 + '.,(x) = (n + 1/2)h,.,(x). (2-44)
In the large n = N 1 limit, the Fermi momentum can be read off locally at x
PF = /2N- 1 -x2. (2-45)
The density of one dimensional fermions is related to the Fermi momentum PF by
1 Pp PF
S(x) ldp=- h=l. (2-46)
27hx) J p2 7-
Combining Eq. 2-45 and Eq. 2-46, we get
o(x) v 2NV -x2. (2-47)
7-
2.3.2 Sine Kernel
In order to calculate the unfolded two level kernel, we consider the spectrum around
the origin in the large N limit, so-called double scaling limit (x 0 and N oo limit)
such that xv/N is finite. The proper unfolded variable in this limit du = KN(x, x)dx =
d2Ndx. Using the definition of the kernel,
N-1
KN(u, v)= lim (- u)i( ). (2-48)
Recalling the Christoffel-Darboux formula,
i(x) (y) = (2-49)
2 x y
= 0
-(u, ) = l im (2-50)
N ooV2 u V
Since the determinant does not change by the adding to any one row or column to
multiples of the other rows or columns, the matrix Q can be rewritten as
po(x) V (x
P1(xi) w(x1)
po(x2) wX2)
p1(x2) w(X2)
(... N)p (X)
.. N(XN) WXN)
PN 1(X1)wI V X1) PN-1(X2)V ) ...w PN-1(XN)VW(XN)
where pi(x) denotes an arbitrary polynomial of degree i. And
matrix multiplication QTQ, we can obtain
1) =
PN({xi}) = Idet
N!
KN(xl, x1)
KN (2, x 1)
KN (xN, x1)
KN(X1, x2)
KN (X2, x2)
KN (XN, X2)
after considering the
... KN(x, xN)
... KN(x2, xN)
... KN(xN, xN)
where the two-level kernel KN(x, y) is defined as
N-1
KN(X, y) ,i(x)i(y),
i=o
where the "wave functions" /bi(x) pi(x) (x = pi(x)e-v(x)2
Here the choice pi(x) is arbitrary but if pi(x) is chosen a system of the polynomials
that are orthogonal with respect to the weight function w(x) = ev(x), i.e.,
Pn(x)pm(X)W(x)dx = 6mn.
The kernel KN(x, y) satisfies following important properties.
SKN(x, x)dx = N
and
KN(x, z) = K(X, y)KN(y, z)dy.
(2-18)
(2-19)
(2-20)
(2-21)
(2-22)
(2-23)
ensembles, the non-extensive q ensembles require the following condition
f dH trH2 [P(H)] 2 (1-11)
f dH [P(H)]
with a constant a. Using the Lagrange multiplier method, it can be shown that the P(H)
that maximizes the non-extensive entropy Sq subject to the above constraints is given by
P(H) ~ expq(-AtrH2) (1-12)
where the q-exponential function exp,(x) is defined as
exp,(x) {[1 (1 q)x]+}, (1-13)
with
[...]+ = max(..., 0). (1-14)
Note that for q=1, the exp,(x) reduces to the usual exponential function thus
leading to the usual Gaussian ensembles. The study shows that the non-extensive q
ensembles exhibit characteristically different behavior depending on the parameter q;
for q > 1, the distributions of eigenvalue density show true long tails and for q < 1,
the distributions have compact support. However, the variability of the parameter q in
this case depends on the dimensionality N of the ensemble such that in the large N
limit where universal behavior is expected, the maximum q allowed for the nonextensive
ensembles approaches unity.
1.3.4 Ensembles Based on Superpositions of Classical or Gaussian Ensembles
There are several RM models categorized under this class of ensembles [24-
28]. The main idea is to construct generalized ensembles with the superposition (or
deformation) of the Gaussian or Wishart ensembles. To illustrate the details of the idea,
the approach in Ref. [24] will be discussed in the following.
The underlying idea of the approach in the Ref. [24] is to work with an associated
characteristic function, rather than a probability distribution, to discover a novel random
3. The numerically-obtained nearest neighbor spacing distribution is found to be
non-universal 1, which depends on the location of the eigenvalue spectrum
considered.
1.3.2 Free Levy Matrices
The free Levy matrices, introduced by Z. Burda et al. [22], are constructed based
on the theory of free random variable (FRV) that offers the probability theory of
non-commuting variables as a generalization of the classical probability theory. The
correspondence between the classical probability theory and the FRV theory is made in
Ref. [33]:
1. Probability distribution P(x) : Spectral density p(x) = -ImG(x + iO)/7
2. Characteristic function P(k) : Green function G(z) -< Tr(z ') >
3. Logarithm of characteristic function In P(k) : R-transform R(G(z)) z -
In the classical probability theory, the logarithm of characteristic function is
additive (the additivity of the cumulants) under convolution operation of two random
variables. Analogously, in the FRV theory, the R-transform of two independent random
matrix-valued numbers, M, and M2, is additive, i.e., RMI+M2(Z) = RMI(z) + RM2(Z)
[22, 33]. The free Levy random matrix approach takes advantage of the fact that one
can reconstruct the Green function, and the spectral density, and finally identify a
particular probability measure that characterizes a random matrix ensemble from the
known R-transforms that correspond to all free stable probability distributions under FRV
calculus: e.g., R(z) = a + bza- where 0 < a < 2 (ca 4 1)2 which determines the
asymptotic power-law behavior of the stable distributions. The parameters a and b are
1 It should be emphasized that the numerical unfolding procedure is non-trivial in
this model so that it requires further investigation regarding the universality of the
spacing distribution. In fact, our numerical investigation shows possible universality of
the spacing distribution.
2 For a = 1, R(z) = a i'(1 +) -2 In yz.
: = I.U -. .
i I I = 0 .9 ................
X=0.7 -
x=0.5 -------
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Figure 3-4.
Eigenvalue density (x = 0 104) for A-ensembles for A
(7 = 0.50), 0.5 (7 = 0.25) as well as 1 (7 = 0.75).
0.9 (7 = 0.75), 0.7
0 5 10 15 20 25 30 35 40
Figure 3-5. Eigenvalue density (x
= 0.50), 0.5 (7 = 0.25)
= 0 40) for A-ensembles for A = 0.9 (' = 0.75), 0.7 (7
as well as 1 (7 = 0.75)
BIOGRAPHICAL SKETCH
Jinmyung Choi was born in Pusan, South Korea in November 1973. He received his
Ph.D. from University of Florida in the summer of 2010.
is made strong enough, the eigenstates become localized and thus the eigenvalues
become uncorrelated. Especially at the delocalization-localization transition, it has
been established that the correlations of the eigenvectors exhibit novel features [8,
36-42] such as multi-fractality and the correlations of the eigenvalues lead to a level
compressibility that is intermediate between WD and Poisson statistics. Similarly in
the studies of quantum chaos, energy level statistics of systems that are intermediate
between chaotic and regular states also require generalization beyond WD and Poisson
statistics [3, 43-47].
In these contexts, extensive studies have been carried out to construct a parametric
generalization of RM models that cross over from WD to Poisson [3, 8, 36-47] as a
function of the parameter. Some of these generalizations indeed capture the essential
features of the critical statistics, among which the family of q-RMEs [48-50] provides
a particularly valuable insight. Within the common framework of rotationally invariant
RM models [4] the q-RMEs show how the universality of the Gaussian ensemble
characterized by the zero parameter two-level sine kernel breaks down and eventually
gives rise to a different kind of universality for the critical ensembles, characterized by a
one-parameter two-level sinh kernel. In particular, the rotationally invariant RM models
are characterized by a "confining potential" which defines the weight function of a set
of orthogonal polynomials; the key difference between the Gaussian and the critical
ensembles comes from the fact that the corresponding orthogonal polynomials, namely
classical vs. q-orthogonal polynomials, respectively, possess qualitatively different
asymptotic properties [51, 52].
2.4.2 Critical Statistics
The conjecture of the novel universality at the Anderson transition is based on the
studies of the spacing distribution at the critical point of the tight binding hamiltonian on
q-polyonomials in Eq. 3-18. Thus, our results seem to imply that the results of Ref. [62]
can be extended to an arbitrary real A > 0.
Eq. 3-20 can be verified in the numerical calculation as well. However, since our
V(x) is regularized near the origin unlike the choice V(x) = [In x]1+' used in the
mean-field approach [60] and in Ref. [62], we expect that the exact form of p(x) will
show agreement only for large x limit. To investigate this, we consider that the density is
given by
p(x) = (3-22)
x+A
where f(x; A) is a logarithmically slowly varying function and A is a constant arising due
to regularization of the density at the origin. Thus, we can expect xp(x) to behave in the
following way.
X
xp(x) = f(x; A), (3-23)
x+A
X
f(x; A) x>A, (3-24)
A
Sf(x; A) x
Fig. 3-4 and Fig. 3-5 show this behavior for A < 1 (A = 0.5, 0.7, 0.9, and 1) and the Fig.
3-6 and Fig. 3-7 for A > 1 (A = 1.1, 1.3, 1.5, andl) respectively.
For all the cases, we chose y = 0(1), which ensures the cut-off A = 0(1). To
further investigate if f(x; A) oc [In x]A-1 for large x (x > 1), we plotted In[xp(x)] vs. In In x
and fitted it in the range of 10 < x < 104. Fig 3-8 and Fig. 3-9 show the expected linear
behavior.
3.2.3 Two-level Correlation Function
The numerical calculations of the cluster function are performed based on
Y(u, v) [K(u, v)]2 (3-26)
18
16
14
-" 12 .
1 2 -
08
----- --- ----- -- --
-- -- --- -"" ~
06 -=0 5 ......- 5
:=0 8 ---
Critical
04 X- 1 5 .
X=2 0 -----
Gaussian --
02 I I
0 5 10 15 20 25 30 35 40
L
Figure 3-18. Number variance for A-ensembles for A =0.5, 0.8, 1,0, 1.5, 2.0 (7 = 0.5).
accurately by a power-law of the form p(x) o [In x]A-'/x. This result is suggested by the
mean-filed approach and it can be checked by the normalization condition of the spectral
density, the edge of which is determined by the coefficient Rn of the A-generalization of
q-polynomials.
Second, we show that the two-level kernel of the A-ensembles has normal/anomalous
structure, which is characteristic of the critical ensembles. The anomalous component
arising due to the sum rule violation is dependent on the parameter A for a fixed value
of 7 ~ 0(1); As the value of the A decreases, the deficiency of the sum rule becomes
larger.
Third, we identify the normal and anomalous components of the two-level kernel
in the semi-classical regime, which are given by Eq. 3-45 and Eq. 3-46 that reduce to
those of the critical ensembles for A = 1. Further, we show that the two-level kernel
of the A-ensembles exhibit novel universal asymptotic behavior, shown by Eqs. 3-53,
which includes the Gaussian ensembles (A oo limit), the critical ensembles (A = 1)
as well as the free Levy matrices (A -+ 0 limit). In particular, the large s behavior of the
two-level kernel is governed by exp[-s1/], which is a novel feature of the A-ensembles.
size) limit but it is not clear if there exists a similar universal two-level kernel for the
power-law or the fat-tail ensembles as well.
1.3 Fat-tail RM Models
1.3.1 Levy Matrices
The Levy matrices [21] are introduced by P. Cizeau and J.P. Bouchaud in consideration
of constructing a RM model based on the Levy probability distribution. In particular, they
considered a Nx N real symmetric matrix ensemble of independent indentically-distributed
(i.i.d.) random numbers of asymptotic Levy distributions given by the following:
P(H) H" O <<2 (1-4)
IHU.I1+p ,
where Hy denotes the matrix elements. By construction, H,- = Hji and Ho is set to be
N11/ in order to ensure that the typical largest element of a row is 0(1). Since the matrix
elements are drawn from the distributions with divergent moments, there is no direct
way to construct an analytical model within the standard RMT techniques. Thus, they
developed a novel analytical technique based on the cavity method [21] to evaluate
the eigenvalue density and the inverse participation ratio in parallel with numerical
simulations. The main results of their work are summarized in the following:
1. The eigenvalue density converges to a limiting form of distribution in the large N
limit that has asymptotic power-law distribution. e.g. p(x oo) ~ which has
the same scaling power as that of the matrix elements.
2. From careful examinations of the inverse participation ratio, it is shown that the
eigenstates undergo a non-trivial localization-delocalization transition within
the eigenvalue spectrum at a certain critical value of x = x depending on
the parameter p: for p > 1, there exists a critical value x, that differentiate the
extended states (x < x) from the algebraically localized states (x > xc), which
is an unusual type of localization since this type of localization allows non-zero
conductivity. For p < 1, all the states are localized (finite participation ratio) except
ground states. The states below x, are localized in a usual sense. However, the
finite fraction of the states above x, are still extended over O(N) site. In the limit
p -- 1 or 2, the x, diverges.
CHAPTER 3
A-ENSEMBLES
In the motivation to investigate the universality associated with fat-tail or power-law
RMEs, we introduced a family of U(N) invariant random matrix ensembles characterized
by an asymptotic logarithmic potential V(H) = A[In H]1+X with A > 0 [55, 56],
named "A-ensembles" 1 The reason for such suggestion is based on the following
few observations. First, it is known that for V(H) oc [In H]2 corresponding to A = 1
limit (the critical ensemble), the eigenvalue spectrum is given by inverse power-law
distribution, which is known by the mean-field theoretic approach [57]. Second, for
V(H) N In H corresponding to the A -+ 0 limit with the constant A being order of N
(free Levy matrices), the spectral density is given by the fat-tail distributions. Third, in the
limit A > 1, it is expected that the confinement potential may grow sufficiently strong,
thereby approaching the Gaussian limit. Therefore, we can speculate that the parameter
A is a controlling parameter of the power-law behavior. The fact that such parametric
generalization (generalizing the power of the logarithm to arbitrary real value larger than
1) connects the existing RM models equipped with rotational invariance is interesting
since the model allows us to explore any possible novel universality associated with
fat-tail RMEs as well as the logarithmic soft-confinement potential within the framework
of the rotationally invariant RMT.
The generic choice of the confining potential V(x) that gives asymptotic logarithmic
behavior is V(x) = A[Inx]1+'. However, it has a unphysical singularity at the origin
so that we need to regularize it in certain way. One possible way to do it is choosing,
i.e., V(x) = A[In(l + x)]'+' but there are a variety of other forms that differ by the
regularization behavior in the vicinity of origin, which will not change the characteristics
1 The earlier name 'Levy like ensembles' in Ref. [55] reflects the motivation of the
study.
0.99/ \C /
0.98
0.97
.96
S 0.95
0.94 =1.0 -
X=1.1 (fit)
0.93 l=1.2 (fit)
k=1.3 (fit)
= 1.1 ..............
0.92 =1.2 ------
k=1.3 --------
0.91 I I I
-8 -7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4
v
Figure 3-17. Fitting results for anomalous component of the cluster function for
A-ensembles for A =1.1, 1.2, 1.3 as well as 1.0 (7 = 1.50).
Thus in the limit A -- oo, we get back the well-known asymptotic sine kernel decay. This
is consistent with the fact that the magnitude of the ghost correlation peaks become
smaller as A increases, presumably disappearing in the Gaussian-ensemble limit of
very large A. For A = 1, the kernel shows the expected exponential decay of the critical
ensembles. For A -+ 0 limit, the asymptotic tail is given by the infinitely fast exponential
decay, leading to the uncorrelated Poisson-like behavior. In general, for all values of A,
the large s behavior is governed by an exponential decay of e-s"'
In the same token, we can study the two-level correlation function or density-density
correlation function defined by
R(x, y) (p(x)p(y)) 1 (3-54)
(p(x))(p(y))
6(x y) (p(x)p(y))
= Y+ -1 (3-55)
(p(x)) (p(x)) (p(y)) xy
In the unfolding scale p(u) = p(x)dx where p(u) = 1, we can rewrite
R(u, v) = 6(u v) (p(u)p(v)),}v 1 = 6(u v) Y(u, v) (3-56)
precise, it is shown in Ref. [20] that the cumulative distribution of the stock price returns
has an asymptotic fat-tail distribution, e.g. P(ri(t) > r) ~ r-a with a 3. Thus,
the application of the classical Gaussian RM models requires reconsideration in this
example and more generally, in the complex systems where the fluctuation is better
characterized by fat-tail noises. This situation motivates search for suitable RM models
that can incorporate the fat-tail noises.
In this context, many attempts have been made to construct generalized random
matrix ensembles that incorporate power-law or fat-tail distributions [21-28]. The
significance of such generalization beyond the Gaussian ensembles is mainly two
fold. First, there are numerous complex systems that exhibit fat-tail noises, notably,
financial markets, earthquakes, scale-free networks etc. [29-32]. As pointed out earlier,
the relevance of the classical RM models in these systems seems questionable. It
is because the classical or Gaussian RM models are based on the assumption that
the systems are characterized by the Gaussian noise, which is not appropriate for
systems with fat-tail noise where the occurrence of extreme events are not as rare
as expected from normal distributions. Second, it is conceived that the universality
of classical or Gaussian RM models is closely linked to the prevalence of normal or
Gaussian distributions in nature, a consequence of the central limit theorem [21]. It has,
however, not been fully investigated if there is a counterpart of the Gaussian ensemble
as implied by the structure of the generalized central limit theorem [21], comparing e.g,
the Gaussian and the Levy basins (refer to APPENDIX A for more details).
So far, fat-tail distributions in random matrix ensembles have been carefully
incorporated in some limited cases and the calculation of the correlation functions of
the eigenvalues have been carried out for certain special cases [21-28]. However,
the question regarding the universality of the correlations of the eigenvalues remains
unresolved. For Gaussian ensembles, it is the well known two-level sine kernel that
establishes the universality of the correlations in the properly scaled large N (matrix
TABLE OF CONTENTS
ACKNOWLEDGMENTS ..........................
LIST O F FIG URES .. .. .. .. .. .. ..
ABSTRACT ................... .........
CHAPTER
1 M OTIVATIO N . .
1.1 Random Matrix Theory and Complex Systems .....
1.2 Complex Systems with Power-law Distributions .....
1.3 Fat-tail RM Models .....................
1.3.1 Levy Matrices ..................
1.3.2 Free Levy Matrices .................
1.3.3 Nonextensive q Ensembles .. ..........
1.3.4 Ensembles Based on Superpositions of Classical
Ensem bles . .
2 RANDOM MATRIX THEORY .. ................
2.1 Random Matrix Ensembles .. ..............
2.2 Orthogonal Polynomial Method .......... ....
2.2.1 Joint Probability Distribution Function (JPDF) ..
2.2.2 Determinant Form of JPDF and Kernel ......
2.2.3 Correlation Functions and Cluster Functions ...
2.2.4 Unfolding . .
2.3 Gaussian Ensembles .. ................
2.3.1 Sem i-Circle law ..................
2.3.2 Sine Kernel . .
2.3.3 Gap Probability and Spacing Distribution .....
2.3.4 Number Variance ..................
2.4 Critical Ensem bles .. ..................
2.4.1 Anderson Transition and Critical Ensembles .
2.4.2 Critical Statistics . .
2.5 Coulomb Gas Analogy ...................
3 A-ENSEMBLES ............... ............
3.1 M ethod . .
3.2 R results . .
3.2.1 A-generalization of q-polynomials .........
3.2.2 Eigenvalue Density .................
3.2.3 Two-level Correlation Function ............
page
. 4
or Gaussian
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
In[In(x)]
Figure 3-8.
Eigenvalue density for A-ensembles for A
(7 = 0.25) as well as 1 (7 = 0.75).
0.9 (7 = 0.75), 0.7 (7 = 0.50), 0.5
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
In[In(x)]
Figure 3-9.
Eigenvalue density for
(7 = 4.00) as well as 1
A-ensembles for A
(7 = 0.75).
= 1.1 (7 = 0.75), 1.3 (7 = 2.00), 1.5
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real valued and are associated with the shift, the slope (a) and the skewness (3) and
the range (7) of the stable distributions. Note that R(z) is given without any particular
realization of matrix ensembles.
In order to find a particular realization of a random matrix ensemble characterized
by the probability measure of the form
e-NTr V(M) dM (1-5)
where Tr is the trace and V(M) is specific to the realization of the ensembles and is
related to the confining potential in the Coulomb gas analogy of the classical RM models
developed by Dyson [34], Z. Burda et al. [22] first calculated the spectral densities 3
from the stable R-transforms for some exactly solvable cases of parameters. They
showed that the asymptotic form of spectral densities displays power-law distributions
with the exponents within the Levy stability regime. They then showed that in all such
cases, the asymptotic form of the confining potential V(x) is given as V(x) = In x2 +
O(1/x) by using the relation between the potential and the spectral density arising in the
Coulomb gas analogy.
After the identification of the confining potential and thus the matrix ensemble (free
Levy matrices), they further studied the eigenvalue correlations based on a standard
RMT technique, namely the orthogonal polynomial method. This model successfully
incorporates the asymptotic Levy distributions and also allows the calculations of
eigenvalue correlations within the standard RMT techniques due to the rotational
invariance of the probability measure of the ensemble. It, however, suggests that the
3 To give a quick illustration of the method, consider the trivial choice of R(z) = a,
then Green function G(z) = a and thus the spectral density p(x) = 6(x a). Refer to
Ref [22] for more details.
1e+25 I I I I
k=0.6 ------- /
k=0.8 --------
le+20 =1.0
= 1 .2 ................
k=1.8 ----- /
le+15
c) le+10
100000
1 e -0 5 '*------------------------'
le-05
5 10 15 20 25
n
Figure 3-1. Log Rn as a function of n for different values of A. Solid line corresponds to
the critical ensemble A = 1.
polynomials. As described earlier, we can determine Rn with 2N + 1 integrals of Qo,m
recursively.
It turns out that the Rn obtained in this way shows an intriguing behavior that is
depicted in Fig 3-1. After careful examination, we found that the behavior of Rn for large
n should be of the form
Rn oc e (3-18)
where
a(A) (3-19)
The fitting result for a(A) is provided in Fig. 3-2. As A = 1, the a(A) is equal to 1 and
thus In Rn grows linear in n, which coincides with the well-known Rn behavior for the
q-polynomials (see Eq. 3-6) and for A # 1, In Rn grows in n1'.
We recognized that this is a novel behavior thus named the orthogonal polynomials
of the A-ensembles as A-generalizaton of q-polynomials, which is one of the central
results of our work. Note that this is dramatically different from that for all Freud-like
where n,
function. For Gaussian ensembles, w(x) = exp(- x2). For more general potential V(x),
w(x) = exp(- V(x)).
2.2.2 Determinant Form of JPDF and Kernel
The form of JPDF obtained in Eq. 2-14 can be greatly simplified by using matrix
determinant. In particular, the unitary class (3 = 2) is the simplest and thus, in the
following we will focus on the unitary case of JPDF. To this end, we first observe that the
Vandermonde determinant n,< Ix j = det[x'-l]. More explicitly,
1 1 ... 1
X1N-1 X2 ... XN-1
i
N-l xN- N-1
After absorbing the weight function into the determinant above, JPDF can be written
down as
PN({x,}) = det[QTQ] (2-16)
ZN
where ZN is a normalization constant and the matrix Q is defined as
1 1w(xw(2) ... W) 1W(X)
W(X1) X2 (2 ... XNW(X) (2-17)
xN-1 /wMw( x'\ N-1 w( N-1 w/w
N2 ViWX2) ... XN / W(XN)
two-level correlations of the ensembles have a non-trivial N-dependence that cannot be
simply scaled out.
1.3.3 Nonextensive q Ensembles
One way of constructing the classical RM models is to maximize the Shannon
entropy S
S = dHP(H)In P(H), (1-6)
subject to the condition that the probability is normalized to 1,
I dHP(H) = 1. (1-7)
Here dH is the measure associated with the matrix elements of H. In particular,
requiring that the variance of the matrix elements be finite, (equivalently assuming
that extremely large matrix elements are improbable)
J dHTr(H2)P(H) < o, (1-8)
it can be shown by using the Lagrange multiplier (A) method that the P(H) that
maximizes the entropy S subject to the the above conditions is given by
P(H)dH e-TrH2 dH. (1-9)
Thus, the maximization of the entropy with the finite-variance constraint on P(H) leads
to the Gaussian ensembles.
In a similar manner, the non-extensive q ensembles [23] can be constructed from
maximizing the non-extensive entropy Sq
S[P(H)] f dH[P(H) (1-10)
q-1
with the usual normalization condition, f dH P(H) = 1 and q is a parameter. Similar
to the constraint that requires the finite variance of P(H) leading to the Gaussian
APPENDIX B
N-DEPENDENCE PROBLEM IN SPREAD FUNCTION APPROACH
The "Spread function approach" is one of the generalization strategies of the
Gaussian unitary ensemble [24]. The underlying idea of this approach is to work with
associated characteristic function, not directly probability distribution itself, to discover
a novel random matrix ensemble. In this framework that the general n-point correlation
function are given by Eq. 1-15. Here we will show how the non-trivial N dependence
arises when the spectral density designed to give a fat-tail distribution. As was observed
in our numerical study 1 we need to search for the novel universality in x oo and
N oo limit where the fat-tail spectral density is expected to occur. We find [72] that
there is an asymptotic expression for harmonic oscillator function V(x) in the limit where
x 0- o and N o so u = finite, e.g., x = 2Ncos .
21/4 N-1/4 \ N 1\ 3r 11
\VN(x) 2- ""n sin + ')(sin 2 2) + + )} (B-1)
\vin (2 4 4 N
Using the Christoffel-Darboux formula, we can write down the two-level kernel.
1
(x- y)K(x, y) = H(x, y) (B-2)
27r sin 0sin "
where x = v/2Ncos y = v/2cos and u(0) = sin(20) 20 and.
H(x,y) = 2sin (u() u()) sin (u()- u())
2 cos N(u() + u())] sin [(u() -u())
1 We investigated Levy matrices discussed in Chapter 1 numerically. It turns out that
the density oN ~ N for large x. So it is expected that in a double scaling limit, e.g.,
x o and N oo such that u -= finite, the density can be unfolded -(u) = 1 and a
novel universality may show up in terms of this variable u.
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statistical physics to risk management (Cambridge University Press, 2003).
[72] M. E. H. Ismail, private communication (2006).
3.2.3.1 Observation of Normal/Anomalous Structure of Two-level
Kernel . .. .. 50
3.2.3.2 Two-level Kernel of A-ensembles .... 52
3.2.3.3 Universality of A-ensembles .... 57
3.2.4 Number Variance ...... ..... ........ ....... .. 60
3.3 D discussion . .. 60
4 CO NC LUSIO N . . 63
APPENDIX
A CENTRAL LIMITTHEOREM .......... ...... ........... .. 65
B N-DEPENDENCE PROBLEM IN SPREAD FUNCTION APPROACH 68
R EFER EN C ES . . 71
BIOGRAPHICAL SKETCH ................... ............. 75
The significance of Rn is that it determines the upper bound of spectral density
DN oc n/R and thus scaling behavior of the bulk of the spectrum in the large N limit.
For example, for the Gaussian ensembles characterized by V(x) = x2 (Hermite
polynomials), the upper bound of the spectral density is /N in the large N limit.
Thus, the normalization condition of the spectral density requires the bulk of the
spectrum to grow at an order of /N as in the semi-circle law. While for the logarithmic
soft-confinement potential, the spectral edge grows at an exponential rate e"nl' (which
will be shown in the next section), the bulk of the spectrum does not scale as N. i.e.,
for V(x) oc [In x]2, the spectral edge grows at ve-, the bulk of the spectrum is given by
p(x) oc 1/x which do not depend on N.
One way to determine the orthogonal polynomials for an arbitrary weight function is
to use the Gram-Schmidt determinant formula, e.g.,
a0 ,1 ... an
a,1 2 ... an+1
1
n (X) (3-8)
Gn-1
an-1 an ... a2n-1
1 X ... Xn
where the Gn stands for the Gram-Schmidt determinant.
a0 a1 ... an
a1 a2 ... ,:n
Gn = a a2 an+ (3-9)
an an ... a2n
and a, are the moments given by
a, = xw(x)dx (3-10)
J/
ensembles for A = 1. Forth, we show that the number variance is linear in L for all A and
the slope (the level compressibility) is increasing as A decreases, which is consistent
with the A-dependence of sum rule violation 0 < X(A) < 1. Finally, we will discuss
the novel universality of the A-ensembles, which interpolates the Gaussian ensembles
(A oo limit), the critical ensembles (A = 1), the free Levy matrices (A -+ 0 limit).
It turns out that for A 4 1, A-ensembles also possess such normal/anomalous structure.
Figures 3-10 and 3-11 show the comparison between the normal part of cluster function
for A 4 1 and that for A = 1 as well as that for the Gaussian. While the nodes of the
cluster function remains the same (occurring at the integer value on the horizontal axis)
for all A values as well as the Gaussian case, the peak height and position between the
nodes show an interesting behavior in the change of A; as the A value is decreased for a
given 7 value (7 = 0.5 in Fig. 3-10 and 7 = 2 in Fig. 3-11 ), the peak height and position
gradually reduces and shifts toward the node on the left. The corresponding behavior is
observed in the anomalous part as well. The figures 3-12 and 3-13 show the numerical
evaluation of 1- Ya(u, v) (u > 0 and v < 0) for a symmetric range around v = -u for
varying A values for a fixed 7 (7 = 0.5 in Fig. 3-12 and 7 = 2 in Fig. 3-13). As the figure
shows clearly, the magnitude of the ghost peak depends on A in a significant way; for
A < 1, the peak is more pronounced than that for A = 1 and for A > 1 it is the opposite.
The observation that such long range correlation leading to the ghost peak is
preserved for all A 4 1 seems to suggest that such features, to some degree, are
common to all logarithmic confinement potentials. In other words, once the critical
ensembles break the U(N) symmetry of the Gaussian ensembles with the introduction
of the parameter q, the A ensembles remain in this broken symmetry family. The fact
that as A becomes large, the ghost peak shrinks seem to imply that the U(N) symmetry
might become fully restored in the limit of A oc. This expectation seems consistent
with the asymptotic behavior of two-level correlation in the limit of A oo that will be
shown later.
The A-ensembles are all "critical" in the sense that the two-level kernel violates
the sum rule that can be associated with the characteristics of the critical statistics
such as the level compressibility and the multi-fractality. In particular, the fact that
the violation of the sum rule is controlled by the parameter A, e.g., 0 < x(A) < 1 is
intriguing. As mentioned above, A seems related to the degree of the U(N) symmetry
ACKNOWLEDGMENTS
First of all, I would like to thank my advisor, Khandker Muttalib for all the help. I
would also like to thank him for all the valuable discussions, which led me to understand
various aspects of random matrix theory and to develop my interests around the
subject. Second, I would like to thank my family and friends for their constant love and
encouragement. And lastly, I would like to thank Department of Physics at University of
Florida for offering me the valuable teaching experiences.
a cubic lattice (L x L x L) given by
H= Z ,c c, c c, (2-61)
where c] and c, are the creation and annihilation operators at site i, andj denotes
the neighbor sites of i thusj = 1,..., 6 on the cubic lattice; ci is the random energy of
the site / in unit of overlap energy of neighboring sites that is uniformly distributed in
the range [- W/2, W/2]. In particular, Ref. [36] showed that the spacing distribution
Pc(s) is scale-invariant (does not depend on the system size L) at the transition point
W = We by investigating the quantity 7(W, L) 1; the sclae-invariant Pc(s) is
characterized by a linear slope in s for s < 1 and an exponential decay for s > 1. Thus
the novel form of P(s) is hybrid of the WD statistics and the Poission statistics.
Another characteristic of the critical statistics is the finite "level compressibility"
or sub-Poissinan number variance, namely ZC(s) = Xs with 0 < X < 1, which is
intermediate between the WD statistics EG(s) oc Ins and Poisson one EP(s) oc s. The
origin of this behavior is known to be the sum rule violation. e.g.,
d(E(s)) L/2 00
X = d(s lim R(s)ds =1 Y(s) 0. (2-62)
dS L-oo jL12 -O
which is also related to the multi-fractal nature of the wave function correlation at the
critical point [8, 36-42].
2.5 Coulomb Gas Analogy
Dyson suggested that the JPDF of the invariant ensembles can be written as a form
of Gibbs distribution for a classical one-dimensional system of N particles described by
1 where A f2 P(s)ds and lower limit of the integral 26 refers to the crossing point of
P,(s) and Pw(s) occurring at 2 6 2.002
review the procedure to construct orthogonal polynomials for non-trivial arbitrary weight
function.
To this end, we define the orthogonal polynomials O(x) for arbitrary weight function
w(x), given by
dxw(x)(Ox)n(xm) = 6n,mhn (3-2)
where hn is the normalization constant. It is well known that every orthogonal system of
real valued polynomials satisfy a three term recurrence relation [51] 2
X n(X) = 0n +(x) + SnOn(X) + n R-l_(x), (3-3)
where S, and Rn are the real coefficients of the recurrence relation. In particular, Rn is
related to the normalization constant hn by hn+ = Rn+ hn For example, the Hermite
polynomials are determined by the recurrence relation of
,n+1(x) = xn(X) nn-1(x). (3-4)
The recurrence relation of the q-polynomials is given by
1
0n+ 1(xq) = X n(Xlq) -q- (1 qn)~nl(xlq), 0 < q < 1 (3-5)
where q = e-7, 7 > 0. We point out that for these cases, the weight function is an even
function, namely V(-x) = V(x) and thus, all Sn = 0. Therefore, the Rn determines
the properties of the orthogonal polynomials. In particular, by comparing the Hermite
polynomials and q-polynomials, we observe that
Rn oc n (Hermite polynomials) (3-6)
oc e"n (q-polynomials). (3-7)
2 Here, we consider that the On(x) is a monic polynomial.
CHAPTER 4
CONCLUSION
In this work, we study invariant-class of random matrix ensembles characterized
by the asymptotic logarithmic soft-confinement potentials, named A-ensembles. The
suggestion is inspired by the existing RM models such as the critical ensembles (A=1),
the free levy matrices (A -+ 0 limit) and the Gaussian ensembles (A oo limit) in an
effort to investigate the novel universality associated with the fat-tail RMEs within the
rotationally invariant RMT framework. The following is the summary of the main findings:
* The polynomials that are orthogonal with respect to the arbitrary power of the
asymptotic logarithmic potential belong to a novel class of orthogonal polynomial
system, named "A-generalization of q-Hermite polynomials".
The one-level correlation (the spectral density) of the A-ensembles are given by a
power-law form.
The unfolded two-level correlation function shows the normal/anomalous structure,
which is the characteristic of the critical ensembles. In particular, the sum rule
violation is controlled by the parameter A for a fixed value of 7 0(1). As the
value of A decreases, the sum rule violation becomes greater.
The asymptotic tail of the density-density correlations show a novel form shown
in Eq. 3-58; for 1 < A < oo, the asymptotic tail interpolates between the critical
ensembles (A = 1), the Gaussian ensembles (A oo), and the uncorrelated
Poisson-like statistics (A -+ 0).
These results seem to have interesting applications/implications:
* It is of immediate interest whether these results are relevant in the context of the
Anderson transition in disordered systems as well as in a broader context of the
localization-delocation problems.
It would also be interesting to see if the these results are applicable in the
mixed states of the quantum chaos systems where the spectral properties are
intermediate between WD and Poisson statistics.
These results strongly impliy that the non-trivial N-dependence of two-level kernel
of the fat-tail RMEs is a generic feature within the framework of the rotationally
invariant RMT. In the A -- 0 limit of the A-ensemble, 7 is required to be a
N-dependent parameter to have the probability measure to be normalizable,
which is the case for the free-levy matrices. Thus, the N-dependence of the
two-level kernel in this limit can be understood as a consequence of the presence
005 I
Gausslan ---
0045 =2 ---- -
i,. = 1 0-
004
0035
003
S0025
002
0015 -
001 ,
0005 .
0
1 15 2 25 3 35 4 45 5
s
Figure 3-11. Normal component of the cluster function for A-ensembles for A = 1.1, 1.2,
1.3 as well as 1.0 ( = 1.50).
I -' I lI
0.97 -
0.96 -
0.95 \ /
0.94
X=1.0
0.93 k=0.9 ................
X=0.7 -------
k=0.7 --------
0.92 I I I1
-9 -8.5 -8 -7.5 -7 -6.5 -6 -5.5 -5
v
Figure 3-12. Anomalous component of the cluster function for A-ensembles for A =0.9,
0.8, 07 as well as 1.0 (7 = 0.50).
From the kernel, density of states reads directly as o-N(x) = limyx K1(x, y).
(B-3)
For the density Eq. B-3, it turns out that choosing the spread function f(b)
(b-(~ +l)e-p/b give rise to a fat-tail density, e.g.,
,-N(x) ~- F 2 dM2e-- sin +1/cos- -
aJxN-V 2N(8N) \e-asi cos-f1 12V'e _2]
F (A) \x J [ J L J
(B-4)
where = finite in the limit x oo, N o and b ~ N and a So if one
V8Nb X2
choose A = and /N, then O-N(X) goes as 7.
In order to study unfolded correlation function, we define a scaling variable u = N
such that -(u) = 1. In particular, the unfolding variables p aN and = aN where a is
some constant. Then the unfolded cluster function
some constant. Then the unfolded cluster function
Y2(P, ) lim T2 (xr, x,).
N-too dp dx
Since T2 = T2 6T2, similarly we define Y2 = Y2 -_ Y2 so that
yO(,Y) lidx dy dbf(b)G
N-oocdp dX J 4b
and
Y dx dy dbf(b) G G)
6Y2(p,X) Ilm -- (x1)_(x2) 1
N-o dp dX Jo 4b
For further calculation, we begin with 6Y2. First, we rewrite o-1(x) in terms of the
unfolding variable p,
(B-5)
(B-6)
(B-7)
G( ) G= aNg) sin co1 a28b p
' )2bN( ( 2COs -8b p
a2 n a2N 1
-8b p 8b p2J
Plugging this into eq. B-7, we get
SY2(PN a2N2 / dbf(b) a aN G aN
6Y2(p, X) lim b 2 2aI -/ -
N/- p2X Jo 4b p 247
(B-8)
o-(x) -~ Msin(O sin dOcos 0)
(B-9)
Using asymptotic solution for harmonic oscillator wave function,
lim (-1)"m 1/4 2m( ) -1/2 COs(u) (2-51)
mmoo 2Ni
lim (-1)m 1/4 2m+l ) = -1/2 sin(7u) (2-52)
m-oo 2/V
where N = 2m. We get the unfolded kernel
G(u sin(w(u v)) (2-53)
KG (u, V)= (2-53)
v (u v)
which is the well-known sine kernel.
2.3.3 Gap Probability and Spacing Distribution
The probability that there is no eigenvalue in the interval of s is called gap
probability defined as a Fredholm determinant
E(s) = det[1 G] (2-54)
where the kG refers to the sine kernel over the interval of -s/2 and s/2. The spacing
distribution is related to the gap probability by
d2 E(s)
p(s) =ds (2-55)
which is commonly used to study short range fluctuations in the eigenvalue spectrum.
This function measures the probability of finding two neighboring eigenvalues in the
interval of s in the unfolded scale. The analytical calculation is highly non-trivial but the
good approximation of the spacing distribution can be obtained from considering 2 x 2
matrix model suggested by Wigner. The general expression for the spacing distribution,
Wigner's surmise or Wigner-Dyson statistics,
p(s) = cs e-d2 (2-56)
where cp and d3 are constants. The 3 refers to the symmetry parameter of the Gaussian
ensembles. Note that in the small s limit, the spacing distribution is determined by s1
of the N-dependent parameter in the model ensemble, which cannot be simply
scaled out.
For the future, we further need to study the level statistics of the A-ensembles such
as the spacing distribution, the number variance, the spectral form factors, the inverse
participation ratio etc. both analytically and numerically. These will allow much broader
applications of our results. In addition, it is also important to explore if there exist a novel
universality of random matrix ensembles associated with the fat-tail distribution in the
framework of the Wigner-class (non-invriant) random matrix ensembles.
APPENDIX A
CENTRAL LIMIT THEOREM
It is understood that the universality of the Gaussian ensembles has some
connection to the central limit theorem1 The fact that there are two kinds of stable
probability distribution, e.g. the Gaussian/Levy basin according to the central limit
theorem2 inspires us that there would exist the counterpart structure in RMT, namely
* Gaussian distribution Gaussian RMT
* Levy distribution Levy RMT
Yet, this conjecture is not proven although there have been several attempts. Here, we
will review the content of the central limit theorem.
The central limit theorem states that the distribution of the large sum of independent
identically distributed (i.i.d) random numbers reach to certain limiting form of probability
distributions. Suppose we consider a sum X of a large number (N) of i.i.d random
numbers, x, drown from a certain probability distribution,
X = +X2 +x3 -... +XN. (A-1)
The sum X can be considered as the position of a particle undergoing a Brownian
motion, each step size of which is given by a random number xi. After the large sum is
done or the large number of steps are taken, it is proven that the probability distribution
of the sum X, or analogously that of the position of the particle under the Brownian
motion converges to a limiting distribution, namely, the Gaussian distribution as long as
1 In fact, it can be proven that the eigenvalue distribution of large random matrices
becomes the well known-semi circle using the central limit theorem [71].
2 To be precise, it should be referred to as "generalized central limit theorem".
I I I I I I I
0 1000 2000 3000 4000 5000 6000
7000 8000 9000 10000
Figure 3-6.
Eigenvalue density (x = 0 104) for A-ensembles for A
(7 = 2.00), 1.5 (7 = 4.00) as well as 1 (7 = 0.75).
1.1 (7 = 0.75), 1.3
0 5 10 15 20
x
Figure 3-7. Eigenvalue density (x
= 2.00), 1.5 (7 = 4.00)
25 30 35 40
= 0 40) for A-ensembles for A = 1.1 (7 = 0.75), 1.3 (7
as well as 1 (7 = 0.75).
X = 1 1 ...............
II
X=1.3
X=1.5 -------
that indicates that the two neighboring levels repel. The level repulsion is characteristic
of the Gaussian ensembles. This is in contrast to the uncorrelated or Possion case,
p(s) = e-s, in which the level repulsion is absent and therefore the levels can be
bunched up or separated far apart.
2.3.4 Number Variance
The number variance E(s) = (n2) (n)2 provides the measure of the long-range
eigenvalue fluctuation. It is given by
j s/2 s/2
Z(s) = [R(u, v)]dudv (2-57)
J-s/2 J-s/2
/s2 s/2
= s2[(u v) Y(u,v)]dudv (2-58)
J-s/2 J-s/2
where R(u, v) is two-level correlation function defined by R(u, v) (p(u)p(v)) 1. Using
the cluster function of GUE, Y(u, v) = YG( v) = [sn(-v)) the further calculation
gives
ZG(s) = s2 (s -r)Y(r)dr (2-59)
= (In(27s)+ 7 + 1) O(s-1) (2-60)
where 7 is Euler's constant, 7 = 0.5772.... Note that for large s, EG(s) o In s, which
displays another characteristic of the Gaussian Ensembles, the level rigidity. In contrast,
for the Poisson or uncorrelated case, RP(u, v) = 6(u v) and thus EP(s) = s.
2.4 Critical Ensembles
2.4.1 Anderson Transition and Critical Ensembles
The suggestion of a novel universality beyond the Gaussian ensembles comes from
the study of the Anderson transition in the disordered electronic systems[8, 36-42]. In
these systems, the Gaussian ensemble is only relevant in the metallic regime where
all the eigenstates are extended across the entire system and correlations of the
corresponding eigenvalues are well described by the WD statistics. As the disorder
model that can incorporate the desired fat-tail spectral density, it is not yet clear if the
unfolded two-level cluster function can have N-independent asymptotic limit 4
4 It turns out that the two-level correlation function carries non-trivial N dependence.
The details will be provided in APPENDIX B.
powers of N-1/2. It is shown that
e-"2 h3(x) h4 (X)
P(x) ~ [1 + ...] (A-7)
where h,(x) can be written in terms of the hermite polynomials of order n. This
approximation breaks down when h3(x)/vN becomes 0(1). It means that the width
of the Gaussian distribution approximation scales x O(N2/3).
In case the variance of the probability distribution does not exist, the central region
collapses faster than N log N and the probability distribution of large sum converges to
a different limiting function, known as Levy distribution. In particular, in the large x limit,
the distribution shows a power-law tail of the form,
A
P(x 00) ~ -xl; 0 < a < 2 (A-8)
X1 a
matrix ensemble. It is shown in this framework that the general n-point correlation
function can be written down as
dbf(b)
Rn(x ,-- ,xn) = (4b) n/2det[K (;xi, )],i 1,2,...n (1-15)
Jo (4b) /2N'
where f(b) is called the spread function, defined as any non-negative normalizable
function, f f(b)db = 1 and ;x = xi/2v b and KN refers to the well known sine kernel of
the Gaussian ensembles. In particular, the spectral density (1-point correlation function)
can be given by
UN(X) = -j- dbf(b) KG(x,x) (1-16)
27 Jo b/2
and two point cluster function defined as T2(x1, x2) -R2(x, x2) + R1(x1)R1(x2) has the
form T2 = T20 T2, where
f dbf(b)[KN( ) 2 2-
T(xix2) = [ ,) (1-17)
and
f" dbf(b)
4b T-N(x ) -Nb(x) O-N(X-N(x2) (1--18)
Thus, a variety of possible generalized ensembles can be obtained from a proper
choice of superposition (a choice of the spread function f(b)) of the kernel of Gaussian
ensemble. At the same time, it is possible to study the correlation functions for the
choice of f(b) immediately. For example, the Gaussian ensembles can be thought of
as a trivial example corresponding to the choice of f(b) = 6(b bo). For a non-trivial
choice of f(b) = b-(C+ 1e- b, it is shown that the spectral density exhibit asymptotic
power-law distributions. Although this framework allows a successful realization of RM
These imply that
/ det[KN(xi, y)] ij... mdxm = (N m+ 1)det[KN(xi, ,x)],J i... m (2-24)
2.2.3 Correlation Functions and Cluster Functions
The probability of finding any n eigenvalues out of the available N eigenvalues in the
intervals, {x, -+ x, + dx,; i= 1... n} is given by n-level correlation function
Rx,...x) = N I dxn ,... dxN P(x (X, .. ). (2-25)
(N I- n)!J- J--
Using the determinant form of JPDF and the properties of the kernel, the n-level
correlation function can be written in a compact form given by
R,(x,, x, ...., x,) = det[KN(xi, x)]{,(i= ...,n}. (2-26)
Note that the diagonal terms of the determinant are given by the one-level
correlation function Ri(x) (or the eigenvalue density oaN(x))
N-1
R (x) = KN(X, x) >= i(x) i(y). (2-27)
i=-
and the off-diagonal terms of the determinant are given by two-level kernel in terms of
which the two-level correlation function can be written down as
R2(x, y) = KN(x,x)KN(y, y) KN(x, y)KN(y,x). (2-28)
For practical purpose, it is useful to introduce the n-level cluster functions defined by
T, = (-1)n- '(- 1)! n RG (x, with in Gk) (2-29)
G i=1
Here, G stands for any division of the indices (1, 2, ....n) into m subgroups (Gi, G2,..., Gm).
For example, the one level cluster function
T7 (x) = R1 (x) (2-30)
Thus, by taking functional derivative 6/6V(y) in Eq. 2-67, we get
Ijdx(p(x))(p(y))R(x, z) In y -z = --6(x y). (2-69)
In the large N limit, around the region under consideration, if (p(x)) scales as N and
becomes a constant, we can introduce a new variable u = x/A scaled by the mean level
spacing A = p-1 (i.e., for the Gaussian ensembles, p(x) oc 2N in the x 0 limit) and
rewrite Eq. 2-69, which reads
/ dwR(u w) In v w = -0- 6(u- v). (2-70)
This implies that the two-level correlation function does not depend on the specific form
of V(x), which carries microscopic information of the system. Therefore, the two-level
kernel is universal. Within this framework, it is understood that for the soft confinement
potential, the average density is not constant in the N -- oc limit so that it does not
simply scale out. Therefore, the two-level correlation function is expected to be different
from the WD universality [54].
with the unfolding map, e.g.,
u(x) dxp(x). (3-27)
3.2.3.1 Observation of Normal/Anomalous Structure of Two-level Kernel
It is well known that for A =1, the two level cluster function has both normal and
anomalous component. The normal part of the two-level cluster function for the critical
ensembles for uv > 0 and lu v| << u is given, in the 7 < 2r2 limit, by the sinh kernel
[ 7 sin[7(u v)] (3-28)
Y, (u, v) -- s[ )-)] (3-28)
) 27r sinh[ (u v)]
The anomalous part of the cluster function, so called "ghost correlation peak" [54] for
uv < 0 given by
[7 sin[7(u- v)] 2
Y (u, v)= [ csh[( )] 2 (3-29)
The presence of the anomalous component occurring due to long range correlation is
required by the normalization sum rule [57]
1 = du [Y,(u, u')+ YY(u, u')]. (3-30)
The deficiency of the sum rule
S1 duY(u,u')= duY (uu') (3-31)
-0C J -0C
is related to certain characteristics of the critical statistics [8, 36-42]: i) the level
compressibility in the number variance and ii) the multi-fractality of eigenvectors. In
particular,
d ((L)) d- D,
X= = (3-32)
d(L) 2d
where the fractal dimensionality Dp determines the scaling behavior of the moments of
the inverse participation ratio via
(< ddx (x) 2p) O L-D(p-1) (3-33)
(7
while the anomalous part of the kernel (Du < 0) for the A-ensembles can be written as
a(, v) = (, ) sin[( (3-46)
27 cosh[- (u 9v)]
We note that this kernel resembles the usual sinh kernel of the critical ensembles with
more general argument. For A = 1, F(u, v) 7* = 7. Thus it reduces to the sinh
kernel. However, in general for A / 1, the F(u, v) is a non trivial function of u and v. For
example,
F(u, ) = *(D + ) for A = 0.5 (3-47)
and
(u, ) = for A = 2.0 (3-48)
VIU+ vv
It is obvious that the form of the function is not translationally invariant. However, if we
choose v to be a fixed value, i.e, v = v uo a = a, which is the same as choosing
v = uo, then = u uo + a = u v + a s +a. In this way, the function F(u, v) and
g(u, v) can be written in terms of a difference variable s- u v = alone with a
constant v = a that serves as a fixed reference point.
F(s + a, a) F(s, a) (s = a) (3-49)
Then
F(s, a) sin(ws)
K
27 sinh[ s]
The figures 3-14, 3-15 shows the fitting results with the kernel given in Eq. 3-50 for
different A values. They show a fit with the numerically obtained two-level kernels with fit
values 7* 7 = 0.5 and a 2 for A < 1 (Fig. 3-14) and 7* 7 = 1.5 and a 2 for A > 1.
(Fig. 3-15) Finding the exact dependence of a and 7* on the parameter A and 7 requires
to know details of the eigenvalue density p(x) in the vicinity of origin and the cutoff.
This bears no importance in the discussion of the universal feature of the ensembles
because p(x) in the vicinity of the origin is simply a reflection of how one chooses V(x) to
the each individual random variable xi are sampled from the distributions with a finite
variance regardless of other details. 3
For a short proof, we define the characteristic function P(k) of a probability
distribution as
P(k) = dx ekx P(x) (A-2)
After the large sum X = C" x,, the probability distribution of P(X) is given as the
inverse fourier transform of N times convolutions of P(k).
P(X) = e-ik [(k)] (A-3)
27dk
P(k) has the cumulant expansions around k = 0 in the following way,
In (k)n (ik) (A-4)
n=0
where ci denotes the cumulants of distributions. The first few of them are named; cl is
mean, c2 is the variance, c3 skewness and c4 kurtosis etc. Then, we can write P(X) as
P(X) = eNf(k,X) (A-5)
S27dk
where f(k, X) is given by
X n
f (k, X) = -ikn (ik)n (A-6)
N n
n=0
Using the method of steepest decent, we know that in the large N limit, the dominant
contribution of f(k, X) comes from the maximum of f(k, X), which is at k = 0. Thus, in
the rescaled variables x = -Nc and w = c2-Nk, we obtain the expansion of P(x) in
3 The existence of the higher order moments determine how fast it converges to the
limiting form.
