Random matrix universality for classical transport in composite materials
N. Benjamin Murphy, Elena Cherkaev, and Kenneth M. Golden∗
University of Utah, Department of Mathematics,
155 S 1400 E, JWB 233, Salt Lake City, UT 84112, USA
Universality of eigenvalue statistics has been observed in random matrices arising in studies of
atomic spectra, internet signal dynamics, and even the zeros of the famous Riemann zeta function.
Here we consider the flow of electrical current, heat, and electromagnetic waves in two-phase composites. We discover that for a new class of random matrices at the heart of transport phenomena,
phase connectedness determines the nature of the system statistics. A striking transition to universality is observed as a percolation threshold is approached, with eigenvalue statistics shifting from
weakly correlated toward highly correlated, repulsive behavior of the Gaussian orthogonal ensemble.
Moreover, the eigenvectors exhibit behavior similar to Anderson localization in quantum systems,
with ”mobility edges” separating extended states. Our findings help explain the collapse of spectral
gaps and associated critical behavior near percolation thresholds. The spectral transition is explored
in resistor networks, bone, and sea ice structures important in climate modeling.
Abbreviations ACM, analytic continuation method;
ESD, eigenvalue spacing distribution;
GOE, Gaussian orthogonal ensemble;
RME, random matrix ensemble;
RMT, random matrix theory;
RRN, random resistor network
Complex systems with a large number of interacting
components are ubiquitous in the physical and biological sciences, and have a broad range of applications. As
the number of components increases, structure begins to
emerge from the underlying randomness of the system,
and universal behavior can arise. Random matrix theory
(RMT) has been quite successful in modeling universal
features found in statistical fluctuations of characteristic quantities of complex systems, such as atomic nuclei [1], biological networks [2], financial time series [3],
the departure times for bus systems [4], quantum systems
whose classical counterparts are chaotic [1, 5], and mesoscopic conductors [6]. Remarkably, it has been shown
that even the (non-random) zeros of the famous Riemann zeta function have universal features which are
very accurately captured by RMT [5, 7, 8]. Each of
these systems can be described in terms of discrete “spectra” confined to a line. In RMT, the spectra of such
systems are modeled by the eigenvalues of ensembles of
large random matrices called random matrix ensembles
(RME’s). Long and short range correlations of these random eigenvalues are measured in terms of various eigenvalue statistics introduced by Dyson and Mehta [8]. The
localization properties of the associated eigenvectors are
characterized by quantities such as the inverse participation ratio (IPR) [3] and the Shannon information entropy [9]. In many systems, the spectral statistics are
parameter independent when properly scaled [1, 10] and
∗ Electronic
address: golden@math.utah.edu
fall into two universal categories: uncorrelated Poisson
statistics [1] and highly correlated Wigner–Dyson (WD)
statistics [1, 11]. The statistical behavior of the spectrum
is related to the extent that the eigenfunctions overlap.
A key example is the metal-insulator Anderson transition exhibited by noninteracting electrons in a random
potential [10, 12]. For small disorder, the electron wave
functions are extended structures. Their overlap gives
rise to correlated WD energy level statistics with strong
level repulsion [12]. However, for large disorder, electrons are typically localized at different points in the
sample and “do not talk to each other,” resulting in uncorrelated Poisson level statistics [12]. During transitions
from small to large disorder, the wave functions become
increasingly localized and an intermediate Poisson-like
behavior of level statistics arises [6, 10]. There are other
systems which undergo an analogous spectral transition
as a system parameter varies. Among them are: the hydrogen atom in a magnetic field [13], random points on
a fractal set [14], quantum chaos [15], and complex networks [9].
We demonstrate here that transitions in the connectedness or percolation properties of macroscopic composite
media, with microstructural scales spanning many orders
of magnitude, can also be characterized by a transition
in the statistics of eigenvalues and the delocalization of
eigenvectors of a random matrix. This is a new type of
random matrix within RMT that depends only on the
geometry of the composite medium, and not directly on
a probability distribution as usual. While the connectedness driven transition in the statistical behavior of eigenvalues is analogous to that of the Anderson transition and
other systems, the delocalization of eigenvectors reveals
new subtleties that distinguish the behavior we see from
classical Anderson localization. This large family of random matrices arises in the analytic continuation method
(ACM) [16–19] for representing transport in composites.
The method provides Stieltjes integral representations for
the bulk transport coefficients of a two-component ran-
2
A
B
C
D
E
FIG. 1: Connectedness transitions in composite structures. (A)–(E) Increasingly connected composites from left
to right. (A) Realizations of the two-dimensional lattice percolation model, with (black) bond probabilities p = 0.20,
p = 0.30, and p = pc = 0.5. (B) cross-sections from X-ray CT
volume renderings of the brine phase within a lab-grown sea
ice single crystal (H. Eicken), with image brine area fractions
of φ = 0.20, φ = 0.51, and φ = 0.70. (C) Melt ponds on the
surface of Arctic sea ice (D. K. Perovich), with area fractions
φ = 0.09, φ = 0.27, and φ = 0.57. (D) Arctic sea ice pack
(D. K. Perovich), with open ocean area fractions φ = 0.06,
φ = 0.10, and φ = 0.47. (E) SEM images of osteoporotic
(left) and healthy (right) trabecular bone (P. Hansma), with
cross-sectional area fractions φ = 0.26 and φ = 0.55.
dom medium, such as the effective electrical conductivity
~ involving
σ ∗ of a medium immersed in an electric field E,
a spectral measure µ of the random matrix [18, 20]. The
measure µ exhibits fascinating transitional behavior as
a function of system connectivity, which controls critical
behavior of σ ∗ near connectedness thresholds. For example, in the case of a random resistor network (RRN)
with a low volume fraction p of open bonds, as shown
in Fig. 1A, there are spectrum-free regions at the spectral endpoints λ = 0, 1 [21]. However, as p approaches
the percolation threshold pc [22, 23] and the system becomes increasingly connected, these spectral gaps shrink
and then vanish [21, 24], leading to the formation of δcomponents of µ at the spectral endpoints, precisely [21]
when p = pc and p = 1 − pc . This leads to critical
behavior of σ ∗ for insulating/conducting and conducting/superconducting systems [21]. This gap behavior of
µ has led [21] to a detailed description of these critical transitions in σ ∗ , which is directly analogous to the
Lee–Yang–Ruelle–Baker description [25, 26] of the Ising
model phase transition in a ferromagnet’s magnetization
M . Moreover, using this gap behavior, all of the classical
critical exponent scaling relations were recovered [21, 26]
without heuristic scaling forms but instead by using the
rigorous integral representation for σ ∗ involving µ.
Our results here reveal a mechanism for the collapse
in the spectral gaps of µ and illustrate that localized and
extended eigenvectors of the matrix are in direct corre~ that
spondence with components of the electric field E
are localized in, and extended throughout the composite medium. In particular, we demonstrate that eigenvalues associated with a disordered state, such as a low
volume fraction RRN, are weakly correlated and welldescribed by Poisson-like statistics. However, as the
percolation threshold pc is approached and the system
develops long range order, the eigenvalues become increasingly correlated and their statistics approach classical WD statistics, causing the eigenvalues to spread out
due to increased level repulsion, subsequently forming δcomponents in µ at the spectral endpoints. Correspondingly, the eigenvectors become increasingly extended and
those associated with these δ-components are typically
highly extended. These regions of extended states are
separated from each other by “mobility edges” [1] of lo~ involving
calized states. A resolvent representation of E
the random matrix provides a one-to-one correspondence
between localized (extended) eigenvectors and localized
(extended) components of the electric field within the
medium.
We show that this spectral behavior emerges in a variety of composite systems, such as the brine microstructure of sea ice [27–29], melt ponds on the surface of Arctic sea ice [30], the sea ice pack itself, and porous human
bone [31]. Our results indicate that it is pervasive in
such macroscopic systems and arises simply from connectedness – at the most basic level of characterizing any
physical system with inhomogeneities.
The behavior of composite materials exhibiting a critical transition as system parameters are varied is particularly challenging to describe physically, and to predict
mathematically. Here, we discuss composites which exhibit critical behavior in transport properties induced by
transitions in connectedness or percolation properties of
a particular material phase.
Lattice and continuum percolation models have been
used to study a broad range of disordered materi-
3
als [22, 23]. In the simplest case of the two-dimensional
square lattice [22, 23], as shown in Fig. 1A, the bonds
are open with probability p and closed with probability 1 − p. Connected sets of open bonds are called open
clusters. The average cluster size grows as p increases.
When the system size L tends to infinity, there is a critical probability pc , 0 < pc < 1, called the percolation
threshold, where an infinite cluster of open bonds first
appears. In dimension d = 2, pc = 1/2, and in d = 3,
pc ≈ 0.2488 [22]. Now consider transport through the
associated RRN, where the bonds are assigned electrical
conductivities σ1 with probability p and σ2 with probability 1 − p. The effective conductivity σ ∗ exhibits two
types of critical behavior. First, when 0 < σ1 < ∞ and
σ2 → 0, the system is insulating, σ ∗ = 0, for p < pc and
is conducting, σ ∗ > 0, for p > pc . Second, when σ1 → ∞
and 0 < σ2 < ∞, the conductive system becomes superconducting, σ ∗ → +∞, as p → p−
c .
Sea ice is a complex composite consisting of pure ice
with sub-millimeter scale brine inclusions, as shown in
Fig. 1B, whose volume fraction φ, geometry, and connectedness vary significantly with temperature T . The
brine microstructure displays a percolation threshold at
a critical brine volume fraction φc ≈ 5% in columnar
sea ice [27], which corresponds to a critical temperature
Tc ≈ −5◦ C for a typical bulk salinity of 5 ppt. This
threshold acts as an on–off switch for fluid flow through
sea ice, and is known as the rule of fives. It leads to
critical behavior of fluid flow, where sea ice is effectively
impermeable to fluid transport for φ < φc , yet is permeable for φ > φc , with the permeability as a function
of φ above the 5% threshold described by the universal
critical exponent for lattices in three dimensions [27–29].
Fluid flow through sea ice mediates a broad range of
physical and biological processes in the polar marine environment [28, 29], including brine drainage, nutrient replenishment for algal communities in the brine phase,
snow-ice formation, and the evolution of melt ponds on
the surface of Arctic sea ice [30]. Melt ponds (Fig. 1C)
determine sea ice albedo in the Arctic, a key parameter
in climate modeling. Despite its importance, it remains
a major source of uncertainty in climate models. In fact,
the lack of inclusion of melt ponds in previous generations of climate models is believed to partially account
for the inadequacy of these models to predict the dramatic rate of melting of the summer Arctic sea ice pack.
The results here advance our understanding of the effective or homogenized properties of the ice pack, and help
provide a path toward more rigorously incorporating sea
ice into climate models.
Human bone also displays a complex, porous microstructure, as shown in Fig. 1E. The strength of bone
and its ability to resist fracture depend strongly on the
quality of the connectedness of the hard, solid phase.
Osteoporotic trabecular bone can become more disconnected and remaining connections can become more tenuous or fragile [31]. The spectral techniques [32, 33] which
have arisen from the ACM provide important methods
for analyzing microstructural transitions in bone and its
biophysical properties.
I.
MATHEMATICAL METHODS
Random matrices arise naturally in the ACM for representing transport in composites [18, 20]. This method
provides Stieltjes integral representations for the effective
parameters of two-component composite media, such as
electrical conductivity and permittivity, magnetic permeability, and thermal conductivity. The integral representations involve a spectral measure µ associated with
a family of random matrices, which depend only on the
composite geometry [20]. A remarkable feature of this
method is that once the spectral measure is found for a
given composite geometry, by spectral coupling of the
governing equations [34], the effective electrical, magnetic, and thermal transport properties are all determined by µ.
Consider the effective parameter problem for twocomponent conductive media [18, 20]. The electromagnetic transport properties of the composite are governed
by the quasi-static limit [18] of Maxwell’s equations
~ ×E
~ = 0,
∇
~ · J~ = 0.
∇
(1)
~ and J~ are the random electric field and current
Here, E
~ and σ denotes the
density, which are related by J~ = σ E,
electrical conductivity of the locally isotropic, stationary random medium. In the case of a two-component
medium with (complex-valued) component conductivities σ1 and σ2 we write
σ = σ1 χ1 + σ2 χ2 ,
(2)
where χ1 is the characteristic function of medium one,
taking the value 1 in medium one and 0 otherwise, with
χ2 = 1 − χ1 .
The effective conductivity matrix σ ∗ is defined by [18]
~
hJ~ i = σ ∗ hEi,
~ =E
~ 0.
hEi
(3)
~0 =
Here, h·i denotes ensemble averaging and the vector E
E0~ek has magnitude E0 and direction ~ek , taken to be the
kth standard basis vector for some k = 1, . . . , d, which defines the d-dimensional coordinate system. Equivalently,
the effective conductivity σ ∗ may be defined [18, 20]
~ =
in terms of the energy (power) density : hJ~ · Ei
~0 · E
~ 0 = σ ∗ E 2 . For simplicity, we focus on the diagσ∗ E
kk 0
∗
∗
onal coefficient σkk
of the matrix σ ∗ and set σ ∗ = σkk
.
The key step in the method is obtaining the following
Stieltjes integral representation for σ ∗ [16–18]
Z 1
dµ(λ)
∗
σ = σ2 (1 − F (s)), F (s) =
,
(4)
0 s−λ
which follows from a resolvent representation of the electric field (in material phase 1) [20]
~ = sE0 (sI − χ1 Γχ1 )−1 χ1~ek .
χ1 E
(5)
4
B
A
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
0
-0.1
-0.1
FIG. 2: The projection matrix Γ. The matrix Γ with periodic boundary conditions for (A) 2D and (B) 3D networks
with system sizes L = 6 and L = 3, respectively, with corresponding matrix size N = Ld d. Notice that the lengths of the
repeated structures are multiples of the system size L.
Here, s = 1/(1 − σ1 /σ2 ), −F (s) plays the role of the
effective electric susceptibility, µ is a spectral measure
associated with the random operator χ1 Γχ1 [18], and
−1 ~
~
Γ = −∇(−∆)
∇· is a projection onto curl-free fields,
based on convolution with the free-space Green’s function
for the Laplacian ∆ = ∇2 .
In this way, the ACM determines a homogeneous
medium which behaves macroscopically and energetically
like a given inhomogeneous medium. Moreover, parameter information in s and E0 is separated from the geometric complexity of the system, which is encoded in the
spectral measure µ. Geometric information about the
composite is incorporated into the positive Stieltjes
meaR1
sure µ in equation (4) via its moments, µn = 0 λn dµ(λ).
For example, the mass µ0 of the measure is the volume
fraction p of medium one, i.e., µ0 = hχ1 i = p. We may
think of the measure dµ(λ) as m(λ) dλ for some density
m(λ) which is allowed to have δ-components. This characterization of µ is more transparent in the setting of a
finite RRN, where the random operator χ1 Γχ1 can be
represented by a real-symmetric random matrix of size
N × N , where N = Ld d [20]. In this case, χ1 is a diagonal matrix with 1’s and 0’s along the diagonal, corresponding to conductive bonds with conductivity σ1 and
σ2 , respectively, and Γ is a (non-random) projection matrix [20]. In this case, the spectral measure µ can be calculated directly from the eigenvalues λj , j = 1, . . . , N ,
and eigenvectors ~vj of the matrix χ1 Γχ1 , and is given by
a weighted sum of δ-measures


N
X
dµ(λ) = 
hmj δ(λ − λj ) i  dλ = m(λ) dλ,
(6)
j=1
where h·i denotes ensemble averaging, mj = [~vj · êk ]2 ,
and êk is a lattice basis vector [20]. The matrix Γ encapsulates the lattice topology and transport characteristics of the resistor network and displays a rich geometric,
banded structure, as shown in Fig. 2.
The action of the matrix χ1 in χ1 Γχ1 is to randomly
zero out each row and column of Γ that corresponds to
diagonal components of χ1 satisfying [χ1 ]jj = 0. The
spectral weights mj associated with this null space of
χ1 Γχ1 are identically zero [20] and do not contribute to
the sum in (6). The only eigenvectors that contribute
to this sum are associated with random N1 × N1 submatrices of Γ corresponding to the N1 components of χ1
satisfying [χ1 ]jj = 1, where N1 ≈ pN .
The discretized sea ice and bone composite structures
displayed in Fig. 1 were converted to 2D binary fluid/ice
and bone/marrow representations (the ice pack images in
(D) are the converted versions). The geometry and the
component connectivity of these binary images can be described in terms of high density two-component resistor
networks, with fluid and bone corresponding to component one, and ice and marrow corresponding to component two. In this way, the matrix χ1 Γχ1 was obtained
for these composite structures. In the next section, we
show that its associated eigenvalues and eigenvectors undergo a connectedness driven transition that is analogous
to that of the Anderson transition [1, 10, 12].
II.
RESULTS
In RMT, long and short range correlations of eigenvalues in the bulk of the spectrum [10] are measured in
terms of various eigenvalue statistics. In order for the
fluctuation properties of the eigenvalues about the mean
density ρ(λ) to be compared to the predictions of RMT,
the spectrum has to be unfolded [1, 3, 10]. It is nontrivial to unfold spectra associated with the RRN for
small volume fractions p, due to prominent “geometric”
resonances in ρ(λ) [20, 24], which has limited our analysis
of these systems in the dilute limit, p 1.
The nearest neighbor eigenvalue spacing distribution
(ESD) P (z) is the observable most commonly used to
study short-range correlations [1]. For highly correlated WD spectra, such as that exhibited by the realsymmetric matrices of the Gaussian orthogonal ensemble (GOE) [1, 11], the ESD is accurately approximated
by P (z) ≈ (πz/2) exp(−πz 2 /2), known as Wigner’s surmise [1, 6], which illustrates the phenomenon of eigenvalue repulsion, vanishing linearly in the limit of small
spacings z. In contrast, the ESD for uncorrelated Poisson spectra, P (z) = exp(−z), allows for level degeneracy.
In Fig. 3 we display the behavior of the ESD for
eigenvalues of the matrix χ1 Γχ1 , which correspond to
RRN, melt pond, and Arctic sea ice pack composite
structures displayed in Fig. 1. This figure demonstrates
that for sparsely connected systems, the behavior of the
ESD’s are well described by weakly correlated Poissonlike statistics [10]. The ESD’s increase linearly from zero
at short separation but the initial slope is steeper, implying less level repulsion. As the systems become increasingly connected, and long range order is established, the
ESD’s transition toward highly correlated WD statistics
with high level repulsion. The blue dash-dot curve displayed in Fig. 3 is the ESD for Poisson spectra, while the
green dashed curve is the ESD for the GOE. For the 2D
and 3D RRN, the eigenvalue density ρ(λ, p) displays the
5
A
Arctic sea ice pack
1
Σ2(L)
P(z)
0.8
A
p=0.47
p=0.10
p=0.06
0.6
0.4
0.2
0
B
0
1
2
3
z
4
0
1
2
p=0.09
3
z
4
0
1
2
3
z
4
B
0.6
0.4
0.2
0
C
0
1
2
1
P(z)
4
0
1
2
3
z
4
0
1
2
3
z
4
p=0.1
p=0.9
p=0.3
p=0.7
p=0.5
0.6
0.4
0.2
0
0
1
2
3
z
4
0
1
2
3
4
0
1
2
3
z
4
3D random resistor network
1
p=0.05
p=0.95
0.8
P(z)
z
2D random resistor network
0.8
D
3
p=0.2488
p=0.4
p=0.5
p=0.6
p=0.7512
p=0.13
p=0.87
0.6
0.4
Arctic pack ice
0
1.2
0.8
0.4
0
10
20 L 30 0
Trabecular bone
10
10
20 L 30 0
20 L 30 0
2D RRN
10
Brine inclusions
φ=0.20
φ=0.51
φ=0.70
p=0.1
p=0.3
p=0.5
p=0.7
p=0.9
p=0.26
p=0.55
0
Arctic melt ponds
p=0.09
p=0.27
p=0.57
p=0.06
p=0.10
p=0.47
2
1.6
p=0.57
p=0.27
∆3(L)
0.8
P(z)
1
Arctic melt ponds
30
25
20
15
10
5
0
10
20 L 30
3D RRN
p=0.05
p=0.13
p=0.2488
p=0.5
p=0.7512
p=0.87
p=0.95
20 L 30 0
10
20 L 30
FIG. 4: Long-range eigenvalue correlations. (A) The
eigenvalue number variance Σ2 (L ) and (B) the spectral rigidity ∆3 (L ) for Poisson (blue dash-dot) and Wigner–Dyson
(WD) (green dash) spectra are displayed along with those
of sea ice brine inclusions, Arctic melt ponds and pack ice,
trabecular bone microstructure and random resistor networks
(RRN). As the systems become increasingly connected, these
long-range eigenvalue statistics transition from the linear
Poisson-like behavior toward a logarithmic WD behavior.
0.2
0
0
1
2
3
z
4
0
1
2
3
z
4
0
1
2
3
z
4
FIG. 3: Short-range eigenvalue correlations. The eigenvalue spacing distributions (ESD’s) for Poisson (blue dashdot) and Wigner–Dyson (WD) (green dashed) spectra are displayed in (A)–(D), along with that of (A) the Arctic sea ice
pack, (B) Arctic melt ponds, (C) 2D random resistor network
(RRN), and (D) 3D RRN. As the systems become increasingly connected, the initial slopes of the ESD’s progressively
decrease, indicating an increase in level repulsion and shortrange correlations.
symmetry ρ(λ, p) = ρ(1 − λ, 1 − p) in the bulk of the
spectrum. This is reflected in the ESD’s by the symmetry P (z, p) = P (z, 1 − p), as shown for the 2D and
3D RRN in Fig. 3C and Fig. 3D. The ESD’s displayed in
Fig. 3D for the 3-dimensional percolation model suggest
that the GOE limit is attained for all pc ≤ p ≤ 1 − pc .
The ESD contains information about the spectrum
which involves short scales (a few mean spacings) [1, 10].
Long-range correlations are measured by quantities such
as the eigenvalue number variance Σ2 (L ), in intervals of
length L (not to be confused with the system size L),
and the closely related spectral rigidity ∆3 (L ) [1]. For
uncorrelated Poisson spectra, these long range statistics
are linear, with Σ2 (L ) = L and ∆3 (L ) = L/15, as displayed in blue color with dash-dot line style in Fig. 4. In
contrast, the strong correlations of WD spectra make the
spectrum more rigid [10] so that Σ2 (L ) and ∆3 (L ) grow
only logarithmically [1, 8]. The green dashed curves of
Fig. 4 are numerical computations of the exact solutions
of Σ2 (L ) and ∆3 (L ) for WD spectrum [1].
In Fig. 4, we also display the behavior of these longrange eigenvalue statistics for the matrix χ1 Γχ1 , corre-
sponding to the composite structures displayed in Fig. 1.
For sparsely connected systems, these statistics exhibit
linear Poisson-like behavior away from the origin with
slope less than their Poisson counterparts. This linear behavior has been attributed to exponentially decaying correlations of eigenvalues [10]. These statistics
transition with increasing connectedness toward logarithmic WD behavior typical of the GOE, which has
quadratically decaying eigenvalue correlations [10]. Similar to the ESD’s in Fig. 3, for the RRN these statistics
also display the symmetry Σ2 (L, p) = Σ2 (L, 1 − p) and
∆3 (L, p) = ∆3 (L, 1 − p).
Moreover, Fig. 3D and Fig. 4B suggest that the GOE
limit is attained by the short and long range statistics
for the 3-dimensional RRN for all pc ≤ p ≤ 1 − pc . They
appear to overlie the GOE limit almost exactly for all p
values tested in the range pc ≤ p ≤ 1 − pc . With this
in mind, we can paint a heuristic analogy with Anderson
localization, where low disorder corresponds to extended
states and WD statistics. When disorder exceeds a critical level, the states localize and the eigenvalues become
de-correlated. We view the 3D RRN with pc ≤ p ≤ 1−pc
to be “ordered” with extended states and WD statistics.
As p decreases, the disorder − or blockages to the flow
− increases, and the eigenstates localize (see below) and
the eigenvalue repulsion diminishes.
The eigenvectors ~vn , n = 1, . . . , N1 , associated with
the random N1 × N1 sub-matrices of Γ, discussed above,
also undergo a connectedness driven transition in their
localization properties. Two commonly used quantities
which measure the localization of the eigenvector ~vn are
the inverse participation ratio (IPR) In [3] and the Shan-
6
non entropy Sn [9], defined as
Localized Mode x10 - 12 Extended Mode x10 - 2
Log Electric Field
A
8
-2
N1
X
In =
[vnj ]4 ,
[vnj ]2 ln[vnj ]2 ,
(7)
-8
0
0
j=1
where vnj is the jth component of the eigenvector ~vn .
Eigenvectors of matrices in the GOE are known to be
highly extended and independent of the distribution of
the eigenvalues [35]. In this case, the IPR and entropy are
given by IGOE = 3/N1 [3] and SGOE ≈ ln(N1 /2.07) [9],
respectively. Associated with the Sn is the eigenvector
localization length ` defined as
-14
(8)
where we denote by hSi and hIi ensemble averaging over
all values of the Sn and In . The meaning of In and Sn
can be illustrated by two limiting cases (i) a normalized
vector with only one component vnj = 1 has In = 1 and
Sn = 0, √
whereas (ii) a vector with identical components
vnj = 1/ N1 has In = 1/N1 and Sn = ln N1 . When all of
the vectors in an ensemble are of type (i) then ` ≈ 2.07,
while ` ≈ 2.07N1 when all are of type (ii). If hSi = SGOE
then ` = N1 .
In the matrix setting, the electric field in equation (5)
has the following eigenvector expansion
~ = sE0
χ1 E
N1
X
[(s − λn )−1 (~vn · χ1~ek )] ~vn .
(9)
n=1
This provides a direct link between localized eigenvectors
~ that have significant magni~vn and eigen-modes of χ1 E
tudes in only a few resistors, while extended eigenvectors correspond to electric field components that extend
throughout the network. Displayed in Fig. 5A, from left
~ and examples of
to right, is the total electric field χ1 E
~ for a random
localized and extended eigen-modes of χ1 E
realization of the 2D RRN with p = pc .
III.
DISCUSSION
Our results displayed in Fig. 5B–D show that the eigenvectors ~vn associated with 2D and 3D RRN undergo a
fascinating delocalization as p increases and the system
becomes increasingly connected. For example, Fig. 5B
displays the IPR In of the 3D RRN as a function of the
index n of the eigenvector ~vn , with increasing index corresponding to increasing eigenvalue magnitude λn . This
figure shows that for p pc , the eigenvectors are typically localized, with values In of IPR quite different from
that of the GOE, identified by the red horizontal lines.
Moreover, they have an oscillatory behavior that, when
plotted as a function of λn , follows the peaks and valleys of “geometric” resonances exhibited by the eigenvalue density ρ(λ) for small p [20, 24], with more localized regions corresponding to lower density. This indicates that there are significant correlations between
-1
-4
IPR vs vector index n
B 10 0
p=0.05
p=0.35
p=0.7512
10 -1
10 -2
10 -3
50
` = N1 exp[−(SGOE − hSi)],
1
4
In
j=1
Sn = −
N1
X
150
n 250
400 800 1200 1600 n
Inverse Participation Ratio
C
2D ⟨I⟩
3D ⟨I⟩
2D IGOE
3D IGOE
0.12
0.08
D
1000 2000 3000 n
Entropy and localization length
0.8
0.4
2D ⟨S⟩/SGOE
3D ⟨S⟩/SGOE
2D l/lGOE
3D l/lGOE
0.04
0
0
0
0.2
0.4
0.6
0.8 p 1
0
0.2
0.4
0.6
0.8 p 1
FIG. 5: Delocalization of eigenvectors. (A) Displayed
~ (in log scale)
from left to right are the full electric field χ1 E
~
and examples of localized and extended eigen-modes of χ1 E
(in linear scale) for a realization of the 2D random resistor
network (RRN), with a system size L = 50 and volume fraction p = pc = 1/2. The corresponding localized and extended eigenvectors have inverse participation ratios (IPR’s)
In ≈ 0.081 and In ≈ 0.002, and the DC values of the component conductivities are those for silver and silicon at 20 ◦ C,
σ1 = 6.3×107 S/m and σ2 = 1.56×10−3 S/m. (B) The IPR’s
of eigenvectors ~vn associated with realizations of the 3D RRN
plotted versus eigenvector index n, with L = 12 and increasing values of p from left to right, with 1 − pc ≈ 0.7512. The
vertical lines define the δ-components of the spectral measure
µ at the left and right spectral endpoints, where the associated eigenvalues λn satisfy λn . 10−14 and λn & 1 − 10−14 ,
respectively, while the horizontal lines mark the IPR value
IGOE = 3/N1 for the Gaussian orthogonal ensemble (GOE)
with matrix size N1 ≈ pN , where N = Ld d. (C) The ensemble averaged IPR hIi as a function of p, displaying transitional behavior at the percolation thresholds, pc = 1/2 for
2D and pc ≈ 0.2488 for 3D. (D) The normalized, ensemble averaged entropy hSi/SGOE and associated localization
length `/`GOE as a function of p, where SGOE ≈ ln(N1 /2.07)
and `GOE = N1 . Panels (B)–(D) demonstrate that, as p increases and the systems become increasingly connected, the
eigenvectors become progressively extended, on average, with
decreasing hIi and increasing hSi/SGOE and `/`GOE .
the eigenvalues and eigenvectors, in contrast with
the GOE. As p → p−
c , gaps in the spectrum about the
spectral endpoints shrink and then vanish [20, 24], while
the values In of the IPR continually decrease, approaching the GOE limit, as shown in Fig. 5B.
As p surpasses pc and 1 − pc , δ-components form in the
spectral measure µ at λ = 0 and λ = 1, respectively [21].
Numerically, the δ-component at λ = 0 manifests itself as
7
a large number of eigenvalues with magnitude . 10−14 ,
followed by an abrupt change of magnitude & 10−4 , with
no eigenvalues in the interval (10−14 , 10−4 ), and similarly
for λ = 1. The locations of these changes in magnitude
are identified by red vertical lines in Fig. 5B, demonstrating that the eigenvectors associated with these δcomponents are typically more extended than the others,
with values In of the IPR closer to the GOE limit. The
entropy Sn has a similar behavior.
In Fig. 5C and 5D the p dependence of hIi, hSi/SGOE ,
and `/`GOE are displayed. As p increases and the system
becomes increasingly connected, hIi decreases, with transitional behavior at the percolation threshold pc , while
hSi/SGOE and `/`GOE increase. Each indicate that
the eigenvectors, hence the eigen-modes of the electric
~ become progressively extended throughout the
field χ1 E
network with increasing system connectedness. Fig. 5B
shows that this average delocalization is largely due to
the formation of the δ-components in µ.
Fig. 5B also shows that regions of extended states are
separated from one another by a series of “mobility
edges” with a sudden increase in the number of localized eigenvectors. This behavior in the eigenvectors of
the random matrix χ1 Γχ1 is analogous to that of Anderson localization, where mobility edges mark the characteristic energies of the metal/insulator transition [1]. Remarkably, the mobility edges in Fig. 5B are precisely at
the locations of the δ-components, identified by red vertical lines, which control critical behavior of transport in
insulator/conductor and conductor/superconductor systems. However, the delocalization behavior of the spectral endpoints displayed in Fig. 5B is different from that
of Anderson localization [1], further demonstrating that
the behavior we see in χ1 Γχ1 is new to RMT.
We have demonstrated that the eigenvalues and eigen-
vectors associated with the random matrix χ1 Γχ1 , corresponding to various composite structures, have a statistical behavior that is analogous to, but distinctly different
from that of the Anderson transition. The eigenvalues
of χ1 Γχ1 shift from weakly correlated Poisson-like statistics toward highly correlated WD statistics as a function
of disorder (connectedness). Correspondingly, the eigenvectors undergo a delocalization, with highly extended
states appearing at the spectral edges, separated by mobility edges of localized states. Moreover, the delocalization of eigenvectors corresponds to a delocalization of
~ extending
the transport field, such as the electric field E,
throughout the composite medium near global connectedness thresholds. The disorder driven transition in the
statistical behavior of the eigenvalues also accounts for
the gap behavior of the spectral measure µ, underlying
exact integral representations for the effective transport
coefficients of the composite medium, such as effective
conductivity σ ∗ , which, in turn, governs their critical behavior via the formation of δ-components of the measure
at the spectral endpoints. This provides a novel way of
understanding and characterizing disorder driven transitions in the effective transport properties of composite
media, and opens a new chapter in the application of
RMT to the analysis of complex, macroscopic systems.
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Acknowledgments We gratefully acknowledge support from the Division of Mathematical Sciences and the
Division of Polar Programs at the U.S. National Science
Foundation (NSF) through Grants DMS-1009704, ARC0934721, DMS-0940249, and DMS-1413454. We are also
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