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 ofatomic 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 com-posites. 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 univer-sality is observed as a percolation threshold is approached, with eigenvalue statistics shifting fromweakly 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 spectralgaps and associated critical behavior near percolation thresholds. The spectral transition is exploredin 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 interactingcomponents are ubiquitous in the physical and biologi-cal sciences, and have a broad range of applications. Asthe number of components increases, structure begins toemerge from the underlying randomness of the system,and universal behavior can arise. Random matrix theory(RMT) has been quite successful in modeling universalfeatures found in statistical fluctuations of characteris-tic quantities of complex systems, such as atomic nu-clei [1], biological networks [2], financial time series [3],the departure times for bus systems [4], quantum systemswhose classical counterparts are chaotic [1, 5], and meso-scopic conductors [6]. Remarkably, it has been shownthat even the (non-random) zeros of the famous Rie-mann zeta function have universal features which arevery accurately captured by RMT [5, 7, 8]. Each ofthese systems can be described in terms of discrete “spec-tra” confined to a line. In RMT, the spectra of suchsystems are modeled by the eigenvalues of ensembles oflarge random matrices called random matrix ensembles(RME’s). Long and short range correlations of these ran-dom eigenvalues are measured in terms of various eigen-value statistics introduced by Dyson and Mehta [8]. Thelocalization properties of the associated eigenvectors arecharacterized by quantities such as the inverse partici-pation ratio (IPR) [3] and the Shannon information en-tropy [9]. In many systems, the spectral statistics areparameter independent when properly scaled [1, 10] and

∗Electronic address: golden@math.utah.edu

fall into two universal categories: uncorrelated Poissonstatistics [1] and highly correlated Wigner–Dyson (WD)statistics [1, 11]. The statistical behavior of the spectrumis related to the extent that the eigenfunctions overlap.A key example is the metal-insulator Anderson transi-tion exhibited by noninteracting electrons in a randompotential [10, 12]. For small disorder, the electron wavefunctions are extended structures. Their overlap givesrise to correlated WD energy level statistics with stronglevel repulsion [12]. However, for large disorder, elec-trons are typically localized at different points in thesample and “do not talk to each other,” resulting in un-correlated Poisson level statistics [12]. During transitionsfrom small to large disorder, the wave functions becomeincreasingly localized and an intermediate Poisson-likebehavior of level statistics arises [6, 10]. There are othersystems which undergo an analogous spectral transitionas a system parameter varies. Among them are: the hy-drogen atom in a magnetic field [13], random points ona fractal set [14], quantum chaos [15], and complex net-works [9].

We demonstrate here that transitions in the connected-ness or percolation properties of macroscopic compositemedia, with microstructural scales spanning many ordersof magnitude, can also be characterized by a transitionin the statistics of eigenvalues and the delocalization ofeigenvectors of a random matrix. This is a new type ofrandom matrix within RMT that depends only on thegeometry of the composite medium, and not directly ona probability distribution as usual. While the connected-ness driven transition in the statistical behavior of eigen-values is analogous to that of the Anderson transition andother systems, the delocalization of eigenvectors revealsnew subtleties that distinguish the behavior we see fromclassical Anderson localization. This large family of ran-dom matrices arises in the analytic continuation method(ACM) [16–19] for representing transport in composites.The method provides Stieltjes integral representations forthe bulk transport coefficients of a two-component ran-

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FIG. 1: Connectedness transitions in composite struc-tures. (A)–(E) Increasingly connected composites from leftto right. (A) Realizations of the two-dimensional lattice per-colation model, with (black) bond probabilities p = 0.20,p = 0.30, and p = pc = 0.5. (B) cross-sections from X-ray CTvolume renderings of the brine phase within a lab-grown seaice single crystal (H. Eicken), with image brine area fractionsof φ = 0.20, φ = 0.51, and φ = 0.70. (C) Melt ponds on thesurface of Arctic sea ice (D. K. Perovich), with area fractionsφ = 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), withcross-sectional area fractions φ = 0.26 and φ = 0.55.

dom medium, such as the effective electrical conductivity

σ∗ of a medium immersed in an electric field ~E, involvinga spectral measure µ of the random matrix [18, 20]. Themeasure µ exhibits fascinating transitional behavior asa function of system connectivity, which controls criticalbehavior of σ∗ near connectedness thresholds. For ex-ample, 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 spec-tral endpoints λ = 0, 1 [21]. However, as p approachesthe percolation threshold pc [22, 23] and the system be-comes increasingly connected, these spectral gaps shrinkand then vanish [21, 24], leading to the formation of δ-components of µ at the spectral endpoints, precisely [21]when p = pc and p = 1 − pc. This leads to criticalbehavior of σ∗ for insulating/conducting and conduct-ing/superconducting systems [21]. This gap behavior ofµ has led [21] to a detailed description of these criti-cal transitions in σ∗, which is directly analogous to theLee–Yang–Ruelle–Baker description [25, 26] of the Isingmodel phase transition in a ferromagnet’s magnetizationM . Moreover, using this gap behavior, all of the classicalcritical exponent scaling relations were recovered [21, 26]without heuristic scaling forms but instead by using therigorous integral representation for σ∗ involving µ.

Our results here reveal a mechanism for the collapsein the spectral gaps of µ and illustrate that localized andextended eigenvectors of the matrix are in direct corre-

spondence with components of the electric field ~E thatare localized in, and extended throughout the compos-ite medium. In particular, we demonstrate that eigen-values associated with a disordered state, such as a lowvolume fraction RRN, are weakly correlated and well-described by Poisson-like statistics. However, as thepercolation threshold pc is approached and the systemdevelops long range order, the eigenvalues become in-creasingly correlated and their statistics approach classi-cal WD statistics, causing the eigenvalues to spread outdue to increased level repulsion, subsequently forming δ-components in µ at the spectral endpoints. Correspond-ingly, the eigenvectors become increasingly extended andthose associated with these δ-components are typicallyhighly extended. These regions of extended states areseparated from each other by “mobility edges” [1] of lo-

calized states. A resolvent representation of ~E involvingthe random matrix provides a one-to-one correspondencebetween localized (extended) eigenvectors and localized(extended) components of the electric field within themedium.

We show that this spectral behavior emerges in a va-riety of composite systems, such as the brine microstruc-ture of sea ice [27–29], melt ponds on the surface of Arc-tic sea ice [30], the sea ice pack itself, and porous humanbone [31]. Our results indicate that it is pervasive insuch macroscopic systems and arises simply from con-nectedness – at the most basic level of characterizing anyphysical system with inhomogeneities.

The behavior of composite materials exhibiting a crit-ical transition as system parameters are varied is partic-ularly challenging to describe physically, and to predictmathematically. Here, we discuss composites which ex-hibit critical behavior in transport properties induced bytransitions in connectedness or percolation properties ofa particular material phase.

Lattice and continuum percolation models have beenused to study a broad range of disordered materi-

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als [22, 23]. In the simplest case of the two-dimensionalsquare lattice [22, 23], as shown in Fig. 1A, the bondsare open with probability p and closed with probabil-ity 1 − p. Connected sets of open bonds are called openclusters. The average cluster size grows as p increases.When the system size L tends to infinity, there is a crit-ical probability pc, 0 < pc < 1, called the percolationthreshold, where an infinite cluster of open bonds firstappears. In dimension d = 2, pc = 1/2, and in d = 3,pc ≈ 0.2488 [22]. Now consider transport through theassociated RRN, where the bonds are assigned electricalconductivities σ1 with probability p and σ2 with proba-bility 1 − p. The effective conductivity σ∗ exhibits twotypes of critical behavior. First, when 0 < σ1 < ∞ andσ2 → 0, the system is insulating, σ∗ = 0, for p < pc andis conducting, σ∗ > 0, for p > pc . Second, when σ1 →∞and 0 < σ2 <∞, the conductive system becomes super-conducting, σ∗ → +∞, as p→ p−c .

Sea ice is a complex composite consisting of pure icewith sub-millimeter scale brine inclusions, as shown inFig. 1B, whose volume fraction φ, geometry, and con-nectedness vary significantly with temperature T . Thebrine microstructure displays a percolation threshold ata critical brine volume fraction φc ≈ 5% in columnarsea ice [27], which corresponds to a critical temperatureTc ≈ −5◦ C for a typical bulk salinity of 5 ppt. Thisthreshold acts as an on–off switch for fluid flow throughsea ice, and is known as the rule of fives. It leads tocritical behavior of fluid flow, where sea ice is effectivelyimpermeable to fluid transport for φ < φc , yet is per-meable for φ > φc, with the permeability as a functionof φ above the 5% threshold described by the universalcritical exponent for lattices in three dimensions [27–29].

Fluid flow through sea ice mediates a broad range ofphysical and biological processes in the polar marine en-vironment [28, 29], including brine drainage, nutrient re-plenishment for algal communities in the brine phase,snow-ice formation, and the evolution of melt ponds onthe surface of Arctic sea ice [30]. Melt ponds (Fig. 1C)determine sea ice albedo in the Arctic, a key parameterin climate modeling. Despite its importance, it remainsa major source of uncertainty in climate models. In fact,the lack of inclusion of melt ponds in previous genera-tions of climate models is believed to partially accountfor the inadequacy of these models to predict the dra-matic rate of melting of the summer Arctic sea ice pack.The results here advance our understanding of the effec-tive or homogenized properties of the ice pack, and helpprovide a path toward more rigorously incorporating seaice into climate models.

Human bone also displays a complex, porous mi-crostructure, as shown in Fig. 1E. The strength of boneand its ability to resist fracture depend strongly on thequality of the connectedness of the hard, solid phase.Osteoporotic trabecular bone can become more discon-nected and remaining connections can become more tenu-ous or fragile [31]. The spectral techniques [32, 33] whichhave arisen from the ACM provide important methods

for analyzing microstructural transitions in bone and itsbiophysical properties.

I. MATHEMATICAL METHODS

Random matrices arise naturally in the ACM for rep-resenting transport in composites [18, 20]. This methodprovides Stieltjes integral representations for the effectiveparameters of two-component composite media, such aselectrical conductivity and permittivity, magnetic per-meability, and thermal conductivity. The integral repre-sentations involve a spectral measure µ associated witha family of random matrices, which depend only on thecomposite geometry [20]. A remarkable feature of thismethod is that once the spectral measure is found for agiven composite geometry, by spectral coupling of thegoverning equations [34], the effective electrical, mag-netic, and thermal transport properties are all deter-mined by µ.

Consider the effective parameter problem for two-component conductive media [18, 20]. The electromag-netic transport properties of the composite are governedby the quasi-static limit [18] of Maxwell’s equations

~∇× ~E = 0, ~∇ · ~J = 0. (1)

Here, ~E and ~J are the random electric field and current

density, which are related by ~J = σ ~E, and σ denotes theelectrical conductivity of the locally isotropic, station-ary random medium. In the case of a two-componentmedium with (complex-valued) component conductivi-ties σ1 and σ2 we write

σ = σ1χ1 + σ2χ2, (2)

where χ1 is the characteristic function of medium one,taking the value 1 in medium one and 0 otherwise, withχ2 = 1− χ1.

The effective conductivity matrix σ∗ is defined by [18]

〈 ~J 〉 = σ∗〈 ~E〉, 〈 ~E〉 = ~E0. (3)

Here, 〈·〉 denotes ensemble averaging and the vector ~E0 =E0~ek has magnitude E0 and direction ~ek, taken to be thekth standard basis vector for some k = 1, . . . , d, which de-fines the d-dimensional coordinate system. Equivalently,the effective conductivity σ∗ may be defined [18, 20]

in terms of the energy (power) density : 〈 ~J · ~E〉 =

σ∗ ~E0 · ~E0 = σ∗kkE

20 . For simplicity, we focus on the diag-

onal coefficient σ∗kk of the matrix σ∗ and set σ∗ = σ∗

kk.The key step in the method is obtaining the followingStieltjes integral representation for σ∗ [16–18]

σ∗ = σ2(1− F (s)), F (s) =

∫ 1

0

dµ(λ)

s− λ, (4)

which follows from a resolvent representation of the elec-tric field (in material phase 1) [20]

χ1~E = sE0(sI − χ1Γχ1)−1χ1~ek. (5)

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FIG. 2: The projection matrix Γ. The matrix Γ with pe-riodic boundary conditions for (A) 2D and (B) 3D networkswith system sizes L = 6 and L = 3, respectively, with corre-sponding matrix size N = Ldd. Notice that the lengths of therepeated structures are multiples of the system size L.

Here, s = 1/(1 − σ1/σ2), −F (s) plays the role of theeffective electric susceptibility, µ is a spectral measureassociated with the random operator χ1Γχ1 [18], and

Γ = −~∇(−∆)−1~∇· is a projection onto curl-free fields,based on convolution with the free-space Green’s functionfor the Laplacian ∆ = ∇2.

In this way, the ACM determines a homogeneousmedium which behaves macroscopically and energeticallylike a given inhomogeneous medium. Moreover, param-eter information in s and E0 is separated from the geo-metric complexity of the system, which is encoded in thespectral measure µ. Geometric information about thecomposite is incorporated into the positive Stieltjes mea-

sure µ in equation (4) via its moments, µn =∫ 1

0λndµ(λ).

For example, the mass µ0 of the measure is the volumefraction p of medium one, i.e., µ0 = 〈χ1〉 = p. We maythink of the measure dµ(λ) as m(λ) dλ for some densitym(λ) which is allowed to have δ-components. This char-acterization of µ is more transparent in the setting of afinite RRN, where the random operator χ1Γχ1 can berepresented by a real-symmetric random matrix of sizeN × N , where N = Ldd [20]. In this case, χ1 is a diag-onal matrix with 1’s and 0’s along the diagonal, corre-sponding to conductive bonds with conductivity σ1 andσ2, respectively, and Γ is a (non-random) projection ma-trix [20]. In this case, the spectral measure µ can be cal-culated directly from the eigenvalues λj , j = 1, . . . , N ,and eigenvectors ~vj of the matrix χ1Γχ1, and is given bya weighted sum of δ-measures

dµ(λ) =

N∑j=1

〈mj δ(λ− λj) 〉

dλ = m(λ) dλ, (6)

where 〈·〉 denotes ensemble averaging, mj = [~vj · ek]2,and ek is a lattice basis vector [20]. The matrix Γ en-capsulates the lattice topology and transport characteris-tics of the resistor network and displays a rich geometric,banded structure, as shown in Fig. 2.

The action of the matrix χ1 in χ1Γχ1 is to randomlyzero out each row and column of Γ that corresponds todiagonal components of χ1 satisfying [χ1]jj = 0. Thespectral weights mj associated with this null space of

χ1Γχ1 are identically zero [20] and do not contribute tothe sum in (6). The only eigenvectors that contributeto this sum are associated with random N1 × N1 sub-matrices 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/iceand bone/marrow representations (the ice pack images in(D) are the converted versions). The geometry and thecomponent connectivity of these binary images can be de-scribed in terms of high density two-component resistornetworks, with fluid and bone corresponding to compo-nent one, and ice and marrow corresponding to compo-nent two. In this way, the matrix χ1Γχ1 was obtainedfor these composite structures. In the next section, weshow that its associated eigenvalues and eigenvectors un-dergo a connectedness driven transition that is analogousto that of the Anderson transition [1, 10, 12].

II. RESULTS

In RMT, long and short range correlations of eigen-values in the bulk of the spectrum [10] are measured interms of various eigenvalue statistics. In order for thefluctuation properties of the eigenvalues about the meandensity ρ(λ) to be compared to the predictions of RMT,the spectrum has to be unfolded [1, 3, 10]. It is non-trivial to unfold spectra associated with the RRN forsmall volume fractions p, due to prominent “geometric”resonances in ρ(λ) [20, 24], which has limited our analysisof these systems in the dilute limit, p� 1.

The nearest neighbor eigenvalue spacing distribution(ESD) P (z) is the observable most commonly used tostudy short-range correlations [1]. For highly corre-lated WD spectra, such as that exhibited by the real-symmetric matrices of the Gaussian orthogonal ensem-ble (GOE) [1, 11], the ESD is accurately approximatedby P (z) ≈ (πz/2) exp(−πz2/2), known as Wigner’s sur-mise [1, 6], which illustrates the phenomenon of eigen-value repulsion, vanishing linearly in the limit of smallspacings z. In contrast, the ESD for uncorrelated Pois-son spectra, P (z) = exp(−z), allows for level degeneracy.

In Fig. 3 we display the behavior of the ESD foreigenvalues of the matrix χ1Γχ1, which correspond toRRN, melt pond, and Arctic sea ice pack compositestructures displayed in Fig. 1. This figure demonstratesthat for sparsely connected systems, the behavior of theESD’s are well described by weakly correlated Poisson-like statistics [10]. The ESD’s increase linearly from zeroat short separation but the initial slope is steeper, imply-ing less level repulsion. As the systems become increas-ingly connected, and long range order is established, theESD’s transition toward highly correlated WD statisticswith high level repulsion. The blue dash-dot curve dis-played in Fig. 3 is the ESD for Poisson spectra, while thegreen dashed curve is the ESD for the GOE. For the 2Dand 3D RRN, the eigenvalue density ρ(λ, p) displays the

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p=0.2488p=0.4p=0.5p=0.6p=0.7512

p=0.3p=0.7

p=0.5

p=0.09 p=0.27 p=0.57

p=0.06 p=0.10 p=0.47

3D random resistor network

2D random resistor network

Arctic melt ponds

Arctic sea ice packP

(z)

P(z

)P

(z)

P(z

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z0 1 2 3 4 0 1 2 3 4 z0 1 2 3 4

z0 1 2 3 4 z0 1 2 3 4 z0 1 2 3 4

z0 1 2 3 4 z0 1 2 3 4 z0 1 2 3 4

z0 1 2 3 4 z0 1 2 3 4 z0 1 2 3 4

p=0.1p=0.9

p=0.05p=0.95

p=0.13p=0.87

FIG. 3: Short-range eigenvalue correlations. The eigen-value spacing distributions (ESD’s) for Poisson (blue dash-dot) and Wigner–Dyson (WD) (green dashed) spectra are dis-played in (A)–(D), along with that of (A) the Arctic sea icepack, (B) Arctic melt ponds, (C) 2D random resistor network(RRN), and (D) 3D RRN. As the systems become increas-ingly connected, the initial slopes of the ESD’s progressivelydecrease, indicating an increase in level repulsion and short-range correlations.

symmetry ρ(λ, p) = ρ(1 − λ, 1 − p) in the bulk of thespectrum. This is reflected in the ESD’s by the sym-metry P (z, p) = P (z, 1 − p), as shown for the 2D and3D RRN in Fig. 3C and Fig. 3D. The ESD’s displayed inFig. 3D for the 3-dimensional percolation model suggestthat the GOE limit is attained for all pc ≤ p ≤ 1− pc.

The ESD contains information about the spectrumwhich involves short scales (a few mean spacings) [1, 10].Long-range correlations are measured by quantities suchas the eigenvalue number variance Σ2(L ), in intervals oflength L (not to be confused with the system size L),and the closely related spectral rigidity ∆3(L ) [1]. Foruncorrelated Poisson spectra, these long range statisticsare linear, with Σ2(L ) = L and ∆3(L ) = L/15, as dis-played in blue color with dash-dot line style in Fig. 4. Incontrast, the strong correlations of WD spectra make thespectrum more rigid [10] so that Σ2(L ) and ∆3(L ) growonly logarithmically [1, 8]. The green dashed curves ofFig. 4 are numerical computations of the exact solutionsof Σ2(L ) and ∆3(L ) for WD spectrum [1].

In Fig. 4, we also display the behavior of these long-range eigenvalue statistics for the matrix χ1Γχ1, corre-

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φ=0.51φ=0.20

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Σ2(L)

∆3(L)

L0 10 20 30 L0 10 20 30L0 10 20 30

L0 10 20 30 L0 10 20 30L0 10 20 30

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p=0.13p=0.05

p=0.2488p=0.5p=0.7512p=0.87p=0.95

FIG. 4: Long-range eigenvalue correlations. (A) Theeigenvalue number variance Σ2(L ) and (B) the spectral rigid-ity ∆3(L ) for Poisson (blue dash-dot) and Wigner–Dyson(WD) (green dash) spectra are displayed along with thoseof sea ice brine inclusions, Arctic melt ponds and pack ice,trabecular bone microstructure and random resistor networks(RRN). As the systems become increasingly connected, theselong-range eigenvalue statistics transition from the linearPoisson-like behavior toward a logarithmic WD behavior.

sponding to the composite structures displayed in Fig. 1.For sparsely connected systems, these statistics exhibitlinear Poisson-like behavior away from the origin withslope less than their Poisson counterparts. This lin-ear behavior has been attributed to exponentially de-caying correlations of eigenvalues [10]. These statisticstransition with increasing connectedness toward loga-rithmic WD behavior typical of the GOE, which hasquadratically decaying eigenvalue correlations [10]. Sim-ilar to the ESD’s in Fig. 3, for the RRN these statisticsalso 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 GOElimit is attained by the short and long range statisticsfor the 3-dimensional RRN for all pc ≤ p ≤ 1− pc. Theyappear to overlie the GOE limit almost exactly for all pvalues tested in the range pc ≤ p ≤ 1 − pc. With thisin mind, we can paint a heuristic analogy with Andersonlocalization, where low disorder corresponds to extendedstates and WD statistics. When disorder exceeds a crit-ical level, the states localize and the eigenvalues becomede-correlated. We view the 3D RRN with pc ≤ p ≤ 1−pcto 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) andthe eigenvalue repulsion diminishes.

The eigenvectors ~vn, n = 1, . . . , N1, associated withthe random N1 ×N1 sub-matrices of Γ, discussed above,also undergo a connectedness driven transition in theirlocalization properties. Two commonly used quantitieswhich measure the localization of the eigenvector ~vn arethe inverse participation ratio (IPR) In [3] and the Shan-

6

non entropy Sn [9], defined as

In =

N1∑j=1

[vjn]4, Sn = −N1∑j=1

[vjn]2 ln[vjn]2, (7)

where vjn is the jth component of the eigenvector ~vn.Eigenvectors of matrices in the GOE are known to behighly extended and independent of the distribution ofthe eigenvalues [35]. In this case, the IPR and entropy aregiven by IGOE = 3/N1 [3] and SGOE ≈ ln(N1/2.07) [9],respectively. Associated with the Sn is the eigenvectorlocalization length ` defined as

` = N1 exp[−(SGOE − 〈S〉)], (8)

where we denote by 〈S〉 and 〈I〉 ensemble averaging overall values of the Sn and In. The meaning of In and Sn

can be illustrated by two limiting cases (i) a normalizedvector with only one component vjn = 1 has In = 1 andSn = 0, whereas (ii) a vector with identical componentsvjn = 1/

√N1 has In = 1/N1 and Sn = lnN1. 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 〈S〉 = SGOE

then ` = N1.In the matrix setting, the electric field in equation (5)

has the following eigenvector expansion

χ1~E = sE0

N1∑n=1

[(s− λn)−1(~vn · χ1~ek)] ~vn. (9)

This provides a direct link between localized eigenvectors

~vn and eigen-modes of χ1~E that have significant magni-

tudes in only a few resistors, while extended eigenvec-tors correspond to electric field components that extendthroughout the network. Displayed in Fig. 5A, from left

to right, is the total electric field χ1~E and examples of

localized and extended eigen-modes of χ1~E for a random

realization of the 2D RRN with p = pc.

III. DISCUSSION

Our results displayed in Fig. 5B–D show that the eigen-vectors ~vn associated with 2D and 3D RRN undergo afascinating delocalization as p increases and the systembecomes increasingly connected. For example, Fig. 5Bdisplays the IPR In of the 3D RRN as a function of theindex n of the eigenvector ~vn, with increasing index cor-responding to increasing eigenvalue magnitude λn. Thisfigure shows that for p � pc, the eigenvectors are typi-cally localized, with values In of IPR quite different fromthat of the GOE, identified by the red horizontal lines.Moreover, they have an oscillatory behavior that, whenplotted as a function of λn, follows the peaks and val-leys of “geometric” resonances exhibited by the eigen-value density ρ(λ) for small p [20, 24], with more local-ized regions corresponding to lower density. This indi-cates that there are significant correlations between

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Log Electric Field Localized Mode Extended Mode

2D ⟨I⟩

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I nFIG. 5: Delocalization of eigenvectors. (A) Displayed

from left to right are the full electric field χ1~E (in log scale)

and examples of localized and extended eigen-modes of χ1~E

(in linear scale) for a realization of the 2D random resistornetwork (RRN), with a system size L = 50 and volume frac-tion p = pc = 1/2. The corresponding localized and ex-tended eigenvectors have inverse participation ratios (IPR’s)In ≈ 0.081 and In ≈ 0.002, and the DC values of the compo-nent 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’sof eigenvectors ~vn associated with realizations of the 3D RRNplotted versus eigenvector index n, with L = 12 and increas-ing values of p from left to right, with 1 − pc ≈ 0.7512. Thevertical lines define the δ-components of the spectral measureµ at the left and right spectral endpoints, where the associ-ated eigenvalues λn satisfy λn . 10−14 and λn & 1 − 10−14,respectively, while the horizontal lines mark the IPR valueIGOE = 3/N1 for the Gaussian orthogonal ensemble (GOE)with matrix size N1 ≈ pN , where N = Ldd. (C) The en-semble averaged IPR 〈I〉 as a function of p, displaying transi-tional behavior at the percolation thresholds, pc = 1/2 for2D and pc ≈ 0.2488 for 3D. (D) The normalized, ensem-ble averaged entropy 〈S〉/SGOE and associated localizationlength `/`GOE as a function of p, where SGOE ≈ ln(N1/2.07)and `GOE = N1. Panels (B)–(D) demonstrate that, as p in-creases and the systems become increasingly connected, theeigenvectors become progressively extended, on average, withdecreasing 〈I〉 and increasing 〈S〉/SGOE and `/`GOE .

the eigenvalues and eigenvectors, in contrast withthe GOE. As p → p−c , gaps in the spectrum about thespectral endpoints shrink and then vanish [20, 24], whilethe values In of the IPR continually decrease, approach-ing the GOE limit, as shown in Fig. 5B.

As p surpasses pc and 1−pc, δ-components form in thespectral measure µ at λ = 0 and λ = 1, respectively [21].Numerically, the δ-component at λ = 0 manifests itself as

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a large number of eigenvalues with magnitude . 10−14,followed by an abrupt change of magnitude & 10−4, withno eigenvalues in the interval (10−14, 10−4), and similarlyfor λ = 1. The locations of these changes in magnitudeare identified by red vertical lines in Fig. 5B, demon-strating that the eigenvectors associated with these δ-components are typically more extended than the others,with values In of the IPR closer to the GOE limit. Theentropy Sn has a similar behavior.

In Fig. 5C and 5D the p dependence of 〈I〉, 〈S〉/SGOE ,and `/`GOE are displayed. As p increases and the systembecomes increasingly connected, 〈I〉 decreases, with tran-sitional behavior at the percolation threshold pc, while〈S〉/SGOE and `/`GOE increase. Each indicate thatthe eigenvectors, hence the eigen-modes of the electric

field χ1~E become progressively extended throughout the

network with increasing system connectedness. Fig. 5Bshows that this average delocalization is largely due tothe formation of the δ-components in µ.

Fig. 5B also shows that regions of extended states areseparated from one another by a series of “mobilityedges” with a sudden increase in the number of local-ized eigenvectors. This behavior in the eigenvectors ofthe random matrix χ1Γχ1 is analogous to that of Ander-son localization, where mobility edges mark the charac-teristic energies of the metal/insulator transition [1]. Re-markably, the mobility edges in Fig. 5B are precisely atthe locations of the δ-components, identified by red ver-tical lines, which control critical behavior of transport ininsulator/conductor and conductor/superconductor sys-tems. However, the delocalization behavior of the spec-tral endpoints displayed in Fig. 5B is different from thatof Anderson localization [1], further demonstrating thatthe behavior we see in χ1Γχ1 is new to RMT.

We have demonstrated that the eigenvalues and eigen-

vectors associated with the random matrix χ1Γχ1, corre-sponding to various composite structures, have a statisti-cal behavior that is analogous to, but distinctly differentfrom that of the Anderson transition. The eigenvaluesof χ1Γχ1 shift from weakly correlated Poisson-like statis-tics toward highly correlated WD statistics as a functionof disorder (connectedness). Correspondingly, the eigen-vectors undergo a delocalization, with highly extendedstates appearing at the spectral edges, separated by mo-bility edges of localized states. Moreover, the delocal-ization of eigenvectors corresponds to a delocalization of

the transport field, such as the electric field ~E, extendingthroughout the composite medium near global connect-edness thresholds. The disorder driven transition in thestatistical behavior of the eigenvalues also accounts forthe gap behavior of the spectral measure µ, underlyingexact integral representations for the effective transportcoefficients of the composite medium, such as effectiveconductivity σ∗, which, in turn, governs their critical be-havior via the formation of δ-components of the measureat the spectral endpoints. This provides a novel way ofunderstanding and characterizing disorder driven transi-tions in the effective transport properties of compositemedia, and opens a new chapter in the application ofRMT to the analysis of complex, macroscopic systems.

Acknowledgments We gratefully acknowledge sup-port from the Division of Mathematical Sciences and theDivision of Polar Programs at the U.S. National ScienceFoundation (NSF) through Grants DMS-1009704, ARC-0934721, DMS-0940249, and DMS-1413454. We are alsograteful for support from the Office of Naval Research(ONR) through Grants N00014-13-10291 and N00014-12-10861. Finally, we would like to thank the NSF MathClimate Research Network (MCRN) for their support ofthis work.

[1] T. Guhr, A. M uller-Groeling, and H. A. Weidenmuller.Random-matrix theories in quandom physics: Commonconcepts. Phys. Rept., 299(4):189–425, 1998.

[2] F. Luo, Y. Yang, J. Zhong, H. Gao, L. Khan, D. Thomp-son, and J. Zhou. Constructing gene co-expression net-works and predicting functions of unknown genes by ran-dom matrix theory. BMC Bioinformatics, 8(1), 2007.

[3] V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Ama-ral, T. Guhr, and H. E. Stanley. Random matrix ap-proach to cross correlations in financial data. Phys. Rev.E, 65:066126, Jun 2002.

[4] M. Krbalek and P. Seba. The statistical properties of thecity transport in cuernavaca (mexico) and random matrixensembles. J. Phys. A: Math. Gen., 33:L229?L23, 2000.

[5] T. Kriecherbauer, J. Marklof, and A. Soshnikov. Randommatrices and quantum chaos. Proceedings of the NationalAcademy of Sciences, 98(19):10531–10532, 2001.

[6] A. D. Stone, P. A. Mello, K. A. Muttalib, and J. L.Pichard. Random Matrix Theory and Maximum EntropyModels for Disordered Conductors, chapter 9, pages 369–448. Elsevier Science Publishers, Amsterdam, Nether-

lands, 1991.[7] F. Mezzadri and N. C. Snaith. Recent Perspectives in

Random Matrix Theory and Number Theory. CambridgeUniversity Press, Cambridge, UK, 2005.

[8] M. L. Mehta. Random Matrices. Elsevier, Amsterdam,third edition, 2004.

[9] J. A. Mendez-Bermudez, A. Alcazar-Lopez, A. J.Martınez-Mendoza, Francisco A. Rodrigues, and ThomasK. DM. Peron. Universality in the spectral and eigen-function properties of random networks. Phys. Rev. E,91:032122, Mar 2015.

[10] C. M. Canali. Model for a random-matrix descriptionof the energy-level statistics of disordered systems atthe Anderson transition. Phys. Rev. B, 53(7):3713–3730,1996.

[11] P. Deift. Random Matrix Theory: Invariant Ensemblesand Universality. Courant Institute of Mathematical Sci-ences, New York, NY, 2009.

[12] V. E. Kravtsov and K. A. Muttalib. New class of randommatrix ensembles with multifractal eigenvectors. Phys.Rev. Lett., 79(10):1913–1916, 1997.

8

[13] D. Delande. Les Houches, ecole d’ete de physiquetheorique. In M. J. Giannoni, A. Voros, and J. Zinn-Justin, editors, Chaos et Physique Quantique—Chaosand Quantum Physics = Les Houches, session LII, 1–31aout 1989, pages 665–726. North-Holland, Amsterdam,1991, 1991.

[14] J. Sakhr and J. M. Nieminen. Poisson-to-Wignercrossover transition in the nearest-neighbor statistics ofrandom points on fractals. Phys. Rev. E, 72:045204, Oct2005.

[15] T. H. Seligman, J. J. M. Verbaarschot, and M. R. Zirn-bauer. Spectral fluctuation properties of Hamiltoniansystems: the transition region between order and chaos.J. Phys. A: Math. Gen., 18:2751–2770, 1985.

[16] D. J. Bergman. Exactly solvable microscopic geometriesand rigorous bounds for the complex dielectric constantof a two-component composite material. Phys. Rev. Lett.,44:1285–1287, 1980.

[17] G. W. Milton. Bounds on the complex dielectric constantof a composite material. Appl. Phys. Lett., 37:300–302,1980.

[18] K. M. Golden and G. Papanicolaou. Bounds for effectiveparameters of heterogeneous media by analytic continu-ation. Commun. Math. Phys., 90:473–491, 1983.

[19] G. W. Milton. Theory of Composites. Cambridge Uni-versity Press, Cambridge, 2002.

[20] N. B. Murphy, E. Cherkaev, C. Hohenegger, and K. M.Golden. Spectral measure computations for compositematerials. Commun. Math. Sci., 13(4):825–862, 2015.

[21] N. B. Murphy and K. M. Golden. The Ising model andcritical behavior of transport in binary composite media.J. Math. Phys., 53:063506 (25pp.), 2012.

[22] D. Stauffer and A. Aharony. Introduction to Percola-tion Theory. Taylor and Francis, London, second edition,1992.

[23] S. Torquato. Random Heterogeneous Materials: Mi-crostructure and Macroscopic Properties. Springer-Verlag, New York, 2002.

[24] T. Jonckheere and J. M. Luck. Dielectric resonancesof binary random networks. J. Phys. A: Math. Gen.,31:3687–3717, 1998.

[25] G. A. Baker. Quantitative Theory of Critical Phenomena.Academic Press, New York, 1990.

[26] K. M. Golden. Critical behavior of transport in latticeand continuum percolation models. Phys. Rev. Lett.,78(20):3935–3938, 1997.

[27] K. M. Golden, S. F. Ackley, and V. I. Lytle. The percola-tion phase transition in sea ice. Science, 282:2238–2241,1998.

[28] K. M. Golden, H. Eicken, A. L. Heaton, J. Miner,D. Pringle, and J. Zhu. Thermal evolution of permeabil-ity and microstructure in sea ice. Geophys. Res. Lett.,34:L16501 (6 pages and issue cover), 2007.

[29] K. M. Golden. Climate change and the mathemat-ics of transport in sea ice. Notices Amer. Math. Soc.,56(5):562–584 and issue cover, 2009.

[30] C. Hohenegger, B. Alali, K. R. Steffen, D. K. Perovich,and K. M. Golden. Transition in the fractal geometryof Arctic melt ponds. The Cryosphere, 6(5):1157–1162,2012.

[31] J. Kabel, A. Odgaard, B. van Rietbergen, and R. Huiskes.Connectivity and the elastic properties of cancellousbone. Bone, 24(2):115–120, 1999.

[32] K. M. Golden, N. B. Murphy, and E. Cherkaev. Spectralanalysis and connectivity of porous microstructures inbone. J. Biomech., 44(2):337–344, 2011.

[33] E. Cherkaev and C. Bonifasi-Lista. Characterization ofstructure and properties of bone by spectral measuremethod. J. Biomech., 44(2):345–351, 2011.

[34] E. Cherkaev. Inverse homogenization for evaluation ofeffective properties of a mixture. Inverse Problems,17(4):1203–1218, 2001.

[35] S. N. Evangelou. A numerical study of sparse randommatrices. J. Stat. Phys., 69(1-2):361–383, 1992.