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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(RMEs). 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 WignerDyson (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) [1619] for representing transport in composites.The method provides Stieltjes integral representations forthe bulk transport coefficients of a two-component ran-







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 = 0.20, = 0.51, and = 0.70. (C) Melt ponds on thesurface of Arctic sea ice (D. K. Perovich), with area fractions = 0.09, = 0.27, and = 0.57. (D) Arctic sea ice pack(D. K. Perovich), with open ocean area fractions = 0.06, = 0.10, and = 0.47. (E) SEM images of osteoporotic(left) and healthy (right) trabecular bone (P. Hansma), withcross-sectional area fractions = 0.26 and = 0.55.

dom medium, such as the effective electrical conductivity

of a medium immersed in an electric field ~E, involvinga spectral measure of the random matrix [18, 20]. Themeasure exhibits fascinating transitional behavior asa function of system connectivity, which controls criticalbehavior of 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 = 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 -components of at the spectral endpoints, precisely [21]when p = pc and p = 1 pc. This leads to criticalbehavior of for insulating/conducting and conduct-ing/superconducting systems [21]. This gap behavior of has led [21] to a detailed description of these criti-cal transitions in , which is directly analogous to theLeeYangRuelleBaker description [25, 26] of the Isingmodel phase transition in a ferromagnets 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 involving .

Our results here reveal a mechanism for the collapsein the spectral gaps of 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 -components in at the spectral endpoints. Correspond-ingly, the eigenvectors become increasingly extended andthose associated with these -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 [2729], 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-


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 a