On random symmetric matrices with a constraint: the spectral density of random impedance networks

4 Pages·0.2 MB·English

by  J. Staering

On random symmetric matrices with a constraint: the spectral density of random
impedance networks
J. St¨aring and B. Mehlig
School of Physics and Engineering Physics, Gothenburg University/Chalmers, SE-41296 G¨oteborg, Sweden
Yan V. Fyodorov
3 Department of Mathematical Sciences, Brunel University, Uxbridge UB83PH, UK and
0
Petersburg Nuclear Physics Institute, Gatchina 188350, Russia
0
2
J.M. Luck

n Service de Physique Th´eorique (URA 2306 of CNRS),
a
CEA Saclay, 91191 Gif-sur-Yvette cedex, France
J

9 We derive the mean eigenvalue density for symmetric Gaussian random N N matrices in the
limitoflargeN,withaconstraintimplyingthattherowsumofmatrixelements×shouldvanish. The

] result is shown to be equivalent to a result found recently for the average density of resonances in
n
randomimpedancenetworks[Y.V.Fyodorov,J.Phys. A:Math. Gen. 32,7429(1999)]. Inthecase
n
of banded matrices, the analytical results are compared with those extracted from the numerical
-
s solution of Kirchhoff equations for quasi one-dimensional random impedance networks.
i
d
.
t The study of random matrices1 has provided insight of N N random matrices M with matrix elements
a
×
m into many physical problems, both in the quantum and
in the classical domain. For example, random matri- N
- M =J δ J , (1)
d ces have been very successfully used to model statistical mn mn− mn ml
n propertiesofdisorderedconductorsandofhighlyexcited Xl=1
o classicallychaoticquantumsystems2. Intheclassicaldo-
c where J is a real symmetric N N matrix with ran-
main random matrices arise, for instance, in the context ×
[ dom entries. Such ensembles have been considered in
ofdiffusioninrandom,directedenvironments(seeforex-
Refs. 6,7,10, and 11. In Ref. 10, M was used to model a
1 ample Ref. 3 and references therein). In most of those
random transition matrix for a model describing glassy
v
applicationstherandommatrixelementsobeyedsymme-
7 relaxation, ∂tu(t) = Mu(t), with matrix elements
try requirements where appropriate, and were otherwise −
2 Jmn distributedindependently(subjecttotheconstraint
taken to be independently distributed random variables.
1 J =J ) according to
mn nm
1 Therearecases,however,whereconstraintsonthema-
0 trix elements must be considered. Such constraints gen- p 1 p
03 erate correlations between the latter: examples are elec- P(Jmn)= N δ(cid:18)Jmn− p(cid:19)+(cid:16)1− N(cid:17) δ(Jmn). (2)
tron hopping in amorphous semiconductors (see Ref. 4
/
t and references cited therein), random impedance net- The form (1) yields M = 0, implying probability
a n mn
m works5,6,7 (see Ref. 8 for a review and Ref. 9 for ap- conservation in the pProblem considered in Ref. 10. The
plications), and random master equations10: in these average density d(λ) of eigenvalues λ,
-
d cases, the random matrices obey the constraints that
n the row sums of matrix elements should be zero. This d(λ)=N−1 trδ(M λ1) , (3)
h − i
o condition implies correlations between diagonal and off-
c diagonal matrix elements. In Refs. 4,6,10, the average was calculated in the limit of large N and p, using the
:
v spectral density and spectral fluctuations of three dif- method ofreplicas. is anaverageoverthe ensemble
h···i
Xi ferent random-matrix ensembles of this type were calcu- defined by P(Jmn).
lated,usingthemethodofreplicasandthesupersymme- In Ref. 6, eigenvalues of matrices M with elements
ar try approach. similarto(1)wereshowntomodelresonancefrequencies
in random impedance networks5, with J =J and
Inthispaperwederivetheaveragedensityofeigenval- mn nm
ues of a suitably modified ensemble of symmetric Gaus-
1 1
sian random matrices. The aim is twofold. First, we P(J )= δ J 1 + δ J +1 . (4)
mn mn mn
2 − 2
wish to show that for full matrices the density obtained (cid:0) (cid:1) (cid:0) (cid:1)
isequivalenttothatfoundinRefs.6and10. Oursecond In Refs. 6 and 7, the average density of resonance fre-
aim is to derive the correspondingresult for banded ma- quencies was calculated, in the limit of large N, using
trices,andtocompareitwiththeresultsofthenumerical a variant of the supersymmetry technique. It was found
solutionoftheKirchhoffequationsonrandomimpedance thattheresultagrees12withthatofRef.10,uptoascale
networks with quasi one-dimensional topology. factor (related to p) and a rigid shift in λ [related to the
Formulation of the problem. We consider an ensemble fact that J 0 in (2)].
mn
≥
2
1
In the following we calculate the ensemble-averaged
density of eigenvalues of M, treating both off-diagonal +
P(aS)frag rheMpliaj
Memjiien=ts
and diagonal entries of J as independent, identically
i j j i i i i i
distributed Gaussian real variables. The corresponding
tshymogmoneatrliecnrsaenmdbolme(mGaOtEri)x1JwbitehlojnoignsttporothbeabGilaituyssdiaennsoitry- (bP)SfragGr(je0jp)l(cid:17)a
e(cid:21)m(cid:0)1en=tsPSfjrag repla
eGmjjen=tsj (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) j
1
P(J)dJ exp trJ2 dJ. P(S5f)rag repla
ements
∝ (cid:18)−4σ2 (cid:19) (
)
j i i i i i j k j
The averagedensity of eigenvalues E of such matrices in
thelimitN 1isgivenbythesemi-circularlawd(E)=
(2πσ2N)−1√≫4σ2N E2 for E 2(σ2N)1/2 and zPeSrofrag repla
(edm)ents
otherwise13. In th−e followin|g |we≤ask how imposing a j j j j j
”constraint”(thattherowsumofmatrixelementsshould
be zero)modifies the meaneigenvaluedensity ofM with
PSfrag repla
(eem)ents
respectto thatofJ. Inthelimit oflargeN,the problem
j j j j j
may be solved using diagrammatic perturbation theory
(see for instance Ref. 3 and references cited therein), as FIG. 1: (a) Contraction hMijMjii: thedashed line carries a
factorof1/N andthewavylinecarriesafactorofunity. The
shown below.
second term contributes only when i=j. (b) Graphical rep-
Ourresultsmaybesummarisedasfollows. Inthelimit
of large N, the averaged eigenvalue density for M coin- reelesmenetnattioofnthoef aGv(je0jr)a≡geEre−s1olvaenndtGGj.j,(ct)heExjathmdpilaegoofnaaldimagartarmix
cides with the result derived in Refs. 6,7. In this limit,
contributingtoGjj. Internalindicesaresummedover(iand
correlationsbetweendiagonalandoff-diagonalmatrixel- k). (d,e) Diagrams of higher order in N−1.
ements are found to be irrelevant, and the result can be
1
understood in terms of an averaged Pastur equation14.
Furthermore,wehavealsoconsideredthe caseofbanded
symmetric random matrices. This case is of interest for (a) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) +(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
randomimpedance networkswith quasione-dimensional
topology5,11. Our results are in good agreement with
tthioonssefoofresxuacchtnneutmweorrikcsa.lsolutions ofthe Kirchhoffequa- + (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
Method. The average eigenvalue density may be ob-
tained from the trace of the averaged resolvent G =
h(E1−M)−1i, + (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)+. . .
d(E)= (Nπ)−1ImtrG. (6)
− PSfrag repla
(ebm)ents (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) = + (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Expanding the resolvent, Wick’s theorem may be em-
j j j j k j
ployed for performing the average,using
PSfrag(
r)epla
eHmjjen=ts (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
M M =σ2 (δ δ +δ δ ) (7) j j
ij mn im jn in jm
h i h
FIG. 2: (a,b) Self-consistent equations for the average of
δ (δ δ +δ δ ) δ (δ δ +δ δ )
− mn im jl il jm − ij im kn in km the resolvent. (c) Graphical representation of the auxiliary
+δijδmn(Nδim+1) . functions Hjj.
i
In the limit of large N, adopting the scaling σ2 = N−1,
(7) can be simplified to of large N, only diagrams with no intersections between
dashed or between dashed and wavy lines contribute.
1
M M (δ δ +δ δ )+δ δ δ . (8) Thus,toleadingorderinN,thecontribution(c)inFig.1
ij mn im jn in jm ij mn im
h i≃ N must be considered, but not (d) or (e), for example.
It is convenient to keep track of the contributions with All diagrams contributing to the resolvent to leading
the help of diagrams. To this end, a graphical represen- order in N−1 may be summed, resulting in a set of two
tation of the contraction (8) is introduced in Fig. 1(a). coupled equations shown in Fig. 2(a) and (b). The di-
TheaveragedresolventGturnsouttobeadiagonalma- agonal elements G and H are independent of j and
jj jj
trix. Fig. 1(b) defines a graphical representation for the denotedby G andH inthe following. The firstequation
diagonal elements G of G. Terms contributing to G [Fig. 2(a)] contains an infinite sum of diagrams. This
jj jj
are shown in Fig. 1(c-e). One observes that, in the limit summaybeperformedexactly. Itisdenotedbyg(H−1),
3
0.3
0.25
n b/2
−
b
√ 0.2 n 1
) −
E
d( 0.15
n
0.1
n+1
0.05
0
0
n+b/2
E/√b
FIG. 3: Density of eigenvalues of exact diagonalisations of
random matrices of the form (1,5) for σ =1, N =500,1000,
and b = 50,100 (symbols), together with the prediction (13)
FIG. 4: Topology of a (quasi) one-dimensional lattice with
(solid line).
periodicboundaryconditions. Each site nisconnectedtoits
b=4 neighbours.
where
g(z)=(2π)−1/2 ∞dJ exp(−J2/2). (9) ThebandwidthofJisthusb. InthelimitoflargeN and
Z−∞ z J large b, the spectral density is given by a slight modifi-
−
cation of (10),
The result for the resolvent scales with σ2 and N as
(σ2N)−1/2g(H−1(σ2N)1/2). Consequently, the equa- G=(σ2b)−1/2g (E G)(σ2b)1/2 . (13)
tions shown in Fig. 2(a,b) imply the following self-con- −
(cid:0) (cid:1)
sistent equation Diagrammatically,the necessarychangesare most easily
G=(σ2N)−1/2g (E G)(σ2N)1/2 , (10) derivedby letting σ2 =N−1 and assuming that b=BN
− (with fixed B 1). Then the wavy line in Fig. 1(a)
(cid:0) (cid:1) ≪
equivalent to Eq. (51) in Ref. 6 for the average density acquires a factor of B. Furthermore, the dashed line
ofresonancesinanensembleofhighlyconnectedrandom in Eq. (8) also acquires a factor of B. Hence the self-
impedance networks15. consistency equation in the banded case becomes (13).
The form of the simplified contraction (8) implies an Let us also note that the same formula is actually valid
interpretation of the result (10) in terms of an averaged not only for b N, but more generally for 1 b N.
∼ ≪ ≪
Pastur equation: consider a random matrix M = J+ This result implies that densities for different val-
V, where J is distributed according to (5), and V is a ues of b can be scaled on one single curve by plotting
diagonal matrix with Gaussian random entries vk with √bd(E/√b). In Fig. 3, solutions of (13) for σ = 1 are
zero mean and unit variance, independent of Jmn. For a compared with results of exact diagonalisations of ran-
givenrealisationofV,Pastur’sequation14 isG=(E1 dom matrices with N = 500,1000 and b = 50,100. We
V G)−1 (see also Ref. 16 and references therein). I−n observeaverygoodagreement. Theresultsconfirmthat,
−
this equation G is the J-averaged resolvent, keeping V for large N and b, the average density of eigenvalues is
fixed. One obtains Eq. (10) after observing that G is independent of N, and that it scales with b as expected.
diagonal, by averaging over the matrix elements of V. In the following we show that (13) also describes the
This interpretation implies that, in the limit of large N, density of resonancesfor certainrandomimpedance net-
thecorrelationsbetweendiagonalandoff-diagonalmatrix works with a large, but finite connectivity.
elements of M [as seen in Eqs. (1,7)] are irrelevant. Random impedance networks. Random networks of
The above procedure is easily extended to the case complex impedances are currently used to model electri-
of banded matrices, also of interest in random impe- cal and optical properties of disordered inhomogeneous
dance networks5,11. In the banded case, JmnJkl = media8. The most common situation is that of a binary
h i
σ2(δmkδnl+δmlδnk) [which follows from (5)] is replaced composite medium, modeled by attributing a random
by conductance to each bond (x,y) of a lattice, according
to the binary law:
J J =σ2(m n) (δ δ +δ δ ). (11)
mn kl mk nl ml nk
h i | − |
The function σ2(x) is given by σ = σ0 with probability p, (14)
x,y (cid:26)σ1 with probability q =1 p.
σ2 for 0 x b/2, −
σ2(x)= ≤ ≤ (12)
(cid:26)0 otherwise. ThehomogeneousKirchhoffequationsfortheelectricpo-
4
0.3
(∆ V) = (V V ), (17)
P x y x
−
y∈XP(x)
0.25
(∆ V) = (V V ),
Q x y x
√b 0.2 y∈XQ(x) −
)
E
d( 0.15 where y(x) are all the sites connected to site x, whereas
y P(x) (resp. y Q(x)) are those connected by a
0.1 con∈ductance σ (resp∈. σ ), so that ∆=∆ +∆ . Res-
0 1 P Q
onances appear as non-trivial solutions to Eq. (16), for
0.05 0<λ<1.
An efficient algorithm allowing for an exact determi-
0
-4 -2 0 2 4 nation of all the resonances of a finite sample has been
E/√b developed in Ref. [5]. We have adapted this algorithm
tothesimplestgeometryallowingforlong-rangedbonds.
FIG. 5: Plot of the density of resonances for a quasi one- Eachsitenofaverylongchainisconnectedtoitsbneigh-
dimensional random impedance network (see text). Data for bors (n b/2,...,n 1,n+1,...,n+b/2), as shown in
three different ranges (b = 60, 100, and 120) (symbols) are Fig. 4. −Periodic bou−ndary conditions are assumed. For
compared with thetheoretical prediction (13). definiteness we choose p = q = 1/2, so that all the res-
onances are expected to be located at λ = 1/2 in the
b limit.
tentials, →∞
Our numerical results are shown in Fig. 5, for very
long periodic chains with ranges b = 60, 100, and 120.
σ (V V )=0, (15)
x,y y− x For each value of b, we have accumulated a number of
Xy resonances of order 107. After rescaling the resonances
according to λ = (1+E)/2 the density is given by (13)
can be recast as
with σ = 1. A very satisfactory quantitative agreement
(∆ λ∆)V =0, (16) with the theoretical prediction is observed.
Q
−
Acknowledgements. B.M. gratefully acknowledges dis-
with λ=σ0/(σ0 σ1), and cussions with J.T. Chalker. This work was in part sup-
−
ported by Vetenskapsr˚adet (J.S. and B.M.), and by EP-
(∆V) = (V V ), SRC Research Grant No. GR/13838/01(Y.V.F).
x y x
−
yX(x)
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(1999); C.R. Offer and B.D. Simons, J. Phys. A: Math. shown6,7 that,asaconsequenceoftheparticula−rstructure
Gen. 33, 7567 (2000). [seeeq.(1)]ofM,andoftheform ofW,thelattercanbe
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3687 (1998). average density of eigenvalues.
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7 Y.V. Fyodorov, Physica E9, 609 (2001). Tocomparewith Refs.5and8,thevarianceshouldbeset
8 J.P. Clerc, G. Giraud, J.M. Laugier, and J.M. Luck,Adv. unity; for the perturbative calculations described below it
Phys. 39, 191 (1990). is convenient to takeσ2 =N−1.
9 F. Brouers et al., Physica A 241, 146 (1997); Phys. Rev. 14 L.A. Pastur, Theor. Math. Phys. (USSR)10, 67 (1972).
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