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VOLUME 76, NUMBER 7 PHYSICAL REVIEW LETTERS 12 FEBRUARY 1996 Universal Properties of Spectral Dimension Raffaella Burioni* Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France Davide Cassi ² Dipartimento di Fisica, Università di Parma, Viale delle Scienze, 43100 Parma, Italy (Received 20 July 1995) The infrared singularities of a Gaussian model on a general network are invariant under a local rescaling of the masses. This exact result leads to some interesting rigorous relations concerning diffusion and harmonic oscillations on fractals and inhomogeneous structures. We show that a generic distribution of waiting probabilities does not affect the spectral dimension in diffusive problems, neither does a change of masses in an oscillating network. In particular, we prove an exact relation between random walks and vibrational spectrum showing the possibility of noncoincidence of vibrational and usual diffusive spectral dimensions. PACS numbers: 63.50.+x, 05.40.+j, 47.53.+n The study of statistical properties in real structures by means of model systems is greatly stimulated and guided by the idea of universality. According to this idea, only a few very general features are sufficient to determine the physical behavior in a wide class of phenomena. In such a way we can group our systems in a limited number of universality classes. This is the case for crystalline solids undergoing magnetic phase transitions, where the lattice dimension and the symmetry of the Hamiltonian are the only information needed to determine critical exponents. In the same way many long range or low frequency properties depend only on the Bravais lattice dimensionality. This phenomenon allows us to study the simplest system in a given universality class to obtain general physical results common to all other members. These universal properties would be even more impor- tant in noncrystalline and disordered structures, as amor- phous solids, polymers, glasses, and fractals, for which a simple geometrical characterization based on dimension is not evident. In this case the determination of univer- sality classes would suggest the most natural geometrical parameter generalizing the concept of dimension. In recent years various definitions of generalized di- mensions have been proposed, starting from Mandelbrot’s fractal dimension [1]. The most useful in the study of dy- namics and critical phenomena turned out to be the spec- tral dimension [2]. However, some different definitions of this parameter have been given and its universality prop- erties are far from being evident. In this Letter we ana- lyze these definitions and rigorously prove their universal features. The latter allow one in addition to deeply ex- plore the relation between the definitions and to point out a highly nontrivial difference which could be fundamental in the study of inhomogeneous structures. The idea of an anomalous dynamical dimension was first proposed by Dhar [3] in 1977, in connection with the behavior of statistical models on networks. Then in 1982 Alexander and Orbach [2] introduced the spectral dimen- sion ˜ d to describe low frequency vibrational spectrum and long time random walks (RW) properties on fractals, ac- cording to the asymptotic power laws rsvd, v ˜ d21 , (1) where rsvd is the density of harmonic vibrational modes with frequency v and P 0 st d, t 2 ˜ dy2 , (2) where P 0 st d is the probability of returning to the starting site after t steps for a random walker. These definitions were considered to be equivalent by physical arguments. Then it became clear that the spectral dimension can be defined not only for fractals, but for generic networks. In this framework Hattori, Hattori, and Watanabe (HHW) [4] gave a rigorous mathematical definition of ˜ d for an infinite discrete structure (graph) based on the infrared singularities of a Gaussian model defined on the same structure. Let us recall in a simplified way the main point of this definition. Given a connected graph G, its adjacency matrix A ij has all elements equal to 0 except when the sites i and j are nearest neighbors (nn), where A ij 1. Then we can define a Gaussian model on G by the Hamiltonian Hshm 2 i jd 1 4 X nn sf i 2f j d 2 1 X i m 2 i f 2 i 1 2 X ij f i sL ij 1 m 2 i d ij df j , (3) where the square masses m 2 i are all bounded from above and from below by positive numbers and L ij is the Laplacian matrix defined by L ij ; z i d ij 2 A ij , with z i P j A ij being the coordination number of site i . The correlation functions kf i f j l hm 2 i j sL 1 Md 21 ij , (4) 0031-9007y 96y 76(7) y1091(3)$06.00 © 1996 The American Physical Society 1091

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Page 1: Universal Properties of Spectral Dimension

VOLUME 76, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 1996

France

Universal Properties of Spectral Dimension

Raffaella Burioni*Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05,

Davide Cassi†

Dipartimento di Fisica, Università di Parma, Viale delle Scienze, 43100 Parma, Italy(Received 20 July 1995)

The infrared singularities of a Gaussian model on a general network are invariant under a localrescaling of the masses. This exact result leads to some interesting rigorous relations concerningdiffusion and harmonic oscillations on fractals and inhomogeneous structures. We show that a genericdistribution of waiting probabilities does not affect the spectral dimension in diffusive problems, neitherdoes a change of masses in an oscillating network. In particular, we prove an exact relation betweenrandom walks and vibrational spectrum showing the possibility of noncoincidence of vibrational andusual diffusive spectral dimensions.

PACS numbers: 63.50.+x, 05.40.+j, 47.53.+n

bdlyt

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.o

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The study of statistical properties in real structuresmeans of model systems is greatly stimulated and guiby the idea of universality. According to this idea, ona few very general features are sufficient to determinephysical behavior in a wide class of phenomena. In sua way we can group our systems in a limited numbof universality classes. This is the case for crystallisolids undergoing magnetic phase transitions, wherelattice dimension and the symmetry of the Hamiltoniare the only information needed to determine criticexponents. In the same way many long range or lfrequency properties depend only on the Bravais lattdimensionality. This phenomenon allows us to studysimplest system in a given universality class to obtageneral physical results common to all other members

These universal properties would be even more imptant in noncrystalline and disordered structures, as amphous solids, polymers, glasses, and fractals, for whicsimple geometrical characterization based on dimensis not evident. In this case the determination of univsality classes would suggest the most natural geometrparameter generalizing the concept of dimension.

In recent years various definitions of generalizedmensions have been proposed, starting from Mandelbrfractal dimension [1]. The most useful in the study of dnamics and critical phenomena turned out to be the sptral dimension [2]. However, some different definitionsthis parameter have been given and its universality prerties are far from being evident. In this Letter we anlyze these definitions and rigorously prove their univerfeatures. The latter allow one in addition to deeply eplore the relation between the definitions and to point oa highly nontrivial difference which could be fundamentin the study of inhomogeneous structures.

The idea of an anomalous dynamical dimension wfirst proposed by Dhar [3] in 1977, in connection with thbehavior of statistical models on networks. Then in 19

0031-9007y96y76(7)y1091(3)$06.00

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hechrehenl

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in

r-or-

aonr-cal

i-t’s-c-

fp--

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utl

s

2

Alexander and Orbach [2] introduced the spectral dimesion d to describe low frequency vibrational spectrum anlong time random walks (RW) properties on fractals, acording to the asymptotic power laws

rsvd , vd21, (1)

wherersvd is the density of harmonic vibrational modewith frequencyv and

P0std , t2dy2, (2)

whereP0std is the probability of returning to the startingsite aftert steps for a random walker. These definitionwere considered to be equivalent by physical argument

Then it became clear that the spectral dimension cbe defined not only for fractals, but for generic networkIn this framework Hattori, Hattori, and Watanabe (HHW[4] gave a rigorous mathematical definition ofd for aninfinite discrete structure (graph) based on the infrarsingularities of a Gaussian model defined on the sastructure. Let us recall in a simplified way the maipoint of this definition. Given a connected graphG, itsadjacency matrixAij has all elements equal to0 exceptwhen the sitesi and j are nearest neighbors (nn), wherAij ­ 1. Then we can define a Gaussian model onG bythe Hamiltonian

Hshm2i jd ­

14

Xnn

sfi 2 fjd2 1X

i

m2i f2

i

­12

Xij

fisLij 1 m2i dijdfj , (3)

where the square massesm2i are all bounded from above

and from below by positive numbers andLij is theLaplacian matrix defined byLij ; zidij 2 Aij, with zi ­P

j Aij being the coordination number of sitei. Thecorrelation functions

kfifjlhm2i j ­ sL 1 Md21

ij , (4)

© 1996 The American Physical Society 1091

Page 2: Universal Properties of Spectral Dimension

VOLUME 76, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 1996

h

n

ies

io

n

en

of

th

, this

hetors.e

r, ites

a

where Mij ; dijm2i , can be defined by averaging wit

respect to the Boltzmann weight exps2Hd. In the infraredlimit s ! 01 the leading singular part of the correlatiofunction kfifilhsm2

i j behaves as

singkfifilhsm2i j , ssdy2d21slnsdIsdy2d, (5)

where Isxd ­ 1 for integer x and 0 otherwise. HHWshowed that this definition has some universal propertindeed it is independent of sitei and, for a restricted clasof graphs withd , 2, also of the particular distributionhm2

i j. In addition, they proved the coincidence of thed with the parameter defined in (2), in the casecontinuous time RW.

Here we prove that the HHW definition is indepedent of thehmij distribution for any graph. This resultallows in turn to prove the following fundamental properties: (i) The coincidence of HHW definition with thcorresponding one for discrete time RW; (ii) the indepedence of RWd of any waiting probabilities distribution;

l

inib

b

-

n

n

1092

s:

rf

-

-

-

(iii) an explicit relation between vibrationald and the av-erage ofP0std over all sites; and (iv) the independencevibrationald of any bounded mass distribution.

Let us begin by the generalization of universality wirespect to mass rescaling. HHW proved that ifm02

i # m2i

for any i, then kfifilhm2i j # kfifilhm02

i j. Because of thedivergence ofkfifjlhsm2

i j for vanishings whend , 2 andto the boundedness conditions on the squared massesallows us to prove the independence ofd of a specificmass distribution. Notice that the divergence of tcorrelation function is necessary in this proof, in orderget significant inequalities between asymptotic behavioIf these functions do not diverge, in the infrared limit thsingular part containing the information aboutd cannot beseparated by a generic nonsingular finite part. Howeveis possible to get divergent quantities by taking derivativup to a suitable order ofkfifjlhsm2

i j with respect tos. Inthis way we obtain a function diverging according topower law with exponent depending ond. Then it hasbeen proven [4] that

µ

2dds

∂N

kfifilshm2i j ­ N!

Xk1...kn

m2k1

· · · m2kN

kfifk1lshm2i jkfk1fk2lshm2

i j · · · kfkNfjlshm2

i j . (6)

s

se-the

the

Now notice that, by straightforward steps, one can aprove that if m02

i # m2i for any i, then kfifjlhm2

i j #

kfifjlhm02i j, for i fi j. Applying the last inequalities to

the correlation functions appearing in (6), and considerthe boundedness by positive numbers of mass distrtions, it is easy to prove the independence ofd of hm2

i j fora generic graph.

Now consider discrete time RW on a graph, definedthe hopping probabilities matrix

Pij ­Aij

zi. (7)

The probabilityPiistd of returning to a starting sitei aftert steps is given by

Piistd ­ sPtdii . (8)

If we introduce the generating functionsPiisld ;Pt ltPiistd, it holds

Piisld ­ s1 2 lPd21ii ­ sL 1 Md21

iizi

l

­ kfifilhm2i j

zi

l, (9)

with m2i ­ zis1 2 ldyl. Considering that, from Taube

rian and Abelian theorems, the conditionsPiistd , t2dy2

for t ! ` and singfPiisldg , s1 2 ldsdy2d21 for l ! 12

are equivalent [5], it follows also that the Gaussian athe discrete time RW definitions ofd are equivalent.

Now let us introduce a waiting probability distributiowi modifying the hopping probabilities matrixP to

P0ij ­

Aij 1 dijwi

zi 1 wi. (10)

o

gu-

y

d

The modified generating functions are given by

P0iisld ­ s1 2 lP0d21

ii ­ sL 1 M 0d21ii

zi 1 wi

l

­ kfifilhm02i j

zi 1 wi

l, (11)

with m02i ­ szi 1 wid s1 2 ldyl and, from the mass in-

dependence, they have the same singular part asPiisld.This proves that the RW asymptotic behavior and conquently the RW spectral dimension are independent ofintroduction of waiting probabilities.

Now let us consider our graphG as an oscillatingnetwork with point massesMi , bounded by positivenumbers, joined by springs of elastic constantk ­ 1 whenAij ­ 1.

The normal modes of this system are the solutions ofeigenvalue equationsX

j

Lijxj ­ v2Mixi , (12)

with v being the frequency andxi the displacement fromequilibrium position at sitei. Let us posev2 ­ l anddefine the densityrlsld of eigenstates with eigenvaluel,and rvsvd the density of modes with frequencyv. Ifrvsvd , vd21 for v ! 0, thenrlsld , ldy221 for l !0. Now notice that, in this case,Z rlsld

l 1 edl , esdy2d21 (13)

for e ! 0. ButZ rlsldl 1 e

dl ­ TrfsM21L 1 ed21g

­ TrfsL 1 Med21Mg , (14)

Page 3: Universal Properties of Spectral Dimension

VOLUME 76, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 1996

es

der

inh-

ohiu

nae

ovsc

phnrye

-

a

adeo

e

Thl-

ahions

wont:icalre

es.eresatndy,ingnye ofe

resealbe

ofterscka

leole

di

r.

with M being the diagonal matrix of oscillating massMi andTrfCg ; limN!` N21

PNi­1 Cii. Now we can ob-

serve that the latter expression, due to the mass bounness, has the same singular part as the average ovesites ofG of kfifilhem2

i j with m2i ; Mi . Because of the

generalized HHW inequalities we considered at the begning of this Letter, it follows the mass independence of tasymptotic behavior ofrvsvd and consequently of the vibrational spectral dimension.

The latter, as one can see from (14), is also relatedRW. However, it depends on the average over all sitesthe probabilitiesPiistd and not on a specific one. Althougall Piistd have always the same asymptotic behavior, itnot possible to conclude that even their average shobehave in the same way. So, in principle, the vibratiod is independent and possibly different from all the othdefinition of spectral dimension.

Although it seems rather difficult to give a general gemetrical condition leading to different asymptotic behaiors of Piistd and TrfPstdg, in some specific structurewhere they can be explicitly calculated such a differenis indeed found. This is the case of the class of graknown as comb lattices [6]. There, by direct computatioone can verify that, due to the very particular geometTrfPstdg coincides with the probability of returning to thstarting point on a linear chain and goes ast21y2 for larget, while Piistd , t2qy2, with q ­ 2 2 22d11, where theinteger numberd is the Euclidean dimension of the natural embedding space [6].

Notice that the boundedness of oscillating masseswell as the boundedness of the weightswi are sufficientconditions for universality. It is a hard task to proveweaker necessary condition, since in absence of bounness the asymptotic behavior could in principle dependthe spatial mass (orwi) distribution. However, on a lin-ear chain one can argue that the boundedness of the mvalue of mi and the finiteness of the mean value ofwi

are necessary and sufficient to preserve universality.validity of this criterion for more complex structures, athough likely, is still an open problem.

In conclusion, the results obtained clarify the universproperties of the spectral dimension and show that tparameter is a good generalization of the usual dimensin the description of a large class of physical phenomeThe coincidence of Gaussian and RW definitions sugge

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that the long range geometry affects in the same way tclasses of problems that are in principle very differeindeed the Gaussian model is deeply related to statistmodels for phase transitions, while discrete time RW ausually introduced to describe diffusion on real structurMoreover, the fact that the possibility for a random walkof staying on each site with different probabilities donot affect its long time properties clearly shows ththe latter are related only to large scale geometry alargely independent of any local detail. In a similar wathe result obtained for the harmonic spectrum, meanthe independence of its low frequencies regime of achanges in the masses (such as, e.g., the presencdifferent isotopes with any distribution), underline thfundamental role played by the geometrical structuwith respect to other physical ingredients. Finally, thpossibility of difference between the vibrational spectrdimension and the RW and Gaussian ones, which willfurther studied in a forthcoming paper [7], is a signthe need to distinguish between local and bulk parameon inhomogeneous structures [8] where, due to the laof invariance properties, local quantities do not containcomplete information about the whole system.

Laboratoire de Physique Théorique de l’Ecole NormaSupérieure is a unité propre du CNRS, associée à l’EcNormale Supérieure et à l’Université de Paris Sud.

*Present address: Dipartimento di Fisica, UniversitàMilano, Via Celoria 16, 20133 Milano, Italy. Electronicaddress: [email protected]

†Electronic address: [email protected][1] B. Mandelbrot, The Fractal Geometry of Nature(Free-

man, San Francisco, 1983).[2] S. Alexander and R. Orbach, J. Phys. (Paris), Lett.43,

L625 (1982).[3] See also D. Dhar, J. Math. Phys. (N.Y.)18, 577 (1977).[4] K. Hattori, T. Hattori, and H. Watanabe, Prog. Theo

Phys. Suppl.92, 108 (1987).[5] W. Feller, An Introduction to Probability Theory and

Its Applications(John Wiley & Sons, New York, 1966),Vol. II.

[6] D. Cassi and S. Regina, Mod. Phys. Lett. B6, 1397 (1992).[7] R. Burioni, D. Cassi, and S. Regina (to be published).[8] D. Cassi (to be published).

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