Fractal growth with quenched disorder
ABSTRACT In this lecture we present an overview of the physics of irreversible fractal growth process, with particular emphasis on
a class of models characterized by quenched disorder. These models exhibit selforganization, with critical properties developing spontaneously, without the fine tuning of external
parameters. This situation is different from the usual critical phenomena, and requires the introduction of new theoretical
methods. Our approach to these problems is based on two concepts, the Fixed Scale Transformation, and the quenchedstochastic
transformation, or Run Time Statistics (RTS), which maps a dynamics with quenched disorder into a stochastic process. These
methods, combined together, allow us to understand the selforganized nature of models with quenched disorder and to compute
analytically their critical exponents. In addition, it is also possible characterize mathematically the origin of the dynamics
by avalanches and compare it with the continuous growth of other fractal models. A specific application to Invasion Percolation will be discussed. Some possible relations to glasses
will also be mentioned.
 Citations (1)
 Cited In (0)

Chapter: The fractal geometry of nature
03/1983: pages 468 p.; W.H. Freeman., ISBN: 0716711869
Page 1
arXiv:condmat/9906443v1 [condmat.statmech] 30 Jun 1999
FRACTAL GROWTH WITH QUENCHED
DISORDER
L. PIETRONERO1, R. CAFIERO1and A. GABRIELLI2
1Dipartimento di Fisica, Universit´ a di Roma ”La Sapienza”,
P.le Aldo Moro 2, I00185 Roma, Italy; Istituto Nazionale di Fisica della Materia,
unit´ a di Roma I
2Dipartimento di Fisica, Universit´ a di Roma ”Tor Vergata”,
Via della Ricerca Scientifica 1, I00133 Roma, Italy
Abstract. In this lecture we present an overview of the physics of irreversible frac
tal growth process, with particular emphasis on a class of models characterized by
quenched disorder. These models exhibit selforganization, with critical properties de
veloping spontaneously, without the fine tuning of external parameters. This situation
is different from the usual critical phenomena, and requires the introduction of new the
oretical methods. Our approach to these problems is based on two concepts, the Fixed
Scale Transformation, and the quenchedstochastic transformation, or Run Time Statis
tics (RTS), which maps a dynamics with quenched disorder into a stochastic process.
These methods, combined together, allow us to understand the selforganized nature
of models with quenched disorder and to compute analytically their critical exponents.
In addition, it is also possible characterize mathematically the origin of the dynamics
by avalanches and compare it with the continuous growth of other fractal models. A
specific application to Invasion Percolation will be discussed. Some possible relations
to glasses will also be mentioned.
1INTRODUCTION
The introduction of the fractal geometry (Mandelbrot (1982)) has changed the
way physicists look at a vast class of natural phenomena which produce irregu
lar structures. Many models have been introduced since the early eighties trying
to relate these structures to well defined physical phenomena. These are the
Diffusion Limited Aggregation (DLA) (Witten and Sander (1981)), the Dielec
tric Breakdown Model (DBM) (Niemeyer and Pietronero (1984)), the Invasion
Percolation (IP) (Wilkinson and Willemsen (1983)), the Sandpile (Bak, Tang
and Wiesenfeld (1987)), the Bak and Sneppen model (BS) (Bak and Sneppen
(1993)), just to give some examples. All these models lead spontaneously (for a
broad range of parameters) to the development of critical properties and fractal
structures.
In the last years these has been a great interest on fractal models charac
terized by quenched disorder. These models are generally characterized by an
intermittent dynamics, with bursts of activity of any size concentrated in a re
gion of the system (avalanches), and by memory effects induced by the presence
of quenched disorder and by the dynamical rules (Paczuski, Bak and Maslov
Page 2
2 L. PIETRONERO, R. CAFIERO and A. GABRIELLI
(1996); Cafiero, Gabrielli, Marsili, and Pietronero (1996); Vendruscolo and Mar
sili (1996); Marsili, Caldarelli and Vendruscolo (1996)). These memory effects
in fractal growth processes with quenched disorder, may resemble an element
characteristic of the spin glasses and glass dynamics. In fact, also in spin glasses,
memory effects (aging) are due to the presence of quenched disorder, and are
relied to the typical Kolrausch stretched exponential relaxation dynamics. How
ever, at the moment it is hard to develop such an analogy in a more concrete
way.
The study of physical phenomena leading to fractal structures can be classi
fied by three different levels:
– Mathematical Level: Fractal Geometry. This is a descriptive level, at which
one simply recognizes the fractal nature of the phenomena and extimates
the fractal dimension D.
– Physical Models. One develops a model of fractal growth based on the phys
ical process. This level is the analogue of the Ising model in equilibrium
statistical mechanics.
– Physical Theories. This level corresponds to a fully understanding of the
origin of fractals in nature, their self organization etc. The corresponding
level for phase transitions is the Renormalization Group.
The analogy we make with phase transitions is quite natural, because, like or
dinary critical phenomena, fractal growth models are scale invariant. However,
some profound differences, like irreversible, nonequilibrium dynamics and SOC,
make unavoidable the development of new theoretical concepts.
Fractal physical models can be classified into two main groups:
1. Irreversible stochastic models
2. Irreversible quenched models
In the next section we discuss these models in relation with standard critical
phenomena.
2
GROWTH
PHYSICAL MODELS FOR SELFSIMILAR
We briefly mention below some examples of these two classes of models, with a
particular emphasis on models with quenched disorder.
1) Irreversible stochastic models
– Diffusion Limited Aggregation (DLA) (Witten and Sander (1981)). This is
the first physical model of fractal growth. Particles performing a Brownian
motion aggregate and form complex fractal structures.
– Dielectric Breakdown Model (DBM) (Niemeyer and Pietronero (1984)). Is a
generalization of DLA via the relation between potential theory and random
walk.
Page 3
FRACTAL GROWTH WITH QUENCHED DISORDER3
– Sandpile Models (Bak, Tang and Wiesenfeld (1987)). These models are in
spired by the marginal stability of sandpiles. The random addition of sand
grains drives the system into a stationary state with a scale invariant distri
bution of avalanches.
2) Irreversible Quenched Models
– Invasion Percolation (IP) (Wilkinson and Willemsen (1983)). This model
was developed to simulate the capillary displacement of a fluid in a porous
medium. The porous medium is represented by a lattice where to each bond
i is assigned a quenched value xi of its conductance. At each time step
the dynamics of the fluid evolves by occupying the bond with the smallest
conductance between all its perimeter bonds. We call this kind of dynamics
extremal dynamics. IP is known to reproduce asymptotically the Percolation
cluster of standard critical Percolation.
The main characteristics of this model are:
1. Deterministic dynamics. Once a realization of the quenched disorder is
chosen, the dynamical rule selects in a deterministic way the bond to be
invaded.
2. Selforganization. The process spontaneously develops scaleinvariantstruc
tures and critical properties. In the limit t → ∞ both long range space
and time correlations appear.
3. Avalanches. The asymptotic dynamical evolution consists of local, scale
invariant macroevents, composed by elementary growth steps spatially
and causally connected, called avalanches. When an avalanche stops, the
activity is transferred to another region of the perimeter.
– Bak and Sneppen Model (BS) (Bak and Sneppen (1993)). This model is
similar to IP and has the same properties exposed in points 13. In fact,
points 13 are characteristic of the whole class of SOC models with quenched
disorder and extremal dynamics ( for a review see Paczuski, Bak and Maslov
(1996)). The BS model has been introduced to describe scale free events in
biological evolution.
– Quenched models with an external modulating field. This is a particular class
of models, the prototype of which is the Quenched Dielectric Breakdown
Model (QDBM (Family and Zhang (1986); De Arcangelis, Hansen, Her
rmann and Roux (1989)). This model is a sort of combination between IP
(quenched) and DBM (stochastic). To each bond of a lattice is assigned a
quenched random number xi, representing the local resistivity. In addition,
an external electric field E is introduced. The ratio yi = xi/Ei between
quenched disorder and local electric field is the inverse of the current flow
ing in the bonds. At each time step the dynamics breaks the bond with the
smallest yi, that is to say with the biggest current. The QDBM modelizes
the dielectric breakdown of a disordered solid. A similar model can be for
mulated to modelize the propagation of fractures in a inhomogeneous solid
(Hansen, Hinrichsen, Roux, Herrmann and De Arcangelis (1990)).
Page 4
4L. PIETRONERO, R. CAFIERO and A. GABRIELLI
All the models in these classes share some important properties, that differen
tiates them from ordinary critical phenomena. First of all they are characterized
by an irreversible dynamics, so that they cannot be described in hamiltonian
terms, and the statistical weight of a configuration depends on the complete
growth history (see. fig. 1). Quenched models have an additional problem, in
t1
t3
t4
t2
t2
t1
t4
t3
a)b)
Fig.1. For systems with irreversible dynamics, the statistical weight of a configuration
depends on the whole growth history. The two identical configuration shown here have,
in general, different statistical weights because their histories, indicated by the times
ti i = 1,2,3,4, are different.
that they have a deterministic dynamics, and the stochasticity enters only in
the choice of the realization of the disorder. So, it is difficult to define transition
probabilities for the dynamics.
Another fundamental difference, is that these models are self organized.
Their dynamics evolves spontaneously in the phase space towards an attractive
fixed point. No fine tuning of any parameter is needed. In addition, quenched
models with extremal dynamics have an avalanche dynamics, that is to say
the system tends to concentrate its activity in a well localized region of the
perimeter, during an avalanche. When an avalanche stops, the activity transfers
to another region of the perimeter and a new avalanche starts. On the contrary,
in stochastic models, like DLA, there is a continuous growth process, that is
to say for large systems the probability to have two nearby subsequent growth
events tends to zero. The dynamical activity is diffused at each time on the whole
growth interface.
In table. 1 we propose a scheme of comparison between the properties of or
dinary critical phenomena and the most popular stochastic and quenched fractal
growth models.
Page 5
FRACTAL GROWTH WITH QUENCHED DISORDER5
3NEW THEORETICAL CONCEPTS
The application of the standard theoretical methods of statistical physics (field
theory and renormalization group) is, in general, not possible for the main fractal
growth problems, like DLA, DBM, Invasion Percolation, the sandpile model,
which are characterized by an intrinsically irreversible dynamics.
Here we discuss two theoretical methods we have developed in the past few
years, as a step towards the construction of a physical theory for selforganized
fractal growth processes. The exposition will be colloquial, and mainly devoted
to the theoretical analysis of quenched models. For more details readers should
refer to the bibliography.
3.1FIXED SCALE TRANSFORMATION (FST)
This approach combines a technique of lattice path integral, to take into account
the irreversible dynamics, with the study of the scale invariant dynamics inspired
by the RG theory. It permits a description of the scale invariant properties of
fractal growth models.
The method focuses on the dynamics at a given scale and analyzes the nearest
neighbours correlations at this scale using lattice path integral approach by which
one can calculate the elements of a probability matrix, the FST matrix. The fixed
point of the FST matrix gives the nearest neighbour correlations, at that scale. If
one uses the scale invariant dynamics of the system, that can be obtained by Real
Space Renormalization Group approaches (Cafiero, Vespignani and Pietronero
(1993)), one can generalize these correlations to all scales and compute the fractal
dimension (Erzan, Pietronero and Vespignani (1995)).
The basic point of the FST is the separation of the long time limit (t → ∞)
for the dynamical process at a given scale, from the large scale limit (r → ∞),
that defines the scale invariant dynamics. The interesting feature of FST is that it
works at a fixed scale, so it is possible to include the fluctuation of the boundary
conditions, that in systems with long range interactions, like Diffusion Limited
Aggregation (DLA), have a great influence on the fractal properties. For this
reason, the FST approach allows to reach a remarkable level of accuracy in the
calculation of the fractal dimension.
At the moment, the FST framework, eventually combined with the RTS
method that we discuss below, seems to be the only general approach to un
derstand the selforganized critical nature of a broad class of models going from
DLA, to Percolation, to sandpile, to Invasion Percolation (Erzan, Pietronero
and Vespignani (1995); Cafiero, Gabrielli, Marsili, and Pietronero (1996)). This
situation therefore supports the idea that these models pose new questions for
which one would like to develop a common theoretical scheme.
3.2QUENCHEDSTOCHASTIC TRANSFORMATION (RTS)
As we mentioned above, quenched models with extremal dynamics, like Invasion
Percolation (IP), have a deterministic dynamics. This makes it impossible to ad
Page 6
6L. PIETRONERO, R. CAFIERO and A. GABRIELLI
Table 1. Comparison between the properties of ordinary critical phenomena repre
sented by the Ising model and the most popular stochastic and quenched models gen
erating fractal or scale invariant structures in a selforganized way.
SELFSIMILARITY: PHYSICAL MODELS
DLADBM (81)IsingType (70's) Invasion Percolation (8283)
Equilibrium
Statistical
Mechanics
Ergodicity
Boltzmann Weight
Standard Critical behaviour
Fine Tuning: T=Tc
Repulsive Fixed Point
ξ = (T  Tc)ν
Approach to the critical point
Γ (r) = 1
r (d2+η)
Anomalous dimension exactly
at T = Tc
Theory: Renormalization
Group
NON LINEAR, IRREVERSIBLE DYNAMICAL
EVOLUTION.
Statistical weight of configurations depend on the whole growth
history. In extremal models they are difficult to compute
because of deterministic dynamics
Asymptotically frozen fractal
structure
Avalanche dynamics driven by
extremal statistics, with
avalanches of all sizes
CRITICAL BEHAVIOUR IS SELFORGANIZED
ATTRACTIVE FIXED POINT
Fractal structure
Long range interactions
(Laplacian)
Cognitive time memory
Problem: understand the origin of
fractal properties and computation
of the fractal dimension D
Problems: origin of SOC
and avalanche dynamics.
Computation of D
distribution of avalanche
sizes P(s) = s τ
THEORY: NEW CONCEPTS ARE NEEDED,
LIKE FST AND RTS
dress directly these class of model with the FST method or any other microscopic
theory. Here we describe a general theoretical approach which addresses the ba
sic problems of extremal models: (i) the understanding of the scaleinvariance
and selforganization;(ii) the origin of the avalanche dynamics and (iii) the com
putation of the independent critical exponents. We will discuss in particular a
specific application to Invasion Percolation (IP) (Cafiero, Gabrielli, Marsili, and
Pietronero (1996)), but these ideas can be easily extended to other models of
this type like the Bak and Sneppen model (Marsili (1994b)).
In order to overcome the problem represented by quenched disorder, we in
troduced a mapping of the quenched extremal dynamics into a stochastic one
Page 7
FRACTAL GROWTH WITH QUENCHED DISORDER7
with cognitive memory, the quenchedstochastic transformation, also called Run
Time Statistics (RTS) (Marsili (1994a)). This approach was improved in various
steps (Pietronero, Schneider and Stella (1990); Pietronero and Schneider (1990);
Marsili (1994a); Marsili (1994b)) and now we can develop it into a general the
oretical scheme, that we call RTSFST method (Cafiero, Gabrielli, Marsili, and
Pietronero (1996)). Its essential points are:
 Quenchedstochastic transformation.
 Identification of the microscopic fixed point dynamics. This point clarifies the
SOC nature of the problem.
 Identification of the scale invariant dynamics for block variables. This eluci
dates the origin of fractal structures.
 Definition of local growth rules for the extremal model. This clarifies the origin
of avalanche dynamics.
 Use of the above elements in a real space scheme, like the FST, to compute
analytically the relevant exponents of the model.
A general stochastic process is based on the following elements: a) a set of
time dependent dynamical variables {ηi,t}; b) a Growth Probability Distribution
(GPD) for the single growth step {µi,t}, obtained from the {ηi,t}; c) a rule for
the evolution of the dynamical variables ηi,t→ ηi,t+1.
Therefore, in order to map IP onto a stochastic process we have to: a) find
the correct dynamical variables (the {ηi,t}’s); b) determine the GPD {µi,t} in
terms of these variables; c) find the evolution rule of the {ηi,t}
A simple example can be useful to get an insight into the essence of the
problem. Consider two independent random variables X1,X2, with uniform dis
tribution p0(x1) = p0(x2) = 1 in [0,1] and let us eliminate the smallest, for
example X2. Clearly the probability that X2< X1is 1/2. At the second “time
step”, we compare the surviving variable X1with a third, uniform, random vari
able X3just added to the game and, again, we eliminate the smallest one. At
first sight one might think that, since both variables are independent, the proba
bility that X1survives again is 1/2, but this is actually incorrect. In this case we
indeed need to calculate the probability µ3that X3< X1given that X2< X1.
This, using the rules of conditional probability, reads:
µ3=˜P(X3< X1) = P(X3< X1X2< X1) =
=P(X3< X1
?X2< X1)
P(X2< X1)
=2
3,(1)
where P(AB) is the probability of the event A, given that B occurred, and
P(A?B) is the probability of occurrence of both A and B. The point is that
the distribution of the variable X1is no longer uniform when it is compared with
X3, even though they are independent. The information that X2< X1changes
in a conditional way the effective probability density p1(x) of X1. Indeed the
probability that x ≤ X1< x + dx must now account for the fact that X2< x.
By imposing this condition, we get: p1(x) = 2x. An analogous calculation for
the distribution of X2 gives p2(x) = 2(1 − x). Qualitatively, the event X2 <
Page 8
8L. PIETRONERO, R. CAFIERO and A. GABRIELLI
X1 decreases the probability that X1 has small values. On the contrary, the
probability that X2is small is enhanced.
The above example contains the essential idea of the quenchedstochastic
transformation. Extremal dynamics establishes, at each time step t, an order
relation between quenched variables (X2< X1in the example). This informa
tion on the statistical properties of the variables involved in the process (active
variables) can be conditionally stored in the form of their effective densities. Vari
ables which have experienced the same dynamical history, will have the same
effective density, irrespectively of their spatial position. This memory is repre
sented by the age k = t−t0, where t is the actual time and t0is the time at which
the variable became active. The effective densities pk,t(x) (p0,t(x) = p0,0(x) = 1)
of variables of age k at time t are the dynamical variables of the stochastic process
we are looking for.
A generalization of our simple example (Eq. 1) leads to the following equation
for the growth probability µk,tof a variable of age k at time t (GPD):
µk,t=
?1
0
dx pk,t(x)
?
θ
(1 − Pθ,t(x))nθ,t−δθ,k,(2)
where Pθ,t(x) =
active variables and nθ,t is the number of active variables of age θ at time t.
The meaning of this expression is that the product inside the integral takes
into account of the competition of the selected variable with each of the other
active variables. The density mk,t(x) of this (smallest) variable after its growth
is conditioned by the information that it has grown, and can be computed from
Eq. 2. The temporal evolution of the densities of the still active variables is then
given by:
?x
0dypθ,t(y), the product is intended over all the ages of the
pθ+1,t+1(x) = pθ,t(x)
?x
0
mk,t(y)
1 − Pθ,t(y)dy. (3)
Equations 2, 3 accomplish our goal to describe a quenched extremal process as
a stochastic process with time memory. The presence of memory is enlighted by
the dependence of the GPD on the parameter k. A mean field like expansion of
Eq.(2) in the limit t → ∞ gives (Marsili (1994a)): µk,∞∼
is also confirmed by simulations (Marsili, Caldarelli and Vendruscolo (1996)).
Memory is at the origin of screening effects in the GPD {µk,t}. The power law
behaviour of µk,tguarantees that screening is preserved at all scales, which is the
condition to generate holes of all sizes in a growing pattern, leading to fractal
structures (Cafiero, Vespignani and Pietronero (1993)).
This mapping, applied to models like IP, allows us to characterize mathe
matically the selforganization. In fact, the following histogram equation can be
derived by the RTS equations 2, 3:
1
(k+1)α. This result
∂xΦt(x) = βΩtΦ2
t(x)
?
1 −
ωt
ωt+ 1Φt(x)
?
(4)
Page 9
FRACTAL GROWTH WITH QUENCHED DISORDER9
where ωt= ?Nt+1− Nt?, Ωt= ?Nt?, Ntis the number of interface variables at
time t, and β is the solution of: β = 1−e−βΩt. This equation describes the time
evolution of the distribution Φt(x) of quenched disorder on the growth interface.
The solution of eq. 4 becomes asymptotically (fig.2) (Marsili (1994a)):
lim
t→∞Φt(x) =
1
1 − pcθ(x − pc)(5)
where pcis a critical threshold of the original extremal dynamics (pc= 1/2 for 2d
bond IP), in agreement with numerical simulations (Wilkinson and Willemsen
(1983)). Note that, in order to obtain the asymptotic behaviour 5, no fine tuning
of any parameter is needed. This clarifies the SOC nature of the problem.
0.00.2 0.40.60.81.0
x
0.0
0.5
1.0
1.5
2.0
Φt(x)
t=10
t=50
t=500
t=infinity
Fig.2. Time evolution of the solution of the equation for Φt(x) (for 2 − d bond IP).
Φt(x) tends asymptotically to a theta function with discontinuity at pc = 1/2.
The scale invariant dynamics can be shown to coincide with the microscopic
one (Cafiero, Gabrielli, Marsili, and Pietronero (1996)). The RTS approach per
mits also to characterize the origin of avalanche dynamics and to write down a
set of equations describing the evolution of a single avalanche, by a straightfor
ward modification of equations 2, 3 (Cafiero, Gabrielli, Marsili, and Pietronero
(1996)). From simulations (Maslov (1995)), and from the histogram equation,
one deduces that, asymptotically, each avalanche starts with a variable (initia
tor) equal to the threshold pc(pc= 1/2 for 2d bond IP). All other variables in
the avalanche have values smaller than pc. In view of the above arguments, the
RTS equations for the local avalanche dynamics are obtained from Eqs. (2, 3) by
Page 10
10 L. PIETRONERO, R. CAFIERO and A. GABRIELLI
taking into account only the variables which become active after the initiator’s
growth and by integrating in Eq.(2) only in [0,pc].
By using the equations for the local avalanche dynamics together with the
FST method we have been able to compute with a very good accuracy (tipically
1−2%, depending on approximations), the relevant critical exponents of Invasion
Percolation, that is to say the fractal dimension and the avalanche exponent
(Cafiero, Gabrielli, Marsili, and Pietronero (1996)). The method can be applied
successfully also to the Bak and Sneppen model (Marsili (1994b)). In table 2 we
show the theoretical values of the exponents of IP, that we have computed with
the RTSFST method, compared with numerical simulations
Table 2. Theoretical values of the fractal dimension of IP, with (Dtrap
trapping (Df) and of directed IP (DDIP
values are compared with numerical simulations.
f
) and without
f
), and of the avalanche exponent τ. These
Df
Dtrap
f
DDIP
f
τIP
τtrap
RTS − FST 1.8879 1.8544 1.7444 1.5832 1.5463
simul.
∼ 1.89 ∼ 1.86 ∼ 1.75 ∼ 1.60 ∼ 1.53
Recently, we have extended the RTSFST scheme to QDBM, and we have
obtained interesting, although preliminar, results, which allow us to elucidate
some important characteristics of the class of models to which QDBM belongs
(Cafiero, Gabrielli, Marsili, Torosantucci and Pietronero (1996)).
4FURTHER DEVELOPMENTS
In this lecture we have discussed two recently introduced theoretical methods,
the FST and the RTS. These approaches have been applied sucesfully to many
models for fractal growth, and allow to make a significative step towards the
formulation of a common theoretical scheme for the physics of selforganized
fractal growth.
At the moment, we are studying the application of the RTS mapping to
interface dynamics in quenched disorder (Sneppen (1992)), and to glassy type
dynamics. An interesting work, for what concerns the last point, is and RTS
type analysis of the statistical and dynamical properties of the random walk
in quenched disorder (RRW) (Vendruscolo and Marsili (1996)), which has been
studied by many authors as a toy model for localization (Tosatti, Zannetti and
Pietronero (1988)), depinning transitions (Bouchaud, Comtet, Georges and Le
Doussal (1990)), and aging effects (Marinari and Parisi (1993)). In this work,
the authors map, by using the RTS method, the RRW dynamics into a stochas
tic dynamics with cognitive memory and recover all the characteristics of the
original model. This suggests a link between stochastic dynamics with memory
and the realizations of a dynamics with quenched disorder.
Page 11
FRACTAL GROWTH WITH QUENCHED DISORDER11
References
Mandelbrot B. B. (1982): The fractal Geometry of Nature. W. H. Freeman, New York.
Witten T. A. , Sander L. M. (1981): Phys. Rev. Lett. 47, 1400.
Niemeyer L. , Pietronero L. , Wiesmann H. J. (1984): Phys. Rev. Lett. 52, 1033.
Bak P. , Sneppen K. (1993):Phys. Rev. Lett. 71, 4083.
Wilkinson D. , Willemsen J. F. (1983): J. Phys. A 16, 3365.
Bak P. , Tang C. , Wiesenfeld K. (1987): Phys. Rev. Lett. 59, 381.
Sneppen K. (1992): Phys. Rev. Lett. 69, 3539.
Family F. , Zhang Y. C. , Vicsek T. (1986): J. Phys. A 19, L733.
De Arcangelis L., Hansen A. , Herrmann H. J. , Roux S. (1989): Phys. Rev. B 40, 877.
Hansen A. , Hinrichsen E. L. , Roux S. , Herrmann H. J. , De Arcangelis L. (1990):
Europhys. Lett. 13, 341.
Cafiero R., Pietronero L., Vespignani A. (1993): Phys. Rev. Lett. 70, 3939.
Erzan A., Pietronero L., Vespignani A. (1995): Rev. Mod. Phys. 67, no. 3.
Pietronero L., Schneider W. R., Stella A. (1990): Phys. Rev.A 42, R7496.
Pietronero L., Schneider W. R. (1990): Physica A 119, 249267.
Marsili M. (1994): J. Stat. Phys. 77, 733.
Gabrielli A. , Marsili M., Cafiero R., Pietronero L. (1996): J. Stat. Phys. 84, 889893.
Marsili M. (1994): Europhys. Lett. 28, 385.
Paczuski M., Bak P., Maslov S. (1996): Phys. Rev. E 53, 414.
Cafiero R., Gabrielli A., Marsili M., Pietronero L. (1996): Phys. Rev. E 54, 1406.
Vendruscolo M., Marsili M. (1996): Phys. Rev. E 54, R1021.
Maslov S. (1995): Phys.Rev.Lett. 74, 562.
Marsili M., Caldarelli G., Vendruscolo M. (1996): Phys. Rev. E 53, 13.
Cafiero R., Gabrielli A., Marsili M., Torosantucci L., Pietronero L. (1996): in prepara
tion.
Tosatti E., Zannetti M., Pietronero L. (1988): Z. Phys. B 73, 161.
Bouchaud J. P., Comtet A., Georges A., Le Doussal P. (1990): Ann. Phys. 201, 285.
Marinari E., Parisi G. (1993): J. Phys. A Math. Gen. 26, L1149.
View other sources
Hide other sources
 Available from Andrea Gabrielli · Aug 21, 2014
 Available from arxiv.org