Complex Exponents and Log-Periodic Corrections in Frustrated Systems

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Complex Exponents and Log-Periodic Corrections in Frustrated Systems

H. Saleur, D. Sornette

To cite this version:

H. Saleur, D. Sornette. Complex Exponents and Log-Periodic Corrections in Frustrated Systems.

Journal de Physique I, EDP Sciences, 1996, 6 (3), pp.327-355. �10.1051/jp1:1996160�. �jpa-00247188�

Complex Exponents and Log-Periodic Corrections in Frustrated

Systems

H. Saleur (~>*) and D. Sornette

(~,**)

(~)

Department

of

Physics

and

Astronomy, University

of Southem

Califomia,

Los

Angeles

CA 90089-0484

(~) Laboratoire de Physique de la Matière Condensée

(***)

,

Université des Sciences, B-P. 70, Parc Valrose, 06108 Nice Cedex 2, France

(Received

là October 1995, received in final form 14 November 1995,

accepted

4 December

1995)

PACS.64.60.Ak Renormalization group, fractal and

percolation

studies of

phase

transitions

PACS.05.70.Jk Critical point phenomena

Abstract.

Recently,

it has been observed that rupture _processes ⁱⁿ

highly

disordered media

and

earthquakes

exhibit universal

log-periodic

corrections to

scahng.

We argue that such _correc- tions should

actually

be present ⁱⁿ a wide class of disordered systems ^and

provide

_a theoretical

framework to handle them.

Ai the naivest level, _a natural

explanation

for

log-periodic

corrections is discrete scale invari-

ance, a notion qualitatively similar to the concept of "lacunarity". However in nature, _any

such structure would be

largely perturbed

by disorder. We therefore

investigate

first the effect

of disorder _on the

log-penodic

corrections.

Remarkably,

_we find that

they

_are

generally

_ro-

bust. We discuss _a variety of disorder associated

effects,

like renormalization of the

log-periodic frequencies.

We then _{propose a} general explanation based _on the fact that _a discrete fractal _is actually a frac-

tal with complex

dimension,

and then that complex critical exponents should

generally

be

expected

in the field theories that descnbe

geometrical

systems, because the latter _{are non}

unitary.

We discuss detailed features of _non

unitary

theones, and present evidence of

complex

exponents in lattice

animais,

_a

simple geometrical generahzation

of

percolation,

which _can be

argued

to be associated with rupture.

Finally,

_we extend

Dur discussion to _more

general

frustrated systems. ^We

reemphasize

that the

non-unitarity,

generated

here by the averaging over disorder, _can lead to

complex

_exponents,

as were

actually

found earlier _{m some E expansion}

approaches.

More

physically,

_smce rephca symmetry

breaking

_is described

by

_an ultrametric tree, ^it _may

natùrally

lead to discrete scale invariance, albeit not in real space but in

replica

_space. We then

study

_a

dynamical

model

descnbing

transitions between states _{in a} hierarchical system ^{of barriers}

modelhng

the energy

landscape

m the

phase

_space of meanfield

spmglasses,

that leads agam ^to

log-periodic

corrections.

We conclude

by mentiomng

_a few

physical

_cases where _we think

log-penodic

corrections should

be observable.

(*)

Packard Fellow

(**)

Author for

correspondence (e-mail: sornette©naxos.umce.fr)

(***) ^{CNRS URA} 190

©

Les Editions de

Physique1996

Résumé. Récemment, des corrections

log-périodiques

universelles _aux lois de

puissance

dé- crivant les

processus de rupture en milieux

hétérogènes

et les statistiques de tremblements de

terre ont été décrites. Nous

suggérons

_{que ces} corrections

log-périodiques

devraient être

pré-

sentes dans

un

grand

nombre de

systèmes physiques

_en milieux désordonnés et _nous proposons

un formalisme

général

_pour les décrire.

De manière naive, ^les corrections

log-périodiques s'expliquent simplement

_par l'existence d'une invariance d'échelle

discrète,

notion

rappelant qualitativement

la notion de fractal "lacunaire".

Dans les

systèmes naturels,

_une telle structure doit forcément être perturbée _par du bruit _ou du désordre. Nous analysons ainsi l'effet du désordre _sur les corrections

log-périodiques.

De manière

assez

remarquable,

_on trouve qu'elles sont _en

général

robustes. Nous discutons différents effets

induit _par le désordre, _comme ^la renormalisation des

fréquences log-périodiques.

Nous _en proposons ^ensuite une

explication générale

basée _sur l'idée

qu'une

fractale discrète n'est

autre

qu'une

fractale

possédant

_{une ou}

plusieurs

dimensions fractales

complexes.

De tels expo-

sants

critiques complexes

sont

généralement

autorisés dans les théories de

champs

_qui décrivent

les systèmes

géométriques

non locaux, _car étant non-unitaires. Nous discutons différentes _pro-

priétés

des théories des

champs

non-unitaires qui sont pertinentes _pour les exposants

complexes.

Nous

présentons

des évidences

numériques d'exposants complexes

dans la statistique des "ani- maux", une

généralisation géométrique simple

de la

percolation

qui peut être associée à certains

problèmes

de rupture.

Nous étendons notre discussion à des systèmes plus

généraux

^frustrés. Ici, la non-unitarité est

produite

_par la _{moyenne sur} le désordre et peut ^à nouveau conduire à des exposants complexes,

comme ceux trouvés _par des calculs passés effectués _par

développements

_{en E.} Physiquement.

nous suggérons _que l'invariance d'échelle discrète

apparaît

dans l'espace des

phases

et est associée à la brisure de

symétrie

des

répliques

donnant lieu à _une structure

ultramétrique.

Nous étudions

un processus

dynamique

dans _un tel _espace de

phase possédant

des barrières

arrangées

selon _une

structure

hiérarchique,

_qui

représente

la structure des

énergies

d'un modèle de _verres de _{spm en}

champ

_moyen. On met _en évidence à _nouveau des corrections

log-périodiques.

Nous concluons brièvement par

quelques exemples

où _{nous pensons que} des exposants

complexes

pourraient

être observables.

1. Introduction

Log-periodic

oscillations in

scaling systems preceding

trie critical point ^{bave been observed} in several

systems. Probably

trie first theoretical

suggestion

of their relevance to

physics

has

been

put

^forward

by

^Novikov

iii

to describe the influence of

intermittency

in turbulent flows.

Prehminary experiments

_[2] seem to observe these oscillations _on the

time-dependent

rate of convergence of structure functions. Shell models of

turbulence,

which have attracted

recently

a lot of interest [3], assume

implicitely

discrete scale invariance. In these

models,

self-similar solutions of the cascade of the

velocity

field and energy in the discrete

log-space

scale have been unravelled

[4],

^whose

scaling

_cari be related to the intermittent corrections to

Kolmogorov scaling.

We note that these solutions

rely

_on the discrete

scahng

shell _structure and would

disappear

in the _continuons limit.

However,

the relevance of these discrete hierarchical models and _more

generally

of

log-periodic

oscillations have trot been

explored systematically

and their

confirmation in turbulence _{remams open.}

The interest in

log-periodic

oscillations has been somewhat revived after the introduction of the renormalization _group

theory

of critical

phenomena. Indeed,

the mathematical existence of such corrections has been discussed

quite early

_in renormalization _group solutions for the

statistical mechanics of critical

phase

transitions [5]. However, these

log-periodic oscillations,

which amount to consider

complex

critical

exponents,

_were

rejected

for

translationally

invariant

systems,

on the

(trot totally correct,

_{as we} discuss

below)

basis that _a

period (even

in _a

logarithmic scale) implies

the existence of _{one or} several characteristic

scales,

which is forbidden in these

ergodic systems

in the critical

regime. Complex

exponents were therefore restricted to

systems

with discrete renormalization _groups.

An _area where

they

were observed at

length

is low-dimensional

dynamical systems. Indeed~

the

Feigenbaum/Coullet-Tresser

_sequence of subharmonic bifurcations to chaos _cari be under- stood from _an

asymptotically

exact discrete renormalization group with _a universal

scaling factor,

therefore

leading

to

complex

exponents and

log-periodic

oscillations around the main

scaling

as the

dynamics

_converges to the invariant Cantor set _measure at

criticality _[6].

Discrete scale invariance and

complex _exponents

have aise been discovered in

dynamical

stochastic _processes in

homogeneous systems

when there exists a

suficiently strong

inter-

mittency,

such _as in mortels of diffusion in

anisotropic quenched

random media

[7].

^{In the} presence of intermittent

amplification

_processes

acting

_{on a}

given variable,

the

probabil-

ity distribution of this variable _cari aise exhibit _a

log-periodic

modulation

[8,9].

This _cari

in fact be _{seen as}

being

the _same mechanism _as in diffusion in

anisotropic quenched

ramdam media I?i: the

amplified

variable is the time necessary ^to diffuse _{over a}

given

distance and the

corresponding probability

distribution of time intervals _cari exhibit

log-periodic

oscillations.

Interplay

between this mechanism and _a

pre-existing

discrete hierarchical _structure has been addressed in

[10].

Another _area

involving

discrete renormalization _group is trie

study

of Boolean

delay

_equa-

tions. These are evolution

equations

^for a vector of discrete

variables,

which mortel trie _evo- lution of

biological

and

physical _systems

with threshold behavior and nonlinear feedback

[11].

It has been found

that,

for boolean

delay equations involving

two time

lags

with _an irrational

ratio,

the cumulative number of

jumps

J presents a

superdiifuse

behavior with

log-periodic

oscillations

[11].

Then of _course, discrete fractals

by

construction lead to discrete renormalization group

equations.

It has been

pointed

eut for instance that vibration and _wave

properties

_on ^discrete

fractal structures should be characterized

by log-periodic

corrections to the

leading singular

behavior for the

density

of states close to the baud

edges [12].

^{This remains} to be observed

expenmentally.

More

recently,

_a clear-cut

experimental

verification of the

generation

^of

log- periodic

oscillations

by

discrete scale invariant fractals has been carried _on man-made

Sierpinski

networks of normal-metal

hnks,

in which the

normal-superconductive

transition temperature presents

log-periodic

oscillations _{as a} function of the

applied magnetic

field. The _{same occurs} for the variation of the network electric resistance _{as a} function of the

magnetic

field

[13].

Interestingly,

it has been

pointed

ont that the

morphology

of the bronchial

airway

of trie mammalian

lung

is

roughly

hierarchical

leading

to a

log-log plot

of the average diameter of _a branch of _a mammalian

lung (for human, dog,

rat and

hamster)

_{as a} function of the branch order which exhibits _a fuit S-oscillation

(log-periodic) decorating

_{an average} linear

(power law) dependence.

This

complex

fractal structure has been

argued

to allow the _organ to be _more stable with respect to disturbance

[14,15]

(~

Notwithstanding

these few

examples, complex exponents

have

always

been considered _as anathema. A first _reason for this is that

they

do not appear in the canonical

exactly

solved mortels of cntical

phenomena

like the square lattice

Ising

model _or Bose Einstein condensation.

This is because such mortels

satisfy

_some sort of

unitarity.

^{We will discuss} this _more

below,

but let _{us give a}

simple example.

From conformai invanance

[16],

it is known that the

exponents

(~) This

stability

_is remmiscent of the robustness of discrete scale _mvariance and

log-penodic

oscilla- tions _in the _presence of

disorder,

that _we demonstrate below

of two dimensional critical mortels _can be measured _as

amplitudes

of the correlation

lengths

m a

strip geometry.

Since the

Ising

mortel transfer matrix _can be written in _a form which is

symmetric,

ail its

eigenvalues

_are

real,

therefore ail its

exponents

_are real. Trie other standard

example

where

exponents

_can be

computed

is _c

expansion.

However in that

context,

there is _an

attitude,

inherited from

particle physics,

to think

mostly

of Minkowski field theories.

For instance in axiomatic field

theory,

Eudidian field theories _are defined

mostly

_as

analytic

continuations of Minkowski field theories

satisfying

a standard set of

"Wightman

axioms"

[17].

Now

complex

exponents, as we argue

below,

make

perfect

_sense for Eudidian field

theories,

but lead to

totally

ill-behaved Minkowski field

theories,

with

exponentially diverging

correlation

functions. An

approach

based _on any ^sort of

equivalence

between the two

points

of view is bound to discard

complex exponents (as

well say as

complex masses).

The current

feeling

_m

the literature until

recently

was that such mathematical monsters coula be observed

only

_in

non

natural,

man-made

systems.

Dur interest in this

problem

has been launched

by

the very ^recent

discovery,

in _a series of _ex-

perimental

observations _on rupture in very

heterogeneous media~

of the existence of

logarithmic periodicities

_on the

approach

to the

major

breakdown

point.

In the first _set of

observations,

the _rate of acoustic emissions which

preceded _rupture

of _pressure vessels

composed

of carbon liber-reinforced resin is found _to increase _{as a} power law of the time to

failure,

decorated

by log-periodic

oscillations

[18].

^{A similar behavior has been documented} on foreshock

activity preceding large earthquakes

in California and trie Aleutian Islands

[19].

^An

exactly

soluble mortel of rupture, trie hierarchical

time-dependent

liber bundle of Newman and collabora- tors [20] bas been shown to

obey

_a discrete renormalization group from which

log-periodic

corrections follow [21] which reinforce trie

experimental

observations. This evidence for

log- periodic

corrections _m

physical systems prompts

a reconsideration of

complex exponents,

^which

is trie purpose of this _paper.

It is

important

to

emphasize

that

log-periodic

corrections bave _a tremendous

practical

in- terest; their

monitoring

coula allow indeed trie

prediction

of trie main

rupture

event

(e.g.

trie main

earthquake). Indeed,

a series of

analysis

bave shown that identification of trie

log-periodic

structure in the acoustic emission rate allowed trie failure _pressure to be

predicted

within less

than 5$io _error when the test pressure had reached

only 85%

failure

[18].

^{Similar tests for earth-}

quakes

show that

"post-diction"

_can be clone with remarkable

precision

in several _cases

[19].

The

point

^is that

log-periodic

oscillations constrain much _more

drastically

the lits to

experi-

mental data than do

simple

_power laws _{smce one} has to

adjust

to the

specific geometrically decaying

local

periodic behavior,

hence the enhancement in fit

quality

and

predictive

_power.

Extreme

caution

should be exercized however before

concluding

that this coula be useful for

earthquake predictive

_purpose

(see

discussion in [19] ^and more work needs to be clone

[22].

As discussed

above,

the

simplest

foreseeable

origin

^of

complex exponents

in rupture

problems

is the existence of _an

underlying

discrete fractal structure

of,

_say, cracks _or faults in the rupture

problem. By

discrete

fractal,

_{we mean a} fractal which

enjoys

discrete scale invariance. Let

us stress that _{one must}

distinguish

between the notion of _a discrete fractal _{on one} hand and

fractals _{on an}

underlying

discrete lattice _on the other hand.

Ising

clusters _{on a} square lattice _at the critical

point

sit _{on a} discrete lattice but _are not "discrete

fractals";

instead

they

become

invariant under any continuous

rescaling (in

the

large

size

hmit)

in contrast to a discrete fractal which

keeps being

invariant

Orly

under

special

discrete values of

rescaling

factors _even in the

large

size limit. We _{can say} that discrete scale invariance _is associated to _a kind of

regularity

ⁱⁿ the

scahng structure;

^for instance, deterministic fractals like the

Sierpinski gasket

are

particularly

clear

examples

of this Mass which

enjoy

discrete scale invariance. This _is also trie mechanism

leading

to

log-periodic

oscillations for spin

systems

close to trie critical

point

on in a hierarchical lattice

[23]. Similarly,

trie existence of _a discrete hierarchical _structure

aise leads to a discrete renormalization group in rupture

[21].

^{On the other}

hard, completely

ramdam fractals like

Ising

clusters _are self-similar

Orly

_{on average} ^and

enjoy

the

property

of continuous scale invariance _on the _average. We note that the

property

^{of discrete scale}

invariance,

which is associated to _a kind of

regularity

in the

scaling structure,

_is reminiscent

of,

albeit _more

specific than,

the

property

of

"lacunarity",

_a

complex

notion

embodying

the ideas of hales and

partial

_coverage

occurring

at ail scales.

If indeed _a

preexisting

crack _or fault

system

_{possesses a} discrete fractal structure, ^{it is} to be

expected

that this structure is net

perfectly regular,

and _a first

key question

is the eifect of the disorder of this structure _on the

log-periodic

oscillations. This is addressed in the first

part

^of this paper, where _we show that

actually

disorder ares trot

destroy

these

oscillations, although

it

might

lead to renormalizations of various

quantities.

^{The influence of disorder} in mortels of intermittent

amplifications,

aise

leading

to

complex exponents,

^{has been addressed}

using

diiferent methods in [8] with similar conclusions.

Of _course,

by invoking

_a

preexisting

^discrete fractal structure, one is

just hiding

trie

problem

under trie

carpet.

What

produced

this structure in trie first

place?

Instead of discrete fractal

structure,

_we

might actually

think that trie

geometry

^{of cracks and faults itself} is described

by

_a

geometrical

critical

phenomena

with

complex exponentsl (see _{[21, 24]}

for _a discussion of

discrete

geometrical

fractals and trie

description

of their

lacunarity

in terms of

complex

fractal

dimensions).

We _now bave _to discuss _a

presumably time-independent geometrical problem,

and _{we cari} gain from trie

experience

accumulated in that field. In

particular,

such

problems

are described

by non-unitary

euclidian fleld

theories,

for _{reasons we} will discuss below. In trie second part ^{of this} _paper, we envision trie formai

question

of

having complex exponents

ⁱⁿ such theories. We find it to be very much

possible indeed,

in

opposition

to _common

tore,

and

even exhibit

exactly

solved

examples.

Moreover _{we argue,} based _on numerical transfer matrix

computations,

that such

complex _exponents

_{are present} in trie

problem

of two-dimensional lattice animais. Note that trie

problem

of lattice animais _is trie most natural

generalization

of trie

percolation mortel,

which itself is trie

prototype

^{of disordered}

systems.

The existence of

complex

_{exponents in} these

paradigms suggest

furthermore their relevance _in other disordered

systems,

^and in

particular provide

the root of _a theoretical framework for rupture

problems.

For instance, it has been

suggested

that

percolation

coula be _a

good

mortel of fault and

earthquake organization

in trie crust

[25].

^{We believe that lattice animais coula be} an even better _approx-

imation,

since

percolation

transforms into the animal

problem

under

generic perturbations.

Trie

physical

mechanism for discrete scale invariance _in fault structures may came from trie fact that if

large

structures

(faults)

start to dominate with _a characteristic distance controlled

by

trie

largest

scale of trie

problem, namely

trie thickness of trie crust, ^trier antithetic faults

must

growth

as trie deformation _occurs to accommodate trie kinematic

compatibility

constrains

developing

_in between trie faults. Trie

secondary

faults should therefore bave _a

length

of the order of the

separation

of the

major

^{faults. At} their

scale,

the process of kinematic

compatibil-

ity

must _occur

again

with the creation of still another

generation

^{of smaller faults} and _{SO on.}

It is thus rather raturai _to envision _a hierarchical cascade of fault

scales,

which ail

together

ensure the

compatibility

of the deformation. This picture has _in fact been advocated

by

several

authors in the

geological

Iiterature and found _to descnbe

adequately

available data

[26].

The existence of

complex exponents

ⁱⁿ rupture ^and

geometrical

mortels of disorder

suggest

that

they

should aise be

present

in other fractal

growth

models

(after aII,

it has been

recognized

that rupture in disordered

systems

is a

particular

realization of

growth

models

[27],

here trie

growth

of cracks _{up to an}

incipient percolating macro-crack).

It is thus

interesting

to mention that _we bave

recently

discovered

complex

_exponents in trie

diffusion-Iimited-aggregation (DLA)

mortel toc

[28].

^The

underlying physical

mechanism is based _on the existence of three

ingredi-

ents 1) ^a characteristic

microscopic

scale

(providing

_an ultraviolet cut-cif of trie

theory), ii)

_an

instability

at short

wavelength (of

trie Mullins-Sekerka

type)

and

iii)

a very eficient

screening leading

to _an almost

decoupling

of diiferent mode instabilities.

Log-periodic

oscillations then

stem from trie

hierarchy developing

in trie _succession of instabilities

[29].

We suspect ^that

complex exponents might

_appear in another kind of _non

unitary theories, replica type

theories associated with disorder.

Indeed, nothing prevents

a

priori quenched

dis- order

systems,

^{with their}

replica-symmetry breaking,

to exhibit

complex

critical

exponents.

^In

fact, they

have been found in

(-expansions

renormalization _group calculations of critical behav-

ior of

spin glasses _[30]

and of random and constrained

dipolar magnets [31], notwithstanding

the fact that these calculations [30] ^have been clone in the

high temperature phase,

1-e- away from the

spin- glass phase.

The authors of these works felt ill-at-ease with this result which has net been considered

seriously.

To _our

knowledge,

there exists _no

experimental

confirmation of this

prediction which,

_we note, would be very dillicult to carry ^eut.

However,

at the theoretical

level,

there is _a

simple

_reason

why _strong

disorder

might generate complex exponents,

and this

will

actually bring

_us back

formally

in the last

part

^of the paper ^to where _we started:

replica

symmetry ^{is broken}

following

_a

discrete,

albeit

probabilistic,

fractal

pattern!

We shall present

a

specific

calculation of the

dynamics

of _a disordered

system

with

replica-symmetry breaking, exhibiting log-periodic oscillatory

corrections to the main In t

dynamics.

It is net clear to _us whether the

probabilistic

structure of the hierarchical _tree of the

rephca _symmetry breaking

will

destroy

_or net these oscillations. This is left for _a future

investigation.

2. Discrete Scale Covariance

In this

section,

we recall how to derive

complex

critical exponents from _a discrete renormahza- tion group. We obtain the relative

amplitudes

of the harmonics of the Fourier series

expansion

of the

log-periodic function,

which

gives

the

leading

correction to

scaling.

We discuss the eifect of _a first non-linear term in the flow _map

in the

generation

of

log-periodic

accumulation of

singular

_points. A naive way of

obtaining

discrete scale _{covariance is} to consider _a discrete

fractal,

where the emergence of

complex exponents

has been known for _a

long

time

[32].

^The

point

^is that _{on a} discrete

fractal, only

discrete renormalizations _are

allowed,

_so these

abjects

have

actually

_a

complex

fractal dimension.

Calling

K the

coupling je-g-

K

=

e~/~

_for

a

spin mortel)

and R the renormalization group map between two successive

generations

of the discrete

fractal,

_one has

AK)

=

glK)

₊

flRlK)1, Ii

11

where

f

is the free _{energy per} lattice site or

bond,

_{g is a}

regular part

which is made of the free energy of the

degrees

of freedom summed _over between two successive

renormalizations, 1/~t

is the ratio of the number of

degrees

of freedom between _two successive renormahzations. The

equation (1)

is solved

recursively by

c~

AK)

=

~j mglR~"~(K)1,

12)

«=o ~

where

R(")

is the n~~ iterate of the renormahzation transformation. For

general ferromagnetic systems,

R has two stable fixed

points corresponding

to K

= 0 and K

= oc fixed

points,

^and

an unstable fixed

point

at K

=

K~

which

corresponds

to the usual critical

point.

It is easy ^to

see that

f

is

singular

at

K~ [33].

Call the

slope

of the RG transformation at

K~

_{((À( >} 1

since the fixed point ^is

unstable).

Then the i~~ term in the series for the k~~ derivative of

f

at the fixed

point

will be

proportional

to (À~

/~t)~, giving

^{rise to} a

divergence

of the series for the k~~ derivative of

f

for k

large enough,

hence the

singular

behaviour. Now _assume that

f

_«

(K K~(~

close to the critical

point. Plugging

this form _in

il) gives

the

constrain,

since g is a

regular function,

~m

=

13)

P This

equation

has _an

infinity

of solutions

given by

mi = m +

iifl, (4)

with

~

~

IÎÎ'

~

ln~'

~~~

and is _an

arbitrary integer. Decorating

the main

algebraic behaviour,

there _is thus the

possibility

of _terms of the form

(K K~(~ cos(iflln (K K~(

+

çii),

which _are the looked for

log-periodic

corrections. Such terms _are not

generally

scale covariant. If continuous

changes

of scale _were

allowed, they

would be

incompatible

with scale

covariance,

and thus forbidden

(unless

the

concept

ofscale covariance is

appropriately refined,

_see the discussion in Sections 4.1 and

4.4).

It is

only

because the

changes

of scale _are discrete that

they

_survive here:

only

when the

change

of scale is of the form

jK _K~(

-

À~(K K~(

do

they

transform

multiplicatively, ensuring

scale covariance.

To find out _more about these correction terms it is very convenient to _use the Mellin trans-

forn1, following [23].

^{Introduce therefore for} any function

f

the Mellin transfornl

/(s)

e

/ ^~ _~~~~f(~)d~

We then consider the free energy in

(2)

in the linear

approximation, replacing R(")(K) _by

K~

+

À"(K K~). Setting

_{~ =} K

K~

_we have

then, by applying

this transformation to bath stries of

(2)

~~~~ ~~~~

lÎ~~~

1' ^~~~

We then reconstruct the

original

^function

by taking

the _inverse transform

c+~c<

f(~)

= =

/(s)~~~ds

2m

~_~~

The usefulness of the Mellin transform is that the _power law behavior

springs

out

1nlmediately

from the

poles

of

lis), _{using Cauchy's}

_theorem. For _a

general

statistical mechanics

model,

_g

being

the

regular _part

of the free energy has

generally

the form of the

logarithm

of _a

polynomial

in z.

Factorizing

the

polynomial,

_we do not lose

generality by considering

_g

given by g(~)

₌

inji

+

~)

for which

à(S)

~

"

~ ~~~~~

In

inverting

the Mellin

transform,

_we have two

types

^of

poles.

The

poles

of

§

_occur for _s

= -n,

n >

0,

and contribute

only

to the

regular

part of

f,

_as

expected

since g is a

regular

contribution.

The

poles

of the

prefactor

_in

(6),

which _stem from the infinite _{sum over successive}

embeddings

of

scales,

_occur at

s = -mi,

(7)

Rjx)

, ' ' ' '

' ' '

' '

x

Fig.

1. In _{some cases,} it is

possible

to have _a second fixed point for _a discrete renormalization

group.

Physical quantities

_are

singular

at this fixed point ^and also at ail its preimages

la

few _are indicated _on the _z

axis).

The latter accumulate

geometrically

towards the main critical point.

with _{mi as} in

(4). They correspond exactly

to the

singular

contributions discussed above.

Their

amplitude

is obtained

by applying Cauchy's

theorem and is of the order of

1

m + iifl sin

~(m

+

iifl)

which behaves at

large

as

e~~~".

_{Hence the}

amplitude

of the

log-periodic

corrections

decays exponentially

fast _{as a} function of the order of the harmonics. This

explains

the fact that

only

the first harmonic has been _{seen so} far in the

analysis

of

experimental

data

[18,19].

Of _course the _previous

computation

suifers from the linear

approximation,

which becomes

deeply

incorrect _{as n}

gets large

in

(2),

hence in the

region determining

the

singularity.

As discussed in

[23],

the crucial

property

missed

by

the linear

approximation

^is that

f

is

analytic

only

in _a sector

(arg~(

_< à while _we treated it _as

analytic

in the cut

plane (argx(

< ~. The

true

asymptotic decay

of the

amplitudes

of successive

log-periodic

harmonics is therefore slower than

mitially found,

and goes as

e~~~~.

The

angle

à

depends specifically

_on the flow _map of the discrete renormalization _group [23] ^{and is}

generally

of order 1.

Assuming

this formula

e~~~~

of the

decay

still

gives

the

right

order of

magnitude

for

=

1,

_{we see} that the

amplitude

of the first mode _can be much

larger

^{than found above} in the linear

approximation.

The linear

approximation

_{misses more}

interesting features,

discussed in

[33]. Beyond

the linear

approximation,

the renormalization transformation R _can be

expanded

_{as a}

polynomial

in _x. Let us

keep

the first two terms

only,

_so

R(~)

= À~

-u~~

_with

u > 0. Such _{a non} monotonic RG transformation _occurs for instance _m frustrated mortels with both

ferromagnetic

and

antiferromagnetic

interactions. An

interesting

consequence is that this transformation has another unstable fixed

point

at ~c =

~j~

The function

f

is

singular

at all the _pre-images of this other fixed

point,

if the absolute value of the

Lyapunov exponent

^of R at this fixed

yoint

is

greater

than _one

namely

if1 < < 1+

[

or < À- _or >

À+

where

À+

₌

~.

This is due to the fact that these preimages all go ^to the fixed

point

after _a finite number of renormahzations. Moreover these _preimages accumulate at the

original

unstable fixed

point

x = 0

(Figure

_1)~ and their accumulation becomes

geometrical

_very close to _{x =} 0. When

crossing

^such a pre-image, there is _a

singularity

in

f, usually

manifested _{as a} kink in the _curve.

This sequence of kinks close to _x

= 0 _con be fitted well with the first harmonic of

log-periodic

oscillations discussed

previously.

But the discussion

suggests

^{that there is} more to be _seen in these oscillations the

geometrical

accumulation of critical points ^towards the "main" critical

point.

In the context of

earthquakes,if

the network offaults under consideration is indeed _a discrete

fractal,

with the critical

point Kc interpreted

_as the main

event,

^{it is}

tantalizing

to think of these other

singularities

_as the

pre-shocks.

Their accumulation to the

original

fixed

point

would

correspond

then to the increased _seismic

activity

close to the _main

earthquake,

which has been documented in _a

variety

of work in the

seismological community. Indeed, _quite

_a few

large earthquakes

have been

preceded by

_an increase in the number of intermediate size events

[34].

It is

interesting

to note that the relation between these intermediate events and the

subsequent

main _event has

only recently

been

fully recognized

because the precursory ^events occur over

such _a

large

_area. Based _{on a}

simple

stress

analysis,

it is dificult to see how such

widely separated

events coula be

mechanically

related.

However,

if the

seismicity

in _a

given region

is viewed _{as a sequence} of seismic

cycles,

and each

cycle

is viewed _{as a}

progressive cooperative

stress

huila-up culminating

in _a critical

point

characterized

by global

^failure in the form of _a

great earthquake,

then the observed increase of

activity

and

long-range

correlation between events _are

expected

_to

precede large earthquakes.

Having

recalled how

log-periodic

corrections follow

naturally

from discrete scale

invariance,

we first discuss what is the eifect of disorder _on this

phenomenon.

This _is of crucial

importance,

since _even if it is confirmed that say a network of faults has discrete scale

invariance, surely

the latter _is not

perfect

and

presents

fluctuations from scale to scale.

3. The Efiect of Disorder

We first present a

general

treatment of disorder in the

coupling

coefficients _on the diamond

lattice,

in terms of the renormalization of the distribution of

couphngs.

This then allows _us to

suggest

a

simple

ansatz for the structure of the disordered renormalization _group, which _can be treated

exactly.

Related issues have been addressed in _a diiferent _context and

using

diflerent methods _in

[8].

3.1. GENERAL FORMALISM. We restrict _our discussion to the

example

of the diamond

lattice,

whose

properties

are quite

generic.

With _a random distribution of "bare"

couphngs

characterized

by

the normalized distribution

Po,

the average free _energy reads

[35].

f ((Pol)

"

~j /

dK

/ dK'An(K)Pn(K')g(K, K')dKdK', (8)

~

ll

where

g(K,K')

=

) In(K

+

K'+ Q 2)

and _~t

= 4.

Calling

R the renormalization group transformation _as

before,

_we have

Pn(K)

=

fl /

Pn-i (K~)à[K R(Ki, K2, K3, K4)]dK~. (9)

Î~

Let _us first discuss the _case of weak disorder. 0ne _can then characterize the distributions

by

their moments, and _reexpress the _sum

(8) using

_an

expansion

_on these _moments.

Conceptually

the results _are the _same if _one

keeps only

the first two moments, which _we will do here.

Introduce

Mn

=

/KPn(K)dK

Vn

₌

/(K Mn)~Pn(K)dK, ₍₁₀₎

and denote

by M',

V* the _same

quantities

^for the fixed

point

distribution. A distribution

P(K)

which diflers

slightly

from P*

(K) corresponds

to

saying

that each bond has _some deviation

AK,

distributed

according

to

ôP(AK). Then, P(K)

=

f dxP*(K x)ôP(x) (note

the convolution instead of _a

simple

additive

perturbation). Up

to second

order, P(K)

_can be written _as

P(K)

m

P*(K)

ôM ^~~~

~~~

+

~~

~~~~~fi~, ^Ill

where ôfi~

= M

M*,

^ôV ₌ V V*.

Using iii

_one finds then

/dKdK'An(I()Pn(K')g(K, ^K')dKdK'

^m

^{g(Mn, Mn)}

⁺

^{g"(Mn, Mn)Vn. (12)}

The renormalization group

equation

for the moments reads

[36].

IÎÎ"

=

TIR) Î~"j~~

,

l13)

where

~~~~

<

ôill Î~Î<

R ><

ôiR

> <

ô2R~

_> -2

ÎIÎ~ÎÎô2R

> ^' ~~~~

~lld

~ /Jn j~

~

~"~

~~

/~J~n ^' ~~~~

il

~

?#l ~

the averages

being

taken _over the fixed

point

distribution.

The

diagonalization

of

T(R) gives

the two most relevant directions.

Usually

for weak disor-

der,

one direction is relevant

(the

thermal

one)

and the other _one, the "disorder

direction",

is

irrelevant if the Harris cnterion is met

[37].

Calhng Ài

the

largest eigenvalue

of

T,

_one has for

large

_n both

ômn

_«

À[

and

ô~[

_ec

À[.

Hence the

leading poles

of the Mellin transform _now follow from the

condition,

obtained

by inserting (11)

in

(8),

ù _~'

~~~~

or

~~

ÎÎÎ

_~ ~~~

n~Ài

~~~~

More

poles

_appear due to the _À2

dependence

and various

higher

order _{terms in} the expansions, which all contribute to corrections to

scahng.

Beyond

the weak disorder

approximation,

we can still write

f dI(dÂ'An(K)Pn (K')

_g

(Ii, K')dKdK'

=

f dKÎ dKmndK[ dK$nPÙ (KÎ PÙ(I(mn )PÙ(I([). PÙ(K$n (18)

g

[R(")j K~

K

n) R(")j

K' K'

~

)j

where

Po(K)

_is the bare distribution and _m describes the

branching

of the fractal

lattice,

m = 4 for the diamond lattice. To find the

leading

exponent, we can use any

generic

^direction of

approach

to the critical point. ^We consider thus the _case of _a distribution

diflering slightly

from P~

(Ii)

_so at dominant order

Po(Â)

m

P*(K) ôm~ (/~

m

P~(K ôm),

_or

Po(K)

=

P*(K ~),

_x

being

_a small

parameter,

^which we use as the distance to the critical

point.

The

singularity

_{as x -} 0 will

give

the

leading exponent,

even with

strong

disorder. Introduce

~~~ ~~~ P~ÎK) Î _{~~~ ~~~~ [ÎÎÎÎÎÎI}

.~~ÎÎÎÎ ^j

~

~

Î~ ^~~~~ ~~~ ~~~ Î

(19)

Then _one finds that

lis)

=

f /~ x~~~dx / _{dKdK'P* (K)P~ (K')g(K}

+ ~,

K'

₊

~)F~,n (K)F~ n(K')

~~~

ll"

o ^'

~

Î~ ⁿ _(à(~,

~Ù>

~Ù'~~s,n(~Ù)~s,n(~Ù'))

(~Ù~

n=0 ~

The term

ôiR(")(Âi,

.,

Kmn

has _in fact the structure of _a

product

of _n correlated terms.

This is

easily

_seen

by taking partial

derivatives with respect to the variables at intermediate

renormalizations,

for instance

m m

/~j~(2)j~

_{1 1".} ~ /~

j~j~l _~m)

_/~ ~,~

j~~)

' m2 # ~,

,. , ~~ _,

~ ~

j

~=l j=1

where _we introduced the notation K~ for

couplings

^after one

renormalization, I(]

for bare

couphngs,

_so K~

=

R(K(,..., K[).

Because of this

product structure,

we

expect

the existence of _a

"Lyapunov"

exponent

Ais)

such that

pis, ii, ~/)F~,~j~)F~,~j~/))

_«

~

~~~,

n - ce.

j22)

The Mellin transform of the average has

poles

at

Ais)

=

~. (23)

11

3.2. THE RANDOM RENORMALIZATION ANSATZ

3.2.1. Results for tlle Ensemble

Average

of tlle Observable. In order to make progress, ^it is necessary ^to add _some further information _on the nature of the disorder. ive shall not

attempt

to solve the

problem

_in full

generahty

but propose a

simple

ansatz which has the double

advantage

of

being exactly

soluble while

capturing

the structure of

(19).

We thus propose that the eflect of

strong

^disorder can be

captured by considering

_a

simplified

model of "random

renormalization" where the

rescaling

factors and _~t used in

(3)

_vary from scale to scale. More

precisely,

_we

replace fig R(")(K)

cf

fig À"z)

,

where _~

= K

Kc,

in

equation (2) by

~

~n~

^g

(fl _Àj )x

The

~tj's

^and

Àj's

_are random numbers

descnbing

^the fluctuations of the j=1">

~=i

scaling

and of the flow _map at each iteration.

Taking

such _a random flow _{map at} each iteration

corresponds

to

changing

the

scaling

^factor of the RG decimation

procedure,

and therefore to

describing

_a hierarchical

system

with disorder _on the scale factor from _one level to the next of the hierarchical structure. Within this ansatz, we

have,

still _in trie hnear

approx1nlation,

flx)

=

~j ~n

^g

lfl

^À~x

,

Îo ^{~otl~ ~o} 124)

where _À~ and

~1~ are random variables taken out of _some

particular

distribution. If _we consider

again

the average of the free energy, one

finds,

the Mellin transform

being

_a linear

operation

that commutes with average,

<

/

_>

_(s)

=

§(s) (1+

< z > + < z

>~

_{+. .)}

=

§(s)

,

(25)

where _z

=

)

and the brackets denote the _{average over many} realizations. We stress that this result is exact due to the

property

^of conlmutation between the Mellin transform and the average. In

particular,

_we have _not

replaced

_a series of the _average of the powers of _z

by

a

series of the _powers of the _average of _{z, as can} be _seen

directly

from

equation (28)

below.

The

poles

of the Mellin transform _are the union of the

poles

of

§

_as in the ordered _case, and the values of _s

satisfying

)

" 1.

(26)

The first consequence one can draw from this

analysis

is that disorder suppresses the harmonics

si " -mi found in the ordered _case, since each diiferent _{power s}

gives

_a diiferent

weight

in the

ensemble average,

ensuring

for instance that _<

À~+~~~ >#<

À~ _>< À~~

>~.

There is another

interesting

consequence of the disorder that _{we now}

analyze.

Let _{us assume} for the sake of

simplicity

that the disorder is

only

_on the À's. Consider the two

leading

terms in the solution

f(x). They corresponds

to _a first

leading

_power law with real

exponent

m

and _a second

log-periodic

term with exponent

m'+

in. The two

exponents

are such that

< À~ _>=<

À~'+~~

_{> since}

_they

_both

satisfy

the

pole flquation

^{< À~~ >= ~1. Let us call}

^p(À)

the distribution of À.

Then,

Re _<

À~'+~~

_>

=

/ dÀp(À)À~'cos(flLogÀ).

Its modulus

~

o

is less than

/ _dÀp(À)À~'

_{and from the above}

_identity

< À~ >=<

À~'+~~

_{>, this}

_implies

o

m < m' if ^~

~~~l'~

> 0 and _{m >} m' _in the _{reverse case.}

Therefore, depending

_on the

specific

form of them distribution

p(À)

^{of the RG flow} _map

eigenvalues

À, ^{the disorder} leads to a renormalization of the real

part

of the critical exponent.

As _a first

illustration,

let _{us assume} that

p(À)

is

log-normal.

An

explicit

calculation shows

that m' _{> m} in this _case. The _reason for this is clear: _a

log-normal

distribution has _a

long

tail

towards

large

values and thus the average is dominated

by

the

large À's,

all the _{nlore so} when m'

gets greater.

^This is a case where the _{average is an}

increasing

function of

m',

hence the

result _{m <} m'. The concrete consequence for _a fit of

experinlental data,

such _as those obtained for

earthquakes

and similar

systenls,

is that the mathematical

expression

used in

fitting

seismic

activity

as clone

previously [18,19, _21]

should be

replaced by fit)

₌ A ₊

B(tf t)~

+

C(tf t)~'

cos

2~~°(~~

~

~~ +

il) j, ⁽²⁷⁾

°g

where the

only

diiference with _our

previous

lits

[18,19, 21]

is that the third term _in the r-h-s- of

(27)

has _{a new} exponent m' with m' _{> m,} which _can be viewed _{as m} renormalized

by

the disorder. This result _means that the relative

strength

of the

log-periodic

correction

compared

to the first

algebraic

term becomes smaller _as t

approaches

the time

tf

of the

large earthquake.

As _a concrete

illustration,

this situation has been found to describe the

log-periodic

correction

to

scaling

of the _mass

M(r)

of _a DLA duster _{as a} function of the radius _r [28]

M(r)

=

cor~U

+

c'r~'cos (2~flogr).

The

problenl

is similar _to the _one discussed above with

tf

^t

replaced by ^r~~ (the

DLA duster becomes critical at infinite

sizes)

and with _m

replaced by Do,