Field heo y o abso bing phase ansi ions wi h a nondi usi e conse ed ield
Romualdo Pas o -Sa o as1,2 and Alessand o Vespignani2
1Depa amen de Fı
´sica Fonamen al, Facul a de Fı
´sica, Uni e si a de Ba celona, A enida Diagonal 647, 08028 Ba celona, Spain
2The Abdus Salam In e na ional Cen e o Theo e ical Physics (ICTP), P.O. Box 586, 34100 T ies e, I aly
共Recei ed 19 June 2000兲
We in es iga e he c i ical beha io o a eac ion-di usion sys em exhibi ing a con inuous abso bing-s a e
phase ansi ion. The eac ion-di usion sys em s ic ly conse es he o al densi y o pa icles, ep esen ed as a
nondi usi e conse ed ield, and allows an in ini e numbe o abso bing con igu a ions. Nume ical esul s
show ha i belongs o a wide uni e sali y class ha also includes s ochas ic sandpile models. We de i e
mic oscopically he ield heo y ep esen ing his uni e sali y class.
PACS numbe 共s兲: 64.60.H , 05.50.⫹q, 05.65.⫹b, 05.70.Ln
The di ec ed pe cola ion 共DP兲关1兴uni e sali y class is ec-
ognized as he canonical example o he c i ical beha io in
he ansi ion om an ac i e o a single abso bing s a e. This
uni e sali y class appea s o be obus wi h espec o mic o-
scopic modi ica ions, and non-DP beha io eme ges only in
he p esence o addi ional symme ies, such as symme ic
abso bing s a es 关2兴, long- ange in e ac ions 关3兴, o in ini ely
many abso bing s a es 关4兴.
Recen ly, a new uni e sali y class o abso bing-s a e
phase ansi ions 共APT兲关1兴coupled o a nondi usi e con-
se ed ield has been iden i ied 关5兴. This class cha ac e izes
he c i ical beha io o se e al models showing APT wi h a
dynamics ha s ic ly conse es he densi y o pa icles, ha
is ep esen ed by a conse ed s a ic 共nondi usi e兲 ield. The
models a e uned o c i icali y by a ying he pa icle den-
si y, and exhibi an in ini e numbe o abso bing s a es. This
uni e sali y class is pa icula ly in e es ing because i em-
b aces also he la ge g oup o s ochas ic sandpile models 关6兴
共and in pa icula , he Manna model 关7兴兲 which a e he p o-
o ypical examples ha illus a e he ideas o sel -o ganized
c i icali y 共SOC兲关8兴. These a e d i en dissipa i e models in
which sand 共o ene gy兲is injec ed in o he sys em and dissi-
pa ed h ough he bounda ies, leading e en ually o a s a ion-
a y s a e. In he limi o in ini esimally slow ex e nal d i ing,
he sys ems app oach a c i ical s a e cha ac e ized by an a a-
lanchelike esponse. Recen ly, i has been poin ed ou ha
his c i ical s a e is equi alen o he APT p esen in he ixed
ene gy case; ha is, in au oma a wi h he same mic oscopic
ules de ining he sandpile, bu wi hou d i ing o dissipa ion
关9–11兴.
The nume ical e idence o he exis ence o such a gen-
e al uni e sali y class 关5兴is co obo a ed by he obse a ion
ha all he models analyzed sha e he same s uc u e and
basic symme ies; namely, a conse ed and s a ic nonc i ical
ield dynamically coupled o a nonconse ed o de pa ame e
ield, iden i ied as he densi y o ac i e pa icles. These ob-
se a ions ha e led o he conjec u e ha , in he absence o
addi ional symme ies, all s ochas ic models wi h an in ini e
numbe o abso bing s a es in which he o de pa ame e
e olu ion is coupled o a nondi usi e conse ed ield de ine
a unique uni e sali y class 关5兴.
In his Rapid Communica ion, we s udy he nondi usi e
ield limi o he wo species eac ion-di usion 共RD兲model
in oduced in Re . 关12兴共see also Re . 关13兴兲. In his limi he
model has a phase ansi ion wi h in ini ely many abso bing
s a es, and i conse es he o al numbe o pa icles ha is
associa ed wi h a nondi usi e conse ed ield. We p esen
ex ensi e nume ical simula ions o he model in wo and
h ee dimensions, and de e mine he ull se o c i ical expo-
nen s. The ob ained alues a e compa ible wi h he new uni-
e sali y class conjec u ed in Re . 关5兴. This de ini ely shows
he exis ence o a b oad uni e sali y class ha includes RD
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o his sec ion a e gi en p io i y ea men bo h in he edi o ial o ice and in p oduc ion, au ho s should explain in hei submi al le e
why he wo k jus i ies his special handling. A Rapid Communica ion should be no longe han 4 p in ed pages and mus be accompanied
by an abs ac . Page p oo s a e sen o au ho s.
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p ocesses, s ochas ic sandpile models, and la ice gases wi h
he same symme ies. Fo he p esen RD model, i is pos-
sible o de i e mic oscopically a ield heo y 共FT兲desc ip-
ion. The esul ing ac ion and Lange in equa ions exhibi he
basic symme ies ha cha ac e ize his uni e sali y class, and
ep esen a mic oscopic de i a ion o a FT o sandpile mod-
els. No ably, he esul ing FT desc ip ion eco e s a phenom-
enological Lange in app oach p oposed o sandpiles 关9,10兴.
The analysis p o ided he e is a e y p omising pa h o a
cohe en desc ip ion o se e al nonequilib ium c i ical phe-
nomena now a ionalized in a single uni e sali y class.
We conside he wo-componen RD p ocess iden i ied by
he ollowing se o eac ion equa ions:
B→Awi h a e k1,共1兲
B⫹A→2Bwi h a e k2.共2兲
In his sys em, Bpa icles di use wi h di usion a e DB
⬅D, and Apa icles do no di use; ha is, DA⫽0. This
co esponds o he limi DA→0 o he model in oduced in
Re . 关12兴. F om he a e Eqs. 共1兲and 共2兲, i is clea ha he
dynamics conse es he o al densi y o pa icles
␳
⫽
␳
A
⫹
␳
B, whe e
␳
iis he densi y o componen i⫽A,B. In his
model, he only dynamics is due o Bpa icles, which we
iden i y as ac i e pa icles. Apa icles do no di use and
canno gene a e spon aneously Bpa icles. Mo e speci ically,
Apa icles can only mo e ia he mo ion o Bpa icles ha
la e on ans o m in o A h ough Eq. 共2兲. In he absence o B
pa icles,
␳
Ais hus a s a ic ield. This implies ha any con-
igu a ion de oid o Bpa icles is an abso bing s a e in which
he sys em is apped o e e .
I is easy o see 关12兴 ha he RD p ocess de ined by Eqs.
共1兲and 共2兲exhibi s a phase ansi ion om an ac i e o an
abso bing phase o a non i ial alue o he o al pa icle
densi y
␳
⫽
␳
c. The c i ical alue
␳
cdepends upon he eac-
ion a es k1,k2. The na u e o his phase ansi ion o DA
⫽0 has been discussed in 关12兴; he s a ic ield case (DA
⫽0), on he o he hand, has ne e been explo ed o ou
knowledge. I is clea ha he s a ic ield conse ed RD 共SF-
CRD兲model allows, o any densi y
␳
, an in ini e numbe 共in
he he modynamic limi 兲o abso bing con igu a ions, in
which he e a e no Bpa icles. This is he key di e ence wi h
espec o he case in which DA⫽0. In he la e case a con-
igu a ion de oid o Bpa icles consis s o many di using A
pa icles. In he long un, all pa icles can isi all si es in he
la ice, and he e o e, in a s a is ical sense, all con igu a ions
wi h a ixed numbe o A’s a e equi alen and he abso bing
s a e can be conside ed unique 关14兴.
The SFCRD model seems o possess all he equi ed sym-
me ies 共s ochas ic dynamics, many abso bing s a es, s a ic
conse ed ield兲 o being pa o he uni e sali y class con-
jec u ed in Re . 关5兴. In o de o es his possibili y, we ha e
pe o med nume ical simula ions o he model in a
d-dimensional hype cubic la ice wi h N⫽Ldsi es. Each si e
can s o e any numbe o Aand Bpa icles; ha is, ou model
can be ep esen ed by bosonic a iables. Ini ial condi ions
a e gene a ed by andomly placing N
␳
A
(0) pa icles Aand
N
␳
B
(0) pa icles B, co esponding o a pa icle densi y
␳
⫽
␳
A
(0)⫹
␳
B
(0) . The esul s a e independen o he pa icula
ini ial a io
␳
A
(0)/
␳
B
(0) , apa om e y ea ly ime ansien s.
The dynamics p oceeds in pa allel. Each ime s ep, we up-
da e he la ice acco ding o he ollowing ules: 共a兲Di u-
sion: on each la ice si e, each Bpa icles mo es in o a an-
domly chosen nea es neighbo si e. 共b兲A e all si es ha e
been upda ed o di usion, we pe o m he eac ions: 共i兲On
each la ice si e, each Bpa icle is u ned in o an Apa icle
wi h p obabili y 1.共ii兲A he same ime, each Apa icle
becomes a Bpa icle wi h p obabili y 1⫺(1⫺ 2)nB, whe e
nBis he o al numbe o Bpa icles in ha si e. This co e-
sponds o he a e age p obabili y o an Apa icle o being
in ol ed in he eac ion o Eq. 共2兲wi h any o he Bpa icles
p esen on he same si e. The p obabili ies 1and 2a e
p opo ional o he eac ion a es k1and k2de ined in Eqs.
共1兲and 共2兲. The o de pa ame e o he sys em is
␳
B, mea-
su ing he densi y o dynamical en i ies.
As we a y
␳
, he sys em exhibi s a con inuous ansi ion
sepa a ing an abso bing phase (
␳
B⫽0) om an ac i e phase
(
␳
B⫽0) a a c i ical poin
␳
c. The o de pa ame e is null
o
␳
⬍
␳
c, and ollows a powe law
␳
B⬃(
␳
⫺
␳
c)
␤
, o
␳
⭓
␳
c. The sys em co ela ion leng h
␰
and ime
␶
, which
de ine he exponen ial elaxa ion o space and ime co ela-
ion unc ions, di e ge as
␳
→
␳
c关1兴. In he c i ical egion he
sys em is cha ac e ized by a powe law beha io ,
␰
⬃
兩
␳
⫺
␳
c
兩
⫺
␯
⬜and
␶
⬃
兩
␳
⫺
␳
c
兩
⫺
␯
储
. The dynamical c i ical ex-
ponen is de ined as
␶
⬃
␰
z, wi h z⫽
␯
储
/
␯
⬜. These exponen s
ully de e mine he c i ical beha io o he s a iona y s a e o
he model 关1兴.
We ha e s udied he s eady-s a e p ope ies o he model
in d⫽2 and 3, by pe o ming nume ical simula ions o sys-
ems wi h size anging up o L⫽512 and L⫽125, espec-
i ely. A e ages we e pe o med o e 104⫺105independen
ini ial con igu a ions. The alues conside ed o he a es i
a e 1⫽0.1 and 2⫽0.5 in d⫽2, and 1⫽0.4 and 2⫽0.5 in
d⫽3. F om he ini e-size scaling analysis o APT 关1兴,we
ob ain he c i ical poin (
␳
c⫽0.3226(1) in d⫽2 and
␳
c
⫽0.95215(15) in d⫽3) and he comple e se o c i ical ex-
ponen s. A de ailed p esen a ion o hese esul s will be e-
po ed elsewhe e. In Fig. 1 we show as an example he o de
FIG. 1. O de pa ame e beha io 共s a iona y densi y o Bpa -
icles兲as a unc ion o ⌬⫽
␳
⫺
␳
c o he eac ion-di usion model in
d⫽2 and 3. The slope o he s aigh lines is
␤
⫽0.65 in d⫽2 and
␤
⫽0.86 in d⫽3.
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pa ame e beha io wi h espec o he con ol pa ame e ⌬
⫽
␳
⫺
␳
c, om which i is possible o calcula e di ec ly he
␤
exponen . The esul s ob ained in d⫽2 and 3 a e epo ed in
Tables I and II and compa ed wi h he Manna sandpile
model in he espec i e dimension.
In APT i is possible o ob ain mo e in o ma ion on he
c i ical s a e by s udying he e olu ion 共sp ead兲o ac i i y in
sys ems ha s a close o an abso bing con igu a ion 关15兴.In
each sp eading simula ion, a small pe u ba ion is added o
an abso bing con igu a ion. I is hen possible o measu e he
spa ially in eg a ed ac i i y N( ), a e aged o e all uns, and
he su i al p obabili y P( ) o he ac i i y a e ime s eps.
Only a he c i ical poin do we ha e powe law beha io o
hese magni udes. In he case o many abso bing s a es, he
choice o he ini ial abso bing s a e is no unique 关16兴. The e
a e se e al me hods o pe o m sp eading exponen s in his
case, and we ha e ollowed he echnique ou lined in Re .
关5兴, which amoun s o he s udy o c i ical sp eading wi h he
so-called ‘‘na u al ini ial condi ions’’ a
␳
⫽
␳
c关16兴. The
p obabili y dis ibu ion Ps(s) o ha ing a sp eading e en
in ol ing ssi es, as well as he he quan i ies N( ) and P( ),
can hus be measu ed. A c i icali y, he only cha ac e is ic
leng h is he sys em size L, and we can w i e he scaling
o ms Ps(s)⫽s⫺
␶
sF1(s/LD), N( )⫽
␩
F2( /Lz), and P( )
⫽ ⫺
␦
F3( /Lz)关15兴. The scaling unc ions Fi(x) a e dec eas-
ing exponen ially o xⰇ1, and we ha e conside ed ha he
sp eading cha ac e is ic ime and size a e scaling as Lzand
LD, espec i ely. In his case simula ions we e pe o med o
sys ems o size up o L⫽1024 in d⫽2 and L⫽200 in d
⫽3, a e aging o e a leas 5⫻106sp eading expe imen s.
The new scaling exponen s
␶
s,D,
␦
, and
␩
a e measu ed
using he now s anda d momen analysis echnique 关17,18兴.
The esul ing exponen s a e summa ized in Tables I and II,
and can be compa ed wi h he a alanche exponen s usually
measu ed in s ochas ic sandpile models. As a u he consis-
ency check o ou esul s, we ha e checked ha ou expo-
nen s ul ill all scaling and hype scaling ela ions in s anda d
APT. Despi e he appa en di e si y in he dynamical ules,
we can sa ely include ha he SFCRD and he Manna mod-
els a e in he same uni e sali y class.
F om a heo e ical poin o iew, he SFCRD allows he
cons uc ion o a ield heo y desc ip ion ha also will ep-
esen he c i ical beha io o all models belonging o he
same uni e sali y class. The cons uc ion o he FT ollows
s anda d s eps 关20兴, and i consis s o ecas ing he mas e
equa ion implici in Eqs. 共1兲and 共2兲in o a ‘‘second quan-
ized o m’’ ia a se o c ea ion and annihila ion bosonic
ope a o s o pa icles Aand Bon each si e. I is hen pos-
sible o map he solu ion o he mas e equa ion in o a pa h
in eg al o e he densi y ields, weigh ed by he exponen ial
o a unc ional ac ion S关20兴. In ou case, we can quo e he
elegan esul s o Re . 关12兴, jus conside ing ha we ha e
DA⫽0. The ac ion o he FT is hus
S⫽
冕
dxd
兵
␺
¯
关
⳵
⫹共 ⫺Dⵜ2兲兴
␺
⫹
␾
¯
关
⳵
␾
⫺␭ⵜ2
␺
兴
⫹u1
␺
¯
␺
共
␺
⫺
␺
¯
兲⫹u2
␺
¯
␺
共
␾
⫹
␾
¯
兲⫹ 1
␺
¯
2
␺
2
⫹ 2
␺
¯
␺
共
␺␾
¯
⫺
␺
¯
␾
兲⫹ 3
␺
¯
␺␾
¯
␾
其
,共3兲
whe e
␺
and
␾
a e auxilia y ields, de ined such ha hei
a e age alues coincide wi h he a e age densi y o Bpa -
icles and he o al densi y o pa icles, espec i ely,
␺
¯
and
␾
¯
a e esponse ields, and he coupling cons an s a e ela ed o
he eac ion a es ki. Namely, D ep esen s he di usion
coe icien o Bpa icles, ␭is ini ially also p opo ional o D,
and is he c i ical pa ame e ha is ela ed o he di e ence
o he o al densi y wi h espec o he c i ical densi y
␳
c.By
s anda d powe -coun ing analysis, one ealizes ha he e-
duced couplings ui/Dha e c i ical dimension dc
(1)⫽4, while
he couplings i/Dha e on hei pa dc
(2)⫽2. This means
ha when applying he eno maliza ion g oup 共RG兲and pe -
o ming a pe u ba i e expansion a ound he c i ical dimen-
sion 4, one could in p inciple d op all he couplings i关21兴.
The c i ical pa ame e o his heo y is he densi y o ac i e
si es
␺
, while
␾
se es jus o p opaga e in e ac ions. We can
exploi some symme y conside a ions o he FT o ela e he
physics o he sys em o he co esponding analy ical de-
sc ip ion. By neglec ing i ele an e ms in he powe -
coun ing analysis, ac ion 共3兲is in a ian unde he shi
ans o ma ion
␾
→
␾
⬘⫽
␾
⫹
␦
, → ⬘⫽ ⫺u2
␦
,共4兲
whe e
␦
is any cons an . This symme y has a e y in ui i e
meaning: I we inc ease e e ywhe e he densi y o pa icles
by an amoun
␦
, we mus be close o he c i ical poin by an
amoun p opo ional o
␦
. In o he wo ds, his symme y ep-
esen s he conse ed na u e o he sys em. I is also in e es -
ing o w i e he se o co esponding Lange in equa ions 共up
o he i ele an e ms i) by in eg a ing ou he esponse
ields
␺
¯
,
␾
¯
in he ac ion S,
TABLE I. C i ical exponen s o sp eading and s eady-s a e ex-
pe imen s in d⫽2. Figu es in pa en hesis indica e he s a is ical
unce ain y in he las digi . Manna exponen s om Re s.
关5,10,18,19兴.
S eady-s a e exponen s
␤␤
/
␯
⬜
␯
⬜z
␯
储
SFCRD 0.65(1) 0.78(2) 0.83(3) 1.55(5) 1.29(8)
Manna 0.64(1) 0.78(2) 0.82(3) 1.57(4) 1.29(8)
Sp eading exponen s
␶
sDz
␩␦
SFCRD 1.27(1) 2.75(1) 1.54(2) 0.29(2) 0.50(2)
Manna 1.28(1) 2.76(1) 1.55(1) 0.30(3) 0.48(2)
TABLE II. C i ical exponen s o sp eading and s eady-s a e
expe imen s in d⫽3. Figu es in pa en hesis indica e he s a is ical
unce ain y in he las digi . Manna exponen s om Re s.
关5,10,18,19兴.
S eady-s a e exponen s
␤␤
/
␯
⬜
␯
⬜z
␯
储
SFCRD 0.86(2) 1.39(4) 0.62(3) 1.80(5) 1.12(8)
Manna 0.84(2) 1.40(2) 0.60(3) 1.80(5) 1.08(8)
Sp eading exponen s
␶
sDz
␩␦
SFCRD 1.41(2) 3.32(2) 1.74(2) 0.16(2) 0.76(3)
Manna 1.43(2) 3.31(2) 1.75(2) 0.16(2) 0.75(3)
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⳵
␺
⫽Dⵜ2
␺
⫺
␺
⫺u1
␺
2⫺u2
␺␾
⫹
␩
␺
,共5兲
⳵
␾
⫽␭ⵜ2
␺
⫹
␩
␾
.共6兲
He e,
␩
␺
and
␩
␾
a e noise e ms wi h ze o mean and co e-
la ions
具
␩
␺
(x, )
␩
␺
(x⬘, ⬘)
典
⫽2u1
␺
(x, )
␦
(x⫺x⬘)
␦
( ⫺ ⬘),
具
␩
␺
(x, )
␩
␾
(x⬘, ⬘)
典
⫽⫺u2
␺
(x, )
␦
(x⫺x⬘)
␦
( ⫺ ⬘) and
具
␩
␾
(x, )
␩
␾
(x⬘, ⬘)
典
⫽0. The noise e ms ha e a mul iplica-
i e na u e 关22兴, ha is he s anda d o m in APT. No e ha
icouplings o Eq. 共3兲con ibu e o noises co ela ions wi h
highe o de e ms. These equa ions ha e a e y clea physi-
cal in e p e a ion. The ield
␾
is conse ed 关23兴and s a ic,
i.e., i only di uses ia he ac i i y o Bpa icles, ep e-
sen ed by he ield
␺
. On i s u n, he ield
␺
is locally
coupled o he ield
␾
, bu is nonconse ed. No iceably, his
se o equa ions eco e s 共up o he disca ded couplings i)
he Lange in desc ip ion p oposed on a phenomenological
le el o he sandpiles in Re s. 关9,10兴, wi h he ex a in o -
ma ion o he c oss-co ela ion e m
具
␩
␺
␩
␾
典
. Indeed, he s o-
chas ic sandpile model has he same basic symme ies o he
p esen RD model, once he local densi y ield
␳
is eplaced
by he local sand-g ain 共ene gy兲densi y and he o de pa am-
e e is iden i ied wi h he densi y o oppling si es ield
关9,10兴. I is hen na u al o expec ha he e y same basic
s uc u e is e lec ed in a unique heo e ical desc ip ion 关24兴.
The comple e RG analysis o he ield heo y would allow
us o ex ac es ima es o he c i ical exponen s o compa e
wi h simula ions in d⫽2 and 3. Un o una ely, some se e e
echnical p oblems a e encoun e ed in his case. In gene al,
as poin ed ou in Re . 关12兴, he couplings ibecome ele an
and should be aken in o accoun in he RG analysis. The
impo ance o he couplings ican be a gued by he change
o he ene gy shi symme y, Eq. 共4兲, in he case o he ull
ac ion Eq. 共3兲. Second, and mo e impo an , is he p esence
o he singula ba e p opaga o o he ield
␾
, which canno
be egula ized by adding a mass e m m2
␾␾
¯
, since i will
ob iously b eak he symme y 共4兲. This singula p opaga o
gi es ise o di e gences in he RG pe u ba i e expansions,
and he esul s o Re . 关12兴canno be ex ended ‘‘ ou -cou ’’
o he limi DA→0. In pa icula , some Feynman diag ams in
he
⑀
-expansion p esen ed in Re s. 关12,13兴a e p opo ional
o 1/DA. Hence, he limi DA→0 in he heo y wi h DA
⫽0 is nonanaly ic; any in ini esimal amoun o di usion in
he ene gy ield eno malizes o a ini e alue, and de ini ely
changes he uni e sali y class o he model. Wo k is in
p og ess o p o ide a sui able egula iza ion ha will allow
an
⑀
-expansion calcula ion o he c i ical exponen s.
This wo k has been suppo ed by he Eu opean Ne wo k
unde Con ac No. ERBFM-RXCT980183. We hank D.
Dha , R. Dickman, P. G assbe ge , H. J. Hilho s , M.A. Mu-
n
˜
oz, F. an Wijland, and S. Zappe i o help ul commen s
and discussions.
关1兴J. Ma o and R. Dickman, Nonequilib ium Phase T ansi ions
in La ice Models 共Camb idge Uni e si y P ess, Camb idge,
England, 1999兲.
关2兴J. L. Ca dy and U. Ta
¨ube , Phys. Re . Le . 77, 4780 共1996兲.
关3兴H. K. Janssen, K. Oe ding, F. an Wijland, and H. J. Hilho s ,
Eu . Phys. J. B 7, 137 共1999兲.
关4兴E. V. Albano, J. Phys. A 25, 2557 共1992兲; I. Jensen, Phys. Re .
Le . 70, 1465 共1993兲.
关5兴M. Rossi, R. Pas o -Sa o as, and A. Vespignani, Phys. Re .
Le . 85, 1803 共2000兲.
关6兴Fo a e iew see H. J. Jensen, Sel -O ganized C i icali y 共Cam-
b idge Uni e si y P ess, Camb idge, England, 1998兲.
关7兴S. S. Manna, J. Phys. A 24, L363 共1991兲; D. Dha , Physica A
263,4共1999兲.
关8兴P. Bak, C. Tang, and K. Wiesen eld, Phys. Re . Le . 59, 381
共1987兲.
关9兴R. Dickman, A. Vespignani, and S. Zappe i, Phys. Re . E 57,
5095 共1998兲; A. Vespignani, R. Dickman, M. A. Mun
˜
oz, and
S. Zappe i, Phys. Re . Le . 81, 5676 共1998兲.
关10兴A. Vespignani, R. Dickman, M. A. Mun
˜
oz;, and S. Zappe i,
e-p in cond-ma /0003285.
关11兴The connec ion be ween SOC and APT has been also dis-
cussed o he Bak-Sneppen model, in M. Paczuski, S. Maslo ,
and B. Bak, Eu ophys. Le . 27,97共1994兲;28, 295共E兲共1994兲;
P. G assbe ge , Phys. Le . A 200, 277 共1995兲.
关12兴F. an Wijland, K. Oe ding, and H. J. Hilho s , Physica A 251,
179 共1998兲.
关13兴R. K ee, B. Schaub, and B. Schmi mann, Phys. Re . A 39,
2214 共1989兲.
关14兴In his sense, he model de ined in Re . 关12兴is simila o he
h eshold ans e p ocess in oduced by J. F. F. Mendes, R.
Dickman, M. Henkel, and M. C. Ma ques, J. Phys. A 27, 3019
共1994兲.
关15兴P. G assbe ge and A. de la To e, Ann. Phys. 共N.Y.兲122, 373
共1979兲.
关16兴I. Jensen and R. Dickman, Phys. Re . E 48, 1710 共1993兲.
关17兴M. De Menech, A. L. S ella, and C. Tebaldi, Phys. Re . E 58,
R2677 共1998兲; C. Tebaldi, M. De Menech, and A. L. S ella,
Phys. Re . Le . 83, 3952 共1999兲.
关18兴A. Chessa, A. Vespignani, and S. Zappe i, Compu . Phys.
Commun. 121-122, 299 共1999兲;S.Lu
¨beck, Phys. Re . E 61,
204 共2000兲.
关19兴K. Nakanishi and K. Sneppen, Phys. Re . E 55, 4012 共1997兲;
E. Milsh ein, O. Biham, and S. Solomon, ibid. 58, 303 共1998兲.
关20兴M. Doi, J. Phys. A 9, 1465 共1976兲; L. Peli i, J. Phys. I 46, 1469
共1985兲; B. P. Lee and J. Ca dy, J. S a . Phys. 80, 971 共1995兲.
关21兴In spi e o he nai e powe -coun ing analysis, he i ele ance
o all e ms mus be checked on he g ounds o a ull RG
analysis.
关22兴J. L. Ca dy and R. L. Suga , J. Phys. A 13, L423 共1980兲;H.K.
Janssen, Z. Phys. B: Condens. Ma e 42, 151 共1981兲.
关23兴I is possible o show ha he noise e m
␩
␾
is equi alen o a
conse ed noise; i.e., i gene a es he same diag ams in a pe -
u ba i e expansion; M. A. Mun
˜
oz and F. an Wijland 共p i a e
communica ion兲.
关24兴I is wo h no icing ha he Bak, Tang, and Wiesen eld sand-
pile model 关6,8兴has de e minis ic dynamics and does no be-
long o his uni e sali y class. Also, he Lange in desc ip ion
p esen ed he e is no alid o de e minis ic models ha
p esen none godic e ec s and ecu en s a es. This poin , dis-
cussed in de ail in Re . 关10兴, has been o e looked in Re . 关9兴,
whe e de e minis ic and s ochas ic models a e no dis in-
guished.
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