1. No category

Anisotropic perturbations of the simple symmetric
exclusion process : long range correlations
Christian Maes, Frank Redig
To cite this version:
Christian Maes, Frank Redig. Anisotropic perturbations of the simple symmetric exclusion
process : long range correlations. Journal de Physique I, EDP Sciences, 1991, 1 (5), pp.669684. <10.1051/jp1:1991161>. <jpa-00246361>
HAL Id: jpa-00246361
https://hal.archives-ouvertes.fr/jpa-00246361
Submitted on 1 Jan 1991
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J.
Phys.
(1991)
J1
669-684
MAi
1991,
PAGE
669
Classification
Physics
Abstracts
05.40
02.90
46.10
Anisotropic perturbations of the shnple symmetric
correlations
process : long range
Christian
(')
(~) and
Maes
Instituut
Frank
Theoretische
voor
exclusion
Redig ~)
Fysica,
Leuven,
K. U.
Celestijnenlaan
200D,
B-3001
Leuven,
Belgium
f)
Instelling
Uriiversitaire
(Received
25
Antwerpen,
J990,
October
accepted
ih
Universiteitsplein
final form
22
I,
B-2610
Wilrijk,
Belgium
January J991)
spin system in which nearest neighbor spins are exchanged at rates
neighboring
configuration. The system has the same symmetry of the
on
correlation
lattice
rotations.
for lattice
We set up a perturbation expansion for the
functions
exclusion
Convergence is proven for
time t around
the simple symmetric
at
process.
limit reproduces the usual high temperature
expansion in the case of
small t and the formal t
co
Abs«act.
which
Consider
lattice
a
weakly depend
except possibly
the
~
detailed
balance.
then, generically,
decays only like
1.
If the
the
a
system is isotropic,
two
power
points function
r~~, where r
then
has
is
the
each
the
term
direction
spatial
expansion
dependence of
in the
separation
and
is
a
strictly local. If
quadrupole field
dm 2 is
the
not,
and
dimension.
Introduction.
have
correlations
decaying in the same
the
at high
temperatures
sense
as
potential. This well known fact from equilibrium
statistical
mechanics
has been
established
methods
by various
such as the
cluster
expansion [1-3], or a
Dobrushin
type
analysis [4-6], and is in last instance a consequence
of the so called local
Markov
property of
Gibbs
interaction, given the configuration
appropriate
inside
the
states : for a finite
range
boundary layer surrounding a volume, the inside and outside are conditionally independent.
It allows
follow the
interdependence of certain
different
regions in space via
in
to
events
intermediate
points (or,
«polymers»)
along which
information
is
transported (and lost).
These «
explicitly
connections
constructed
in the
cluster
expansion, starting from the
are
Boltzmann
factor Z~
(pH),
small,
for
interaction
order
p
H.
For example, to first
exp
an
in p, for a lattice spin system, only the
neighborhood of a site, as
determined
by H, can
influence
the spin
there.
probabilistic
More
methods
the
(quasi-) locality of the
use
conditional
distributions
estimate
the dependence of a spin on its neighboorhood.
This
to
explains in fact why we expect that local
interactions
generically (that is, away from critical
points) give rise to a finite
correlation
length.
Tuming to real non-equilibrium situations, we cannot expect such methods to be applicable
by only considering the spatial structure.
For a general lattice spin dynamics, we will not find a
simple local
mechanism
by which to describe the mutual
influence
of spins in
different
Gibbs
states
interaction
JOURNAL
670
PHYSIQUE
DE
N
I
5
regions. The spatial
of
non-Gibbsian
stationary
best be
studied
by
structure
states
can
embedding it in the
spatio-temporal
Simple examples for which
readily
process.
one
recognizes the advantages of connecting spatially separated spins via their
history
common
interacting particle systems or probabilistic cellular
[7-8]. In fact, such an
automata
are
analysis naturally arises for general non-equilibrium phenomena. This may seem obvious but
it has
important
If the
is not
Gibbsian, and the spatial
stationary
state
consequences.
correlations
intermediated
via events in the past, then the spatial decay
must be thought off as
will be related to the temporal
correlations.
In particular if the latter have a weak decay such
for
diffusive
similar
then
behavior
be
expected for the stationary
systems,
as
a
can
correlations.
the starting point of the analysis in [9] to study the phenomenon of long
was
in
non-equilibrium
dynamics. Here we wish to
spatial
correlations
conservative
range
perturbation expansion around an exactly solvable
investigate in more detail the associated
Z( on the d-dimensional
consists of spins «(x)
lattice gas dynamics. The system
± I, x e
neighbor spins are exchanged with rates
introduced
in which
lattice. A dynamics is
nearest
interested
in the
which depend weakly on the configuration in a finite neighborhood. We are
This, in short,
=
behavior
of the
xe
fl
functions
correlation
«(x)
for
finite
some
A
,
Z(
<
stationary
in the
A
To
states.
study them,
make
we
symmetric simple
so-called
the
an
expansion
exclusion
around
[7]. It
can
be
the
product
viewed
as
state
a
which
system
of
is
invariant
random
for
walkers
potential only. Although the stationary states are trivial for this
fluctuations
correlations
decay weakly. Typically,
in the density of the
system, the temporal
walkers decay diffusively in time like t~ ~'~. This decay enters in the perturbation expansion as
analysis considerably
wl~ile
this
makes
of the free
the
propagator.
our
more
mass
zero
complicated, it has the interesting feature to produce, for generic perturbations, also a weak
and
Putting r~~ t as the usual space-time scaling for diffusive
decay in space.
systems
intermediated
diffusive
decay, we end up
assuming that the spatial decay is
via the temporal
bonstant Q such that
(«(0); «(r))
that
there is
Q (r(-~,
with the prediction
some
jr
for the
stationary two points function. wl~ile, a priori, this scenario is completely
co
general, the details of the dynamics enter in the prefactor Q. We will argue that generically
situations
in which by symmetry Q »0.
Those
Q # 0 but there are special (non-generic)
(overlapping) classes.
be divided
into two
situations
can
The first important special case
when this « self-organized criticality
disappears by
occurs
arranging the dynamics in such a way that it satisfies the condition of detailed balance, and the
appropriate Gibbs states become stationary. In that case we
(in an unusual way) the
recover
usual
high-temperature
expansion. In the light of the
discussion
follows
that
emphasize
we
this is as
expected and is independent of the (an-)isotropy of the
that
corresponding
Hamiltonianthat we need is completely
contained
the
in the
detailed
balance
symmetry
which
interact
via
hard
a
core
=
-
condition,
Secondly,
anisotropy
the
fact
as
that
become
will
dynamics,
of the
the
systems
clear,
All
we
an
essential
(necessary) ingredient in our analysis is
leading to long range
correlations
based
are
arguments
consider
do not
our
possess
the
full
symmetry
of
the
the
on
lattice.
that starting from the formal
expansion for the stationary
correlation
isotropic
have
perturbation
short
correlations.
This is
must
any
range
consistent
with the results of [10] for similar
dynamics, but where it is not excluded to have
than
particle per site.
more
one
Finally, it may still happen that by some other symmetry Q
0 even though the model does
described
classes
above. An example of this is presented at the end of
not fit in one of the two
still have a power law decay with the
section 4 and it is found
there
that the
correlations
Moreover,
functions,
we
will
show
fully
"
N
POWER
5
dependence
direction
Q
DECAY
LAW
of
octopole
an
STOCHASTIC
IN
(instead
field
LATTICE
of
quadrupole
the
671
GASES
decay
for
the
case
0).
#
The
difference
rigorous
analysis
of the
(but,
and
model
In
of
the
details
[9] lies in the
of the
find
we
the
section
weak) results
concerning the
anisotropy. We also add complete
possibility of octopole-like decay.
define
we
expansion. We
is strictly finite,
can
the
model.
only
show
finite
for all
that
times.
3 is
it
convergence
explicit
and
devoted
converges
the
small
formal
derivation
the
to
for
taking
Still
expansion, the
the
thorough
and
results for a specific
t
-
times, or,
co limit,
of
the
when
the
discuss
we
points
decay
is like
term
we
per term
r~~, where r is the spatial separation, and dm 2. We emphasize the role of the anisotropy.
This
behavior
is explicitly
calculated
first
order
expansion for a specific
in the
up to
perturbation. In the Appendix we apply our formal expansion to the Kawasaki dynamics and
give this expansion more significance by recoveringGibbs
without
using the
standard
formalismthe well
known
high temperature
expansion for the equilibrium
correlation
in
section
function
4 the
of
presence
long
Section
perturbation
of the
admittedly,
role
next
perturbation
perturbation
with
work
our
studied
is
correlations
range
expamion,
in the
stationary
in the
find
and
The
measure.
generically
that
two
its
functions.
2.
The
model.
Z~ in which the rates
We
consider
stochastic
for spins «(x)
time
evolution
± I,
a
x e
c(x, y, « ) at which the spin at site x and y, ix y
I are exchanged depend weakly on the
configuration at other sites in a finite neighboorhood of the bond (xy). For a finite volume
Pf(«) to find configuration « e (- I, +
~ in A,
A < Z( the probability
say with periodic
boundary conditions, is govemed by the master equation
=
=
)
PI(")
£
=
«~Y is the
where
«'YJ Pf("~~
y,
c(x,
y,
«
)
PI(«))
(2.I)
,
(xy)e
(xy)
(c(x,
A
configuration
obtained
from
@fter exchanging
«
the
neighbor
nearest
sites
:
«~Y(z)
(z)
~y)
«(x)
«
if
=
z
if
=
«
z
if
=
The
process
which
on
«i, t
local
m
0, in the
infinite
f(«),
functions
Lf(«)
volume
z
=
x
z
,
z
=
x,
=
y.
limit is
(- I, +1)
e
«
#
~
# y
,
(2.2)
conveniently
is defined by
c(x,Y, «) ~f(«~Y~
defined
its
via
generator
-f(«))
L,
(2.3)
(.<y)
The
c(x,
sum
y,
in
«)
formulation
=
(2.3) is over
c~y, x, «) m
is postponed
nearest
0.
Further
until
neighbor sites
assumptions
after
dynamics.
The simplest example
corresponds
the
satisfy the
condition
rates
~0(~,
have
we
to
y,
)
on
"
ye
the
introduced
infinite
an
"
x,
Co (X, y,
Z(
rates
some
temperature
W
~~
ix
y(
must
well
=
be
I,
and
imposed
known
Kawasaki
of
course
but
type
dynamics,
of
their
such
where
(2.4)
672
JOURNAL
DE
Clearly, all
Bemoulli
measures
with
this
evolution.
Finite
temperature
p
time
m
satisfying
0,
the
condition
cp (x,
y,
)
«
«
(«~)~
m
=
M
e
[- I,
dynamics
Kawasaki
detailed
cp (x, y,
=
I
spin
average
of
PHYSIQUE
],
I
+
have
invariant
are
5
for
cp(x,y,«),
rates
balance
(- p (H(«~Y~
exp
~Y~
H(«)))
(2.5)
,
where
H(«)
£
m
fl
JA
A
is
a
local
translation
equilibrium
whenever
0
=
=
from
Hamiltonian,
invariant
that JA
JA, and J~
~ ~
Gibbs-measures
pp for
(2.6)
mechanics
( WA )
as
a
convergent
magnetic
p
=
0,
there
measures,
h3~«(x)
least
how
co(x,
y,
to
are
further
a
rates
perturbed
The
=
the
express
2
c
numbers
say if diam
(A )
dynamics,
and
the
such
The
R.
m
know
we
functions
correlation
dpp(«)
«(x)
(2.7)
course,
always
may
we
verified.
As
corresponding
the
add
the
in
different
to
Gibbs
called
canonical
local
Hamiltonian.
so
a
case
discussion.
in dynamics with
generic perturbations
)
«
for
real
are
large,
=
interested
look
Z~ finite
AC
of A is too
measures
to
fl
m
p
(2.6)
expansion around p
0. Of
(2.6) and (2.5) remains
to
parameter family of Gibbs
one
measures
invariant
for the dynamics. They are
a
which
[11] for
see
are
therefore
rates
temperature
term
is at
magnetizations,
We
high
field
(JA,
reversible
then
are
statistical
I.e.
diameter
the
(x)
«
xeA
(x,
Y,
)
«
satisfying (2.5) for
not
of the
infinite
process
is
i
=
qA
dynamics, having
temperature
then
z
+
any
assumed
(x, Y)
have
to
We
constant
rates
(2.8)
«A
,
A
where
WA
fl
m
«(x).
(qA(x, y))
coefficients
The
taken
be
must
so
that
the
(2.8)
are
(2.13)).
We
rates
xeA
non-negative,
also
local
that
assume
(A (, is odd.
bounded
and
the
rates
Translation
are
of the configuration « (see (2.12)
of
qA(x, y)
0
whenever
the
number
condition
that
is imposed by the
functions
even
:
invariance
any a e
I.e. if
axes,
Z~. Finally,
(e~)~
are
that
assume
we
the
unit
~A
"
the
"
a(X
+
+
model
in
vectors
~0~ A(°, ea')
elements
=
~A(X, Y)
for
and
is
Z(
(2.~)
~)
~, Y +
symmetric
reflection
in A,
over
all d
coordinate
then
~A(°, ea')
If
£Y
# £Y'
,
and
qo~ A(0, ea)
=
qA(0,
e«)
«
,
d,
i,
=
(2,io)
,
where
@~Am
((xi,..,-x~,..,x~):x=
(xi,..,x~,..,x~)eA).
Let
q~Bm~ £
j.<y)
(q~xy~B(x,y)-q~AB(x,y))
(2.ii)
N
POWER
5
with
A
«
the
symmetric
the
»
labels
LAW
of
(xy).
sites
the
difference
STOCHASTIC
IN
DECAY
between
q~~
if
0
=
GASES
obtained
and A~Y is
sets,
require
further
We
LATTICE
from
673
A
by exchanging
that
(A
diam
)
AB
(2.12)
R,
m
and
q~~
for
certain
measures.
perturbation
order
In
in
mind
model
not
oneself
loose
to
dimensional
two
Hamiltonian
«
to
There
I.
«
y
as
y
use
in the
is
no
an
(2.13)
B
priori
a
expansion
and
«
)
which
we
exchange
the
e~,
+
x
for
notations
following simple example
C(x,
for
and
co
~
therefore
try
A,
information
on
and
parameter
the
set
up
stationary
formal
a
theory.
the
is
R
constants
We
for all
y
w
rates
(-
exp
=
will
the
most
take
up
for
are
general
again at
a
model
the
good
it is
end of
keep
to
section
4. The
1, 2 given by
=
p~jH(«~'~~~~)
H(«)j
(2.14)
»
H(«)
£
2
=
«(x
[Ki «(x)
+
ei)
K~ «(x)
+
«(x
+
e~)]
(2.15)
x
pi, p
Here
of
~
detailed
question,
the
3.
they
and if
small
are
(2.5). The
happens
balance
what
both equal to
p, then (2.14) satisfies
some
problem this paper investigates might be
if pi # p~.
stationary
correlations
are
main
the
to
condition
the
by
summarized
expansion.
The
unperturbed
process
tool in its analysis is the
corresponds to the simple symmetric
exclusion
process.
self-duality [7]. In this case, the generator (2.3) is
The
Lo
f(«)=(
£
The
main
jf(«xY~-f(«)j.
(3.i)
(~y)
If
we
define
for
finite
any
Z(
A
<
function
the
D
(A,
)
«
fl
m
check
that
Lo
D
is
same
both
Lo D(A,
«
the
it
when
acts
£
=
on
(D(A~(
A
and
«
)
on
m
WA,
then
it is
easy
to
I.e.
«,
(A,
D
(x)
«
A
xe
«
))
(3.2)
(<Y)
We
Z(
therefore
with
consider
the
dual
called
so
process
A(t),
m0,
t
on
the
finite
subsets
of
generator
Lo f (A )
=
~
£
f (A ~Y~
f (A
(3. 3)
,
jxy)
and
let
pi(A,B)
collection
interaction,
of
be
random
end
corresponding
transition
probability, I.e. the probability that a
starting from the sites in A and subject only to a hard core
unless
A
B (, and
set B at time t. Obviously, pi (A, B ) is zero
the
walkers
up in the
=
formally,
~.
j
£ lm dt~Pi(A,
xy
o
B)
-Pi(A~l
B))
=
3~,B
(3.4)
JOURNAL
674
is
if A
one
and
B
=
otherwise.
zero
Et
where
denotes
(3,2)
From
Et(D(A, WA)
PHYSIQUE
DE
derives
one
zPi(A,
dP («)
=
B
duality
the
D
expectation in the simple symmetric
the
M
I
(B,
«
exclusion
relation
5
:
),
(3.5)
process
started
Li
let I
from
the
p.
measure
Consider
the
now
expectation
with
(with
«rprocess
perturbed
respect
to
By integrating
E"lD(A,
Al
«
(3.6)
"
overs
from
s
£pi(A,
B
)
p
of
the
with
commutes
~ljD(Ai-w
0
"s)I
to
ID (B,
t
and
L
Lo
=
+
(with
and
«
)
denote
the
and
the
Lo)
generator
D(A,
since Lo
D(Ai-s,
ljLi
"
=
Lo D(A,
)
«
"s)1
(3.6)
get
we
)j
«
(2.3) L
Arprocess
generator
product
L). Lo
generator
with
process
the
+
s
i
£
ds
+
~(A,
pi
) E"[Lj
B
D
(B, «~)]
(3.7)
,
o
B
expectation in the perturbed
with initial
«i-process
measure
p.
perturbed process with rates (2.8) has as initial state p the Bemoulli
measure
with
magnetization («~)~
from (3.7), the time
evolved
spin-spin
correlations
0. Then,
satisfy the equation
with
E"
the
Suppose the
=
(«~)
3~
=
j
ii ds
+
£
o
(3.8)
~(A, B) v~(B)
pi
B
with
v~(B)
and
from
(Lj
m
D
(B,
«
(2,8), (2,11),
Us(B)
=
=
,i
(«BXY- «B) lC(x,
z
I WC1~
qBc
Y,
«
~
)
(3.9)
c
Substituting (3.9)
(3.8)
into
we
iterate
can
obtain
to
(«Al
=
z
VI (t)
V$(t)
where
WE AJ,
(3. lo)
,
i
and
VI (t)
ids
z
=
Pi
-s(A,
B
qBc
Vt~
n
»
(3.I1)
,
B,c
The
first
~"A)1'
question
is
to
see
when
(3.10),
(3.ll)
defines
a
convergent
expansion
for
N
LAW
POWER
5
PROPOSITION
evolved
If tyR < I,
For fixed
I.
correlations.
Proof: By locality,
then
time
t,
STOCHASTIC
IN
DECAY
(3.10)-(3. II) 4bfines
(«~ ) is of the
675
GASES
LATTICE
convergent
order y ~
a
expansion for the
'i
y
time
small.
(2.12),
see
£
pj(A,
)
B
if
0
q~c
A
C
m
=
+
(3.12)
R
B
by
Therefore,
induction
(3.ll)
from
V( (t)
0
whenever
A
On
hand,
other
the
(2.13)
from
:
[Vj(t)[
Assuming
for
that
some
from
have
(A]
if
«ty
(3.14)
«R.
m1
n
V(~~(s)[
we
(3.13)
nR
m
=
y~~~ R~~~s~~~,
«
(3.15)
(3.13) that
i'~(~)
ii
~
~S(N~l') (l'~
~
~~
S~
~)
~
l'~ ~~
(~.l~)
~~
0
that
so
By
(3.10)
this
2.
If
PROPOSITION
yRt
for
converges
construction
I.
<
satisfies
series
the
£
sup
A
(3.10)-(3.ll)
then
Proof:
and
Note
that
perturbation
(3.17)
c(x,
y,
invariance
requires
«)
and
times
t
(3.17)
co
<
,
from
follows
be
a
not
x
m
co.
<
finite, it
to
do
we
(q~~
B
only
for
#1/2
all
is
done.
the
are
we
ondition
converges
the
Since
for
equations (3.8).
evolution
consider
number
this
in
finite
trictly
finite
of
ere
(
=
the
sense
as
above
that
that
bonds
case
(«~)
estimates
sinfilar
any
further.
(3.19)
V$
"
n=o
where
V$
w
3 ~j,
Vi
m
£
G
(A,
B
)
q
V)~
~c
n
(3.20)
m
,
s, c
and
£
s
G
(A,
B
) f(B)
i
m
lim
itm
0
£
s
p~(A,
B
) f(B)
ds.
(3.21)
676
JOURNAL
There
of
are
for
First,
(3.19)
q~c.
that
the
series
reflect
the
question
is
They
generally,
More
(3.19). On
meaningful at
for
is
argue
we
coefficients
the
time,
finite
every
result
convergence
indicate
that
no
problems
many
course
expansion (3,10)
the
in
well
we
some
respects.
and
whether
the
f(B)
£
additional
two
due
have
that
may
of
form
present in the dynamics.
for
functions
f
the
of
Kc
qBc
the
law
conservation
(3.21) exists
special
the
to
we
but
results
5
the
control
cannot
we
purely formal,
co
have
termwise
the
limit
=
Since
I
t
hand,
defined
symmetry
1ilnit
is the
other
least
N
I
(3.19)-(3.21).
with
only
not
is
PHYSIQUE
DE
form
(3.22)
c
If
that
assume
we
Kc
coefficients
the
Q~(N, f)
f(B)
is
a
quadrupole
«
£ xl
£
m
is
B:
then,
(3.22)
in
»
=N
£
(3.23), (3.24)
f(B) but we
£
=N
co
<
f(B)
is
=N
a
I,
=
,
2,..,
1,
N
(3.23)
d
,
B
=N
xeB
£
0
=
f(B)
x~
f(B)
x~ x~>
0
=
Q~(N, f) 3~,
=
(3.24)
~,
xes
probably
sufficient
proof
for
general
no
are
have
numbers
s
xe
£
B;
Properties
G (A, B
all
=
f(B)
£
£
for
that
;
B
B-18
such
are
Gf
for
show
now
Gf(A)m
by
defined
be
to
We
A.
sets
how it
be
done
Gf((0,x)
)m
can
s
the
in
where
case
V(x)
(0,x)
A
=
be
can
found
£
consists
unique
the
as
of
at
solution
most
of
Gf ( (0,
(Gf ( (0, x)Y~)
sites.
two
(3.4)
From
equation
the
))
x
(0, x)
f
=
(3.25)
,
~~
or,
~
z
da (v(x)
3x,ov(ea))
+
(3.26)
(x),
p
=
conditions
V(x)
with
boundary
0
(x(
In
(3,26), A~g (x)
g(x + e~) +
as
co,
Laplacian in the direction a and p (x)
g(x
e~)
2 g(x) is the discrete
f( (0, x) is a
quadrupole. Using the properties (3,23), (3.24), we have that
-
-
w
=
for
some
x~
(k)
Here, it
suffices
defined
solution
A
second
measures.
measures
=
bounded
to
note
We
are
~
at
k
0.
I
cos
k~
(k)
X
~
This
will be
the
explicitly in section 4, equation (4.7).
expression in (3.26) allows to find a well
derived
=
substituting
that
z
«
~~
above
V(x).
argument
meaningful for
dk e'~~
(X)
P
certain
tum
yielding
dynamics
therefore
reversible.
In
to
the
substance
on
the
which
to
we
Kawasaki
Appendix
we
the
have
detailed
dynamics
derive
expansion
formal
the
defined
usual
(3.17)-(3.19)
information
in
high
(2.5)
about
for
temperature
is
the
which
it is
that
stationary
the
Gibbs
expansion
for
N
POWER
5
LAW
equilibrium
correlations
fully
relying
but
measures
DECAY
dynamical
syInmetry
«
on
»
simplify
to
of
formal
our
(2.5)
LATTICE
using any properties
expansion (3.19).
without
the
STOCHASTIC
IN
each
the
We
corresponding
find
thus
expansion
the
in
term
677
GASES
how
get
to
to
Gibbs
a
the
use
strictly
local
function.
Long
4.
correlations.
range
Appendix
the
infinite
perturbation
expansion
around
in
dynamics reproduces
local high
expansion
the case of
temperature
temperature
detailed
satisfy
condition.
balance
dynamics. Generic perturbations are not expected to
such a
investigate
the
behavior
is
of
function
predicted
from
each
what
the two points
Here
as
we
in
the
expansion,
for
generic
perturbation.
such
Each
V(o,~j, nml,
term
term
a
Z( in the expansion (3.19) satisfies the equation (3.26) with
x e
the
In
show
we
formal
the
that
the strictly
(x)
P
z
m
(Q j
(Y,
o, xi
Qj
)
z
o, xi
Y~
(Y,
z
(4. i )
)
(Yzi
and
Q~ (Y, Z)
£
"
~A
(Y,
AB
Z
) i'~
B
translation
The
invariance
Qi (Y,
for
any
Z(
as
by
and
reflection
Qi
)
Z
P(~)
(4.I)
consequence,
a
i
(Q(0,-xj
"
+
+
a)
e
rewritten
~,
that
+
Z
(4.2)
)
~
(2.10),
summetry
be
can
a(Y
"
Qi (°,
As
(2.9), implies
model,
the
of
Qi
"
A
(°,
(4.3)
e~
as
Q)0,x+e~j(0,~a)+Q(0,xj (0,~a)
(0,~a)~
~Q)0,-x+e~j(0,~a))~
d
£
A~ 3~
oQ)0
a
The
behavior
functions
Vii
of
~j,
Q(
o,~j
(0, e~)
fixed
for
Q
(0,
o, xi
(x(
as
ea
£
=
the
qj
order
~~
(0,
ea
) VI
I(
correlation
then
m
oj
(4.4)
~
depends on the (n
I )-th
origin. Indeed, if (x(
R,
co
-
around
A
sets
(0, e~)
j
e
'
=1
(4.5)
xi
~
First
least
holds
consider
the
situation
exponentially fast to
for p (x). Taking
where
as
zero
V$j( ~j
the
ix
-
co.
is
That
transforms,
Fourier
are
k
=
4a (k)
"
£ Q( 0, xi
quasi-local in the sense
certainly verified for n
(kj,.. k
k ~) e [,
(0,
~a
~,..,
they decay
that
I.
Then,
this
at
also
=
gr,
gr
]~,
(4.6)
) C'~~,
x
WC
get
> (k)
=
z
P
d
(X) e'~
=
z
(I
cos
k~ )
Xa
(k)
(4.7)
678
JOURNAL
PHYSIQUE
DE
N
I
5
where
e~'~~) 4a(k)
(1
'~«
analytic
is
"
0.
k
at
(i
One
V)o,
the
j
~
neighbor
nearest
£
=
~~
~
+
k~)
cos
find
first
can
=
e'~~) 4a(- k)
(i
+
i°'ear ~~ '~"
~~ ~~
functions
correlation
equal
to
(4.9)
(T~ ~)~~ M~
,
~
i
y
Where
the
is
T~
inverse
the
matrix
3~~
+
of
T~~
«
given by
T
I
(l
_;~
Me
k~)
cos
(4.10)
~
£
(2 gr) ~
(I
k~,)
cos
and
£
M~
Equation (3.26)
order
is
then
dk
m
(2
gr
(1
k~)
cos
2
(4. II)
ia(k)
£
(cos k~
£ Vj
=
2
ix
the
if
system
analytic
co
at
-
least
is
o,xj
e'~~
the
n-th
a
(1
k~) g~(k)
cos
~4 i~~
~
£
=
isotropic,
k=0,
at
obtain
to
j
~
(I
and
exponentially
the expansion,
fast.
the
I«(k)=I(k)
points
two
We
thus
all
for
=
V(o,~j
locality
the
.,d, then S~(k)
decays to
zero
I,
a
function
that
see
k~)
cos
x
is
V)o
Xa(k)
m
£
f(k)
1)
as
S~(k)
Hence,
(k)
X«
~
)~
by putting
solved
function
structure
e~'~~
correlation
of the
=
as
functions
points functions, at order n,
In particular this is true in
assume
we
isotropic perturbation of an
d
I. This is
infinite
dynamics still gives rise to short
spatial
correlations
in the
temperature
range
S~(k)
x~(k)
equal,
then
is
analytic
stationary
If
however
the
state.
not
not
at
are
correlations
strictly
local,
anisotropic
I,
k
0. So, even if at order n
I the
are
e-g- n
correlations
perturbations may lead to long range stationary
order n. From (4.12), the two
at
points function then decays as
order
at
f
n
I in
is
reproduced,
least
at
for
the
two
of the lattice.
system has the full
summetry
consistent
with [10], where it is also the case that an
that
=
the
=
=
V(o~j
~2
m£b]
(X(
(x(
~,
~
(4.13)
-co,
a
for
some
constants
b].
anisotropy is certainly not
that
example happen that x~(k)
2 S~(k)
Note
=
Qj
o,~j
2
e~
)
V)o
J(x)
produce
to
~
J(x
=
sufficient
+
j.
As
an
J(e~)13~,~~
this
weak
illustration,
decay.
suppose
It
may
for
that
(4.14)
30, ~j
,
M
POWER
5
J(x) strictly local,
Then, x~(k)
4
with
)(k).
DECAY
LAW
)(k)
symmetric, J(0)
4J(e~) and V(o,~j
fact
the
reflection
=
isotropic
a
not.
or
I,..,
in
is
anisotropy
generic
However,
multiple
This
we
rates
be
=
c(x,x
The
energy
«
parametrized by the
expansion of the detailed
is
(4.14) has
Q[
As
a
0,x)
on
be
to
"
a
such
whether
J(x) is
dynamics, see (A24).
simplest example is the
Choose p~ close to zero,
balance
relations.
The
announced
in
(2.14).
matter
given by
X«
(k)
[)(k)
p
4
=
pa iJ(X
"
(4.16)
p~
all
p,
=
(Al),
or
p~
+
all
not
are
Kj(3~
J(e~ )]
and
equal.
In
first
in p.
order
+
in
P2) K~
+
(-4
d
with
2
(4.12)
must
K~(3~
nearest
3~
+
neighbor coupling
~~)
~~
(4.18)
,
(4.9)-(4.ll),
(-2gr+12(pi
=
(4.1?)
3x,0)
correspondingly f~ (k)
=
+
W-2
~~j
pa J(~a)(3x,e~
+
~~
~~
(2
W-2
K~)
W
12
ar
~~
Kj
+
+
gr
~~
+18
gr
gr
~~
~~
+
~~((4gr-18+~~)Ki+
K~) mvi
ar
V)o
J(X)i
3~
+
~,
integrals
+
(4.19)
V~.
m
the
f~(k)
4
=
p ~[Kj
ki
cos
K~
+
k~]
cos
+ 2
V~,
K~
p
a
~
explicitly
V)o~j
(4.16) is the
then
to
up
~
the
the
ea)
''
are
(4.15)
«~y)
functions
(°, ~a)
0,x)
doing
after
y) «(x)
J(x
)o~j =(pi+P2)Kj+~~
Therefore,
no
transform
Fourier
=
whenever
a
£
m
=
find,
J(x),
=
balance
J(x)
we
2
having
detailed
J(x) as in (4.14). If
exchange rates (2.5), (2.14)
replaced for n
I by
local
~(
consequence,
depend
and
»
H(«)
Now,
=0,
679
GASES
LATTICE
w( (i -(paiH(«~'~+~a) H(«)1)
«)
ea,
+
satisfy
not
exchange
model
the
let
does
which
temperature
for
scenario
of [9]
d,
and
STOCHASTIC
IN
determined.
~fi-4
p
£
=2p~J(x)+
transforming (4.12) gives
Fourier
Inverse
K
(4.20)
1, 2
=
V
~
A~I(x)+
~
«=1,2
~
+
~
2
~
[Kj hi (I (x
ej
)
+
I
(x
+
ei
))
+
K~ hi (I(x
e~)
+
I
(x
+
e~))]
(4.21)
where
1(x)
«
~°~
dk
(2
gr
)~
l
~
(1/2) [cos kj
~
+
cos
k~]
(4.22)
680
JOURNAL
is the
potential
(4.21)
that
=
derived
be
can
pi
if
simple
kernel
for
p~
p,
=
then
(x(
as
walk
o,~j
the
on
pJ(x).
2
pi
If
=
N
I
plane.
# p~,
We
immediately verify
then
the
asymptotic
5
from
behavior
from
jxj xii
? (xj x])2
+
~
~~~~~~
~~~~~~
Hence,
random
V)
PHYSIQUE
DE
~~
'
~
~~'~~~
"
co,
-
°, Xl
~
(~2
g~
xi)
f~(0) ) (xl
( ii (0)
i
~'(
l
~~'~~~
~
~2)2
2
and
(4.24)
Kj
is
=
K,
m
then
(P
P~)(Ki
i
=
typical
the
K~
~)-/
i~(0)
ii (0)
+
K~)
decay predicted in (4.14).
points function still decays as
field-like
quadrupole
(4.25) is zero but the two
(4.25)
Note
a
that
power
:
if
in this
case,
"2fl21°(3x,el~ ~x,-el~~x,e2~~x,-e2)
fl2) l°hi~(X)
fl2) l°hl~x,0
~(fll
+ (PI
xl + xl 6 xl xi
12
m--(pi-p~)K
(x(-co,
(4+X~)~
i')0,x)
correlations
octopole field. We thus get always long range
that pi # p~.
induction
hypotheses on the V(jj, n m I must be
Summarizing, we find that the correct
is
described
that
their decay, as (x(
slower
than
in (4.14). flu (k) is no longer
not
co,
derivation
of (4.13)
remains
meaningful since the
necessarily analytic at k
0 but
the
corresponding f~ (k) are then still bounded
k
Generic
anisotropy
then gives rise to
0.
near
has
the
for
this
long distance
(4.26)
,
'~
behavior
of
an
provided
model
temperature
two
-
=
=
the
decay (4.14)
law
power
also
for
the
order
next
in
expansion.
the
Acknowledgement.
We
are
indebted
to
Jan
Naudts
for
a
careful
reading
of
the
manuscript,
and
for
useful
criticism.
Appendix.
Derivation
We
choose
of
the
the
high
rates
expansion
temperature
c(x,
y,
«)
c(x,
of the
y,
«
)
=
da
Kawasaki
dynandcs.
form
~P
(p (H(«x~
H(«)))
(Al)
+(- z)
(A2)
with
e~~'~
*(z)
=
+(z),
+
(z)
=
M
5
LAW
POWER
We
P(0)
that
assume
can
DECAY
IN
STOCHASTIC
P(z)
that
and
(z)
I
=
f
cases,
around
for
as
Metropolis
the
=
bi z~
o,
(A3)
even
P(z)
where
rates
0
z
=
P
Other
analytic
is
681
GASES
LATTICE
~' '~,
can
be
satisfy
the
following
e~
=
treated
uniform
via
approximation.
then
We
~P
~~)(0)
that
have
a~,
m
derivatives
the
4l(z)
of
at
0
z
relations
=
:
if
then
(~
(
i )~ ~n
~n
"
(A4)
k
and
~~
£
(~ l)~
(A4)
(A2)
is
immediate
an
(~)
an-k
that
(A5)
is
(_
equivalent
i )n
and
even
£(z)
is
(~ ~~
e~~
~P(- z). As
=
0
(2~5)
Sn
Sm
,
(A5),
for
note
we
that
from
~
0
m
z
~
~
(A6)
1<2
(m~
1/2
z
i
(z
("f ~n
~
this
But
n.
«
(- I)
(Z~)iz
~
d~
k
~n
o
mm
~f
hi
to
(
d~
k
all f
for
an-k
(A3)
and
ak
so
~P(z)
of
consequence
~~
i
"
n-m
(z
2
k
obvious
is
m
~~~
~
1<2
z
the
i
(z
2
m
(n
0
because
i
+
k
function
m
(z
2
i
+
n-m
(A8)
2
odd.
Since
they
we
appear
want
in
to
the
obtain
rates
an
expansion
and
~~ (X, Y)
qic
m
in p,
have
we
the
separate
to
different
orders
in p
as
define
~
~i
"
z (qixy~c (x,
IR.AAf=Ar~l
I
fl
qi~c
y)
~A)Y)
(~A~
f
(x, y))
=
,
1, 2,
(2~~)
(Rio)
xy)
so
that
c(x,
Y,
«
)
i +
=
~£ £ q( (x,
i
The
expansion
(3.19)
now
the
has
y)
«~
l.
(Al i)
A
form
(
(«A)~
"
V$
(A12)
n=o
with
V$
=~
m
~£ £
B,c
and
V$
=
3~j.
G
(A,
B
)
qjB~ VI,
n
m
I
(Al 3)
JOURNAL
682
We
by computing
start
first
the
order
Vj
£
=
PHYSIQUE
DE
M
I
5
term
G
(A,
£
G
)
B
q j~
s
=
(A,
£
)
B
that
so
(J~
(A14)
J~xy)
(~y)
s
by (3.4) Lo V(
Lo JA. Using the boundary
that V( =J~.
We will repeatedly
use
V(
condition
this
0
-
=
conclude
argument
(A
diam
as
in
-
derivation
the
co,
we
of
the
following.
PROPOSITION
3
~/
V$
~'
It
is
that
clear
we
decay
exponential
thus
get
a
Proof.- By induction,
computed
was
£
(A,
G
above.
flJ~,
p~~
i
~
0,..,
gives
calculation
(n~~
~
~
=1
oYl
A n~
~~~
An
r
,
=
with
(A16)
lwkwn-1.
jjJ~ £ I
£
~
~
(g~i(Ai,
x
m
small,
fl
r=1
straightforward
Aj,
for
for
k
AAk=A
'~
where,
expansion
temperature
~
A
qic _~ VC~
B
r
n
high
£
AIR..
I
(Al 5)
J~
that
assume
~'
=
A
AA
A.
correlations.
pk
V(=~
k
~
fl
~
At
converging
spin-spin
of
£
=
f
~~
n-f
)]
~
~~
~~~'
( Al 7)
'~ ~~~
~
n,
gy~ (Aj,
A
~)
(A,
G
m
,
(YA..
A
AA
(YAA~~
j
A..
AA
(A18)
~)
Hence,
~~
p
~
)n
n
~~
n
~~
Al,
+
=~
z (,
an
~l ~~
~A
~) ~~~~ ~~ ~~
()
£
-k
nr~l
fl
p
£ (gin
JA,
~
n
A
V$
pn
=
~
p
(- l)~
£
a~
~'
(~)
fl
A..
AA
A
n
r
~~
m!
(n
m
)I
I
g
im (A
xyj
=~
i (~ ~'~
an-k
pn
n
AAn"Ar=1
n))
j,
)
A
~
,
~n-kl(l~l~)
JA
+
q
~'
[I
(-1)~ an]
I
n
fl
JA
~
AIR..AAn"Ar=1
(A20)
JA
~
AI
A
that
fl
n
£
n
g10 (Al,
n)
,
-1
'
AjA
q
A
I
conclude
to
~'
=
n
r
~o
((-1)~ £
(A5)
and
n
~'m~j
~ ~~
X
(A4)
(Al,
,
Aj,
use
~
jay>
~
we
~
~
n
AI,
and
~~~ ~~ ~~
~ ~~
~
M
POWER
5
local
The
character
of
condition.
balance
of
each
implies
It
DECAY
LAW
the
(c(x,
using (Al I), for
or,
each
n
order
£ £
k
o
n
Vi
terms
for
that
STOCHASTIC
IN
each
y,
(K~ ).
some
separately,
)(«Axy- WA))
«
must
we
detailed
the
have
that
0
(A21)
~A ~1°A'Y
(2~22)
=
~~ h~(X, y)) V~
(~~~Y~B(X, Y)
"
B
Assume
whenever
0
=
case
0, 1.
=
condition
KA
-1
£ £
n
"
k
o
0
-
of the
as
B is
set
(A.22) has
diam (A)
Then
the
-
co.
larger
unique
(A22)
fact, put
In
=
Q~ (X, Y)
diameter
the
for k
=
i'~,
(2~2~)
VixY,
(3.48)
~~ h~(X, Y)
B
that
so
Qi(X, Y)
QixY(X, Y)
and
s
from
I,
m
inductively that Vi
than kR, 0 w n
I, as we know is certainly the
solution K~
V$ by imposing the boundary
implies that V$
whenever
diam (A) mnR.
0
for
(xy)
683
directly
understood
be
can
bond
GASES
LATTICE
(x(I),..,x
let
I,
I,
r
=
(r))
such
be
x(I)
that
e
of
sequence
a
Vi
"
neighbor
nearest
A, x(s) J A for
s
=
,
2,..
I,
r
I)),
(x(s)x(s+
sites
diam (A~(~~~~~~)
and
m
nR.
,
Then
Vi
VI
V$x~i)x~2)+ V(x(1)x~2)
=
Q (x~i)x<2)(x(I ), x(2))
=
+
gives
solution
the
V$x<i)x<3) +
+
Q $x<i)x~r)(x(r
+
Q
I
V$x~i)x<r-i)
(x(I ), x(2)
), x(r))
+
V$x<i)x<r)
Q (x<1)x~3)(x(I), x(3))
Q(x~i)x<r- i)(x(r
Q (x~i)x<2)
(A24)
), x(r))
I
(A22).
of
References
[1]
[2]
RUELLE
[3]
PARK
D.,
GALLAVOTTI
Phys.
Y.
7
[4]
GEORGii
and
(1968)
M., The
(1982)
H.
Mechanics,
Statistical
G.
MiRACLE-SOLE
Rigorous
S.,
Results
Correlation
Benjamin,
QV. A.
Functions
of
a
Inc., New York, 1969).
System, Comm. Math.
Lattice
274-288.
Cluster
Expansion
for
Classical
and
Quantum
Lattice
Systems,
Gruyter
Studies
J.
Stat.
Phys.
27
553-576.
O.,
Gibbs
Measures
and
Phase
Transitions,
De
in
Mathematics
its
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(1988).
[5]
KUNSCH
H., Decay of
Comm.
[6]
[8]
Correlations
84
(1982)
under
Dobrushin's
Uniqueness
Condition
and
207-222.
SCHLOSMAN
S.B.,
Analytic
Constructive
Completely
Interactions:
Phys. 46 (1987) 983-1014.
LiGGETT
T. M.,
Interacting
Particle
Systems (Springer-Verlag, New York, 1985).
of
LEBowiTz
J.L., MAES C, and SPEER E. R.,
Statistical
Mechanics
Probabilistic
Cellular
Automata, J. Stat. Phys. 59 (1990)
l17-169.
DOBRUSHiN
R.L.
Description,
[7]
Phys.
Math.
and
J.
Stat.
684
[9]
JOURNAL
P.L.,
GARRIDO
Conservative
LEBOWITz
J.L.,
Dynamics,
Rutgers
MAES
PHYSIQUE
DE
C,
and
University
SPOHN
M
I
H.,
Long
Range
Preprint (1990) Phys.
Rev.
A
5
Correlations
for
(1990)
1954-
42
l968.
[10]
Correlations
for
REDIG F., Long Range Spatial
University preprint (1990).
Scale
Dynamics of Interacting Particles,
SPOHN H., Large
Preprint (1989).
Mfinchen
MAES
C.
and
Anisotropic
Zero
Range
Processes,
Leuven
ill]
Part
B:
Stochastic
Lattice
Gases,