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Chapter
5
The Cluster Expansion
5.1
Introduction
In this chapter, we expose the cluster expansion method. The latter was introduced by Mayer during the early stages of the study of the phenomenon of
condensation, and remains, nowadays, a powerful tool in the rigorous study of
statistical mechanics.
We will use the cluster expansion in various situations. For example, we
will use it to show that in all dimensions, the pressure of the Ising model with
h = 0, (0), is analytic in when is sufficiently large. Although we would
expect this analytic behavior to hold on the entire subcritical regime, that is
for all > c (d), the method will guarantee analyticity only when > 0 ,
where 0 is some number that depends on d, strictly larger than the critical
value c (d). To distinguish it with the regime > c , which was called low
temperature in earlier chapters, we will call a regime such as > 0 a regime
of very low temperature. The cluster expansion won’t yield analyticity in the
whole subcritical regime, but it will nevertheless allow to compute, in principle,
each coefficient of the expansion
2
(0) = b0 + b1 + b2 + ;
thus giving very precise quantitative information on the pressure of the Ising
model in infinite volume, at least for low enough temperatures.
We will obtain similar results at very high temperature, or at arbitrary
temperature but sufficiently large magnetic field. Additional applications of
the technique will also appear in other chapters, for various types of models.
A general feature of the cluster expansion technique is therefore the following: it is a versatile tool that allows to analyze properties of a model only
in a restricted region of the space of parameters, but provides, in that region,
information which is often unavailable when using other techniques.
5.2
Polymer models
Let us describe the type of models to which the cluster expansion applies.
159
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CHAPTER 5. THE CLUSTER EXPANSION
Remember from Section 3.7.2 that the configurations of the Ising model at
low temperature were more conveniently described using extended geometric
objects, the contours, rather than the individual spins. Once expressed in
terms of contours (see (3.34)), the Boltzmann weight provided insight into the
geometric structure of typical configurations at low temperature. An alternative graphical representation, relying on different objects, was used in the
regime of high temperatures (see (3.43)), providing an efficient way to establish
uniqueness. We will come back to these representations later in the chapter.
The description of a system in terms of suitable geometrical objects happens to be common in equilibrium statistical mechanics. In fact, the partition
functions associated to these systems often have a common structure that is
rich enough to provide, under certain hypotheses, useful information on their
logarithm. This class of models, called polymer models, is precisely the one
for which the cluster expansion will be developed.
Consider a finite (or countable) set , the elements of which are called
polymers. Each polymer 2 has a weight (or activity) ( ), which can
be an either real or complex number. The interaction between two polymers
is measured by a function : ! R satisfying, for later convenience,
(; ) = 0 for all 2 . The (polymer) partition function is then defined
by
o
n
Y
def X Y
Ξ =
( )
(; 0 ) ;
(5.1)
0
0
0
0 2 0
f; g2 where the sum is over all finite subsets of . We allow
the products are, as usual, defined to be 1.
0 = ∅, in which case
It will be convenient to label the polymers of the subfamily 0 and
to distinguish the number of polymers it contains. The partition function will
therefore be written as
Ξ=1+
n
o Y
1 X X nY
(i ; j ) :
(i )
n! 1
n i=1
1i<j n
n1
X
(5.2)
P
(The factor n1! is necessary to avoid
i
P overcounting.) From now on, a sum
should always be understood as i 2 . Even though the partition function will
be a finite sum in all the cases considered later in the book, it does not need to
be. In this chapter, the polymers will be objects living on Zd (or, possibly, the
dual lattice), and the interaction will be related to suitable pairwise geometric
compatibility conditions between the polymers; these conditions will usually
be local, meaning that the compatibility of two polymers can be checked by
inspecting a neighborhood of their positions on Zd . Various examples will be
described in the following sections of the chapter. For the time being, we only
assume that
j (; 0 )j 1 ; for all ; 0 2 :
(5.3)
5.3
The formal expansion
The cluster expansion provides an explicit series expansion for the logarithm
of the partition function of a polymer model of the type described above. Our
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5.3. THE FORMAL EXPANSION
first result is a formal computation that provides the terms of this series. Before
stating the basic result, we introduce a few notations.
Let Gn = (Vn ; En ) be the complete graph on Vn = f1; 2; : : : ; ng. That is, En
contains all simple non-oriented edges (i; j ) between distinct points i; j 2 Vn
(G12 is represented in Figure 2.1). We will write G Gn to indicate that G
is a subgraph of Gn ; that is, G has the same set of vertices Vn , but its set of
edges is a subset of En . Moreover, given a graph G = (V; E ), we will often
write i 2 G, respectively e 2 G, instead of i 2 V , respectively e 2 E .
Let
0 def
0
(; ) = (; ) 1 :
Proposition 5.1. Let the Ursell functions be defined by
'T (1 ; : : : ; m ) def
=
1
X
m! GGm
Y
connected
(i;j )2G
'T ( ) def
= 1, and
(i ; j ) :
Then, as a formal series,
log Ξ =
XX
m1 1
X
m
'T (1 ; : : : ; m )
m
Y
i=1
(i ) :
(5.4)
Observe that, even when the partition function is a finite sum, the resulting
series in (5.4) does not reduce to a finite sum, since the same polymer can
appear multiple times.
In the next section, we will state conditions that ensure that the series
in (5.4) is actually absolutely convergent, which will justify the rearrangements
done in the proof below.
Proof. The proof consists in showing that the terms in the sum (5.2) defining Ξ
can be rearranged, leading to an explicit expression for Ξ, of the form exp( ).
The starting point is to use the “ +1 1” trick and to expand the product (see
Exercise 3.50):
Y
1i<j n
(i ; j ) =
Y
(1 + (i ; j )) =
1i<j n
X
Y
E En (i;j )2E
(i ; j ) :
Since a set E En can be put in one-to-one correspondence with a subgraph
G Gn defined by G def
= (Vn ; E ), we can interpret the sum over E En as a
sum over G Gn . This yields
n
o Y
1 X X X nY
:::
(l )
(i ; j )
n!
n l=1
n1 GGn 1
(i;j )2G
X 1 X
=1+
Q[G] ;
n!
n1 GGn
Ξ=1+
X
e = (Ve ; E
e ), we write
where, for a graph G
Q[Ge] def
=
X Y
(r )r2e
V l2Ve
(l )
Y
(i;j )2Ee
(i ; j ) :
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CHAPTER 5. THE CLUSTER EXPANSION
Consider now the decomposition of G into maximally connected components:
G = (G01 ; : : : ; G0k ). Clearly, in that case,
Q[G] = Q[G01 ] Q[G0k ] :
Now, observe that Q[G] = Q[G0 ] if G and G0 are isomorphic 1 . One can thus
replace the vertex set Vi0 of G0i by f1; : : : ; mi g, where mi = jVi0 j. Therefore,
X
GGn
Q[G] =
=
k
Y
X
r=1
GGn
G=(G01 ;:::;G0k )
Q[G0r ]
n!
X
m1 ;:::;mk
m1 ++mk =n
X
m1 ! mr ! G0 G
1
m1
connected
k
Y
X
G0r Gmr r=1
Q[G0r ] ;
connected
where, in the second identity, the coefficient n!=(m1 ! mk !) takes into account
the number of ways of partitioning Vn into k disjoint subsets of cardinality
m 1 ; : : : ; m k 1.
Finally, observing that, at least as formally,
n
XX
X
1 ;:::;mk
n1 k=1 m m
++m =n
1
k
=
XX
X
1 ;:::;mk
k1 nk m m
++m =n
1
=
k
X
X
k1 m1 ;:::;mk
;
we get, after simplifications and further formal rearrangements,
Ξ=1+
k n
o
X
1
1 X Y
Q[G0r ]
k!
mr ! G0 G
k1 m1 ;:::;mk r=1
r mr
X
connected
m
k
Y
1 X X X T
=1+
' (1 ; : : : ; m ) (j ) :
k! m1 1
m
j =1
k1
X
This finishes the proof, since
1+
P
1 k
k1 k! ()
exp()
Ensuring absolute convergence of the series is essential for the formal operations above to make (analytic) sense. Notice that proving that Ξ has a
well-defined logarithm in particular implies that Ξ 6= 0. We saw when studying uniqueness in the Ising model, that the absence of zeros of the partition
function on a complex domain implies in fact uniqueness of the infinite-volume
Gibbs measure of this model. This indicates that guaranteeing the absolute
convergence of the series in (5.5) is non-trivial in general, and the latter will
usually hold only for some restricted range of values of the parameters of the
underlying model.
Exercise 5.2. In the proof of Proposition 5.1, which were the formal manipulations that are only warranted when the series is absolutely convergent?
1 Two graphs G = (V; E ), G = (V 0 ; E 0 ) are isomorphic if there exists a
V ! V 0 such that an edge e = (x; y) 2 E if and only if e0 = (f (x); f (y)) 2 E 0 .
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bijection
f
:
163
5.4. A CONDITION ENSURING CONVERGENCE
5.4
A condition ensuring convergence
We now impose conditions on the weights that ensure that the series in (5.4)
converges absolutely:
XX
k1 1
X
k
j'T (1 ; : : : ; k )j
k
Y
i=1
j(i )j < 1 :
(5.5)
We will actually show the following:
Theorem 5.3. Assume that (5.3) holds, and that there exists
such that
1.
X
j( )jea() < 1 ;
(5.6)
j( )jea() j (; )j a( ) :
(5.7)
2. for each
2
,
X
Then, for all 1
1+
2
,
X X
k2
k
a : ! R+
2
X
k
j'T (1 ; 2 ; : : : ; k )j
k
Y
j =2
j(j )j ea(1 ) :
(5.8)
In particular, (5.5) holds.
Exercise 5.4. Show that (5.6) and (5.8) imply (5.5).
The choice of a( ) will be problem-dependent; it will usually be related to
some measure of the size of .
The series in (5.6)-(5.7) should remind the reader of those considered when
implementing Peierls’ argument, as in (3.36). Actually, proving that these
conditions hold in a specific situation usually amounts to bounding a sum of
weights of a collection of objects. Proving convergence thus ultimately reduces
to an energy-entropy argument.
Proof of Theorem 5.3: We will fix 1 2 and show that, for all N 2,
N X X
k
X
Y
1 + k j'T (1 ; 2 ; : : : ; k )j j(j )j ea(1 ) :
(5.9)
k
k=2 2
j =2
Clearly, letting N ! 1 in (5.9) yields (5.8). The proof of (5.9) is done by
induction.
For N = 2, the only connected graph G G2 is the one with one edge
1 ( ; ). Therefore, the left-hand
connecting 1 and 2, and so 'T (1 ; 2 ) = 2!
1 2
side of (5.9) is
1+2
X
2
j'T (1 ; 2 )jj(2 )j = 1 +
X
2
j (1 ; 2 )jj(2 )j ea(1 ) ;
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CHAPTER 5. THE CLUSTER EXPANSION
where we used 1 ea(2 ) , (5.7) and 1 + x ex . This proves (5.9) for N = 2.
We now show that if (5.9) holds for N , then it holds also for N + 1.
To do that, consider the left-hand side of (5.9) with N + 1 in place of N ,
take some k N + 1, and consider any connected graph G Gk appearing in
the sum defining 'T (1 ; 2 ; : : : ; k ). Let E 0 6= ∅ denote the set of edges of G
with an endpoint at 1. The graph G0 , obtained from G by removing 1 together
with each edge of E 0 , splits into a set of connected components (G01 ; : : : ; G0l ).
G1
0
7
5
2
6
4
G2
1
G3
0
3
8
9
0
We can thus see G as obtained by first partitioning f2; 3; : : : ; kg into sets
V10 ; : : : ; Vl0 , l k 1, associating to each Vi0 a connected graph G0i , and then
connecting 1 in all possible ways to at least one point in each connected component Vi0 :
'T (1 ; 2 ; : : : ; k ) =
kX1
(5.10)
l n
X Y
on X
X
Y
1
1
( 0 ; 0 )
k! l=1 l! V 0 ;:::;V 0 i=1 G0 :V (G0 )=V 0 (i0 ;j 0 )2G0 i j
1
i
l
i
i
i
connected
Ki Vi0 j 0 2Ki
Ki 6=∅
We then distinguish the number of points in each Vi0 . If
X
Y
G0i :V (G0i )=Vi0 (i0 ;j 0 )2G0i
Y
o
(1 ; j 0 ) :
jVi0 j = mi ,
(i0 ; j 0 ) = mi ! 'T (j 0 ; j 0 2 Vi0 ) :
connected
Moreover,
X
Y
Ki Vi0 j 0 2Ki
Ki 6=∅
(1 ; j 0 ) Exercise 5.5. Assuming
nY
j 0 2Vi0
o
1 + (1 ; j 0 )
1:
(5.11)
j1 + k j 1 for all k 1, show that
n
Y
(1 + k )
k=1
Since (5.3) guarantees that
1 n
X
k=1
jk j :
j1 + j 1, (5.11) and (5.12) yield
X Y
(1 ; j 0 )
Ki Vi0 j 0 2Ki
Ki 6=∅
X
j 0 2Vi0
j (1 ; j0 )j :
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(5.12)
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5.4. A CONDITION ENSURING CONVERGENCE
We now use (5.10) to bound the sum on the left-hand side of (5.9) (with
N + 1 in place of N ). The sum over the sets Vi0 will be made as in the proof
of Proposition 5.1: the number of partitions of f2; 3; : : : ; kg into (V10 ; : : : ; Vl0 ),
with jVi0 j = mi , m1 + + ml = k 1, is equal to m(1k!1)!
ml ! . Now, since the
summands are nonnegative, we can bound
NX
+1 kX1
X
k=2 l=1
m1 ;:::;ml
m1 ++ml =k 1
=
N NX
+1
X
X
l=1 k=l+1
m1 ;:::;ml
m1 ++ml =k 1
N X
N
X
l=1 m1 =1
N
X
ml =1
;
which leaves us with
NX
+1 X
k=2
k
X
j'T (1 ; 2 ; : : : ; k )j
2
k
l
N
n
Y X X
1
l!
l1 i=1 mi =1 10
X
X
0
m
i
k
Y
j =2
j'T (10 ; : : : ; m0 i )j
k=1 1
k
X nX
k
i=1
mi
Y
j 0 =1
j(j0 0 )j
mi
X
j 0 =1
o
j (1 ; j0 0 )j :
(5.13)
Lemma 5.6. If (5.9) holds, then, for all
N X
X
j(j )j
2
o
j ( ; i )j j'T (1 ; : : : ; k )j
,
k
Y
j =1
j(j )j a( ) :
(5.14)
Proof. We fix 2 , and multiply both sides of (5.9) by j ( ; 1 )j j(1 )j,
and sum over 1 . Using (5.7), the right-hand side of the expression obtained
can be bounded by a( ), whereas the left-hand side becomes
N X
X
k=1
k
But clearly, for all
X
1
X
k
1
X
k
j ( ; 1 )jj'T (1 ; : : : ; k )j
k
Y
j =1
j(j )j :
i 2 f2; : : : ; kg,
k
Y
j ( ; 1 )jj'T (1 ; : : : ; k )j j(j )j
j =1
=
X
1
X
k
j ( ; i )jj'T (1 ; : : : ; k )j
k
Y
j =1
j(j )j ;
which proves the claim.
Using (5.14), we can thus bound (5.13) by
1
a(1 )l ea(1 )
l
!
l1
X
1;
which concludes the proof of Theorem 5.3.
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CHAPTER 5. THE CLUSTER EXPANSION
5.5
The case of hard-core interactions
Up to now, we have considered an arbitrary interaction between polymers.
Often in practice, and in all cases treated in this book, takes the particularly
simple form of a hard-core interaction, that is
(; 0 ) 2 f0; 1g
for all ; 0 2 .
In such a case, two polymers and 0 will be said to be (pairwise) compatible
if (; 0 ) = 1 and incompatible otherwise. Obviously, only collections of
pairwise compatible polymers yield a non-zero contribution to the partition
function Ξ in (5.2).
Let us now turn to the series (5.4) for log Ξ. We say that an n-tuple
f1 ; : : : ; n g is decomposable if it is possible to express it as a disjoint union
of two sets, in such a way that (i ; j ) = (i ; j ) 1 = 0 for each i in
the first set and each j in the second set. It follows immediately from the
definition of the Ursell functions that
for all decomposable n- tuplesf1 ; : : : ; n g :
' T ( 1 ; : : : ; n ) = 0
The non-zero contributions to log Ξ in (5.4) therefore come from the nondecomposable n-tuples f1 ; : : : ; n g; we will call these objects clusters and
denote them by the letter X .
Remark 5.7. We emphasize that a cluster X = f1 ; : : : ; n g is an unordered
set of polymers. This is why there is no term n1! in (5.15) below.
We can thus write
log Ξ =
XX
n1 1
X
n
'T (1 ; : : : ; n )
n
Y
i=1
(i ) where the sum is over all clusters of polymers in
f1 ; : : : ; n g,
T (X ) def
=
n
X
Y
GGn (i;j )2G
(i ; j )
connected
5.6
X
X
T (X ) ;
, and for a cluster
n
oY
i=1
(i ) :
X =
(5.15)
When the weights depend on a parameter
Often, the weights of the polymers depend on some real or complex parameter:
z 7! z ( ) :
If each weight depends smoothly (for example, analytically) on z , it can be
useful to determine whether this smoothness extends to log Ξ.
Theorem 5.8. Assume that j j < 1, that for each 2 , the function
z 7! z ( ) is analytic on the domain D C, and that there exists a weight
( ) 0 such that
sup jz ( )j ( ) ; 8 2 ;
(5.16)
z 2D
and such that (5.6)-(5.7) hold with ( ) in place of ( ). Then, (5.8) holds
with ( ) in place of ( ), and z 7! log Ξ is analytic on D.
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5.7. APPLICATIONS
Proof. Let us write the expansion as
log Ξ =
X
n1
fn (z ) ;
where
def X
fn (z ) =
1
X
n
n
Y
'T (1 ; : : : ; n ) z (i ) :
i=1
Since j j < 1, fn is a sum containing only a finite number
P of terms; it is
therefore analytic in D. If we can verify that the series n fn is uniformly
convergent on compact sets K D, Theorem A.37 will imply that it represents
an analytic function on D. We therefore compute
X
sup fn (z )
z 2K n1
N
X
n=1
fn (z ) sup
X
z 2K n>N
X
jfn (z )j
sup jfn (z )j
n>N z 2K
XX
n>N 1
X
n
j'T (1 ; : : : ; n )j
n
Y
i=1
By our assumptions, Theorem 5.3 implies that (5.5) holds, with
of j()j. This implies that (5.17) goes to zero when N ! 1.
5.7
(i ) :
(5.17)
() in place
Applications
The cluster expansion can be applied in many situations. Our main systematic
use of it will be in Chapter 10, when developing the Pirogov-Sinai Theory.
We will also use it to obtain a uniqueness criterion for infinite-volume Gibbs
measures, in Subsection 6.7.4.
Before that, we apply it in various ways to the Ising model (and the corresponding nearest-neighbor lattice gas). We will see that to each region of the
phase diagram (in (; h)) corresponds a well-suited polymer model, and how
the cluster expansion then allows to extract useful information for parameters
in that region.
When checking the conditions (5.6)-(5.7), we will see that the regions for
which the cluster expansion converges for those polymer models are all far from
the critical point or from the first-order phase transition line (h = 0):
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CHAPTER 5. THE CLUSTER EXPANSION
h
large h > 0 (Section 5.7.1)
large (Section 5.7.4)
c
small (Section 5.7.3)
large h < 0 (Section 5.7.1)
We will mainly focus on the properties of the pressure in those regions, but
the decay of the truncated correlation functions at low temperature will also
be considered.
5.7.1
The Ising model in a large magnetic field
Consider the Ising model with a complex magnetic field h 2 C, at an arbitrary
inverse temperature > 0. We saw in Chapter 3 that when d = 1, analyticity of the pressure as a function of h holds everywhere (see the expression (3.12)). In higher dimensions, the Lee-Yang Theorem yielded existence
def
and analyticity of the pressure in the half planes H + = fz 2 C : Re z > 0g,
def
H = fz 2 C : Re z < 0g. Here, we will prove a much weaker result using the
cluster expansion, namely that analyticity holds at least for Re h > x0 > 0 or
Re h < x0 < 0 (see below for the value of x0 ). However, we will also provide
additional information on the pressure.
Consider the Ising model on Zd , d 1. When the magnetic field
h > 0 is large, there is a very strong incentive for spins to take the value
+1. Negative spins should thus be very rare, and it is natural to describe
configurations by only keeping track of the positions of the latter.
As we have seen in Chapter 3, the pressure does not depend on the boundary
condition; we will thus use +-boundary conditions.
Let ! 2 +
. We emphasize the role of the 1 spins by writing the Hamiltonian as follows:
H;h (! ) =
X
fi;j g2Eb
!i !j
= jEb j hjj h
X
i2
!i
X
fi;j g2Eb
(!i !j
1) h
X
i2
(!i
1) :
(5.18)
Since the first term does not depend on h, we shall ignore it from now on.
Identifying ! with the set (! ) defined by
(! ) def
= fi 2 : !i = 1g ;
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(5.19)
169
5.7. APPLICATIONS
we can write
hjj + 2 j@e (!)j + 2hj (!)j ;
where we remind the reader that, for any A ,
H;h (! ) =
@e A def
= fi; j g : i j; i 2 A; j 62 A :
Note that (! + ) = ∅, where the configuration ! + is the unique ground state
of the model when h is real positive. We can then write the partition function
as
Z+
;;h
n
= ehjj 1 +
X
6
∅=
e
2 j@e o
j 2hj j :
Let us declare that two vertices i; j 2 are connected if i j . We can then
decompose into maximally connected components (see Figure 5.1):
= S1 [ [ Sn :
Figure 5.1: A configuration of the Ising model. Each connected component of the shaded area delimits one of the polymers S1 ; : : : ; S17 .
By definition, d(Si ; Sj ) > 1 if i 6= j . The componentsPSi play the role of
n j@ S j, j j =
the
i=1 e i
Pn polymers in the present application. Since j@e j =
j
S
j
,
we
have
i
i=1
n
n
o Y
o
X 1 X
X nY
hjj 1 +
Z+
h (Si )
(Si ; Sj ) ;
;;h = e
n!
n1 S1 Sn i=1
1i<j n
(5.20)
P
where each sum Si is over non-empty connected subsets of (from now
on, all sets denoted with the letter S , with or without a subscript, will be
considered as non-empty and connected), the weights are
h (Si ) def
=e
2 j@e Si j 2hjSi j ;
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CHAPTER 5. THE CLUSTER EXPANSION
and the interactions are of hard-core type:
def
(Si ; Sj ) =
(
1
0
if d(Si ; Sj ) > 1 ;
otherwise:
hjj f1 + g, nicely expresses the fact
The representation (5.20), Z+
;;h = e
that the polymers Si model the perturbations away from the configuration ! + ,
whose contribution to the partition function is ehjj . However, this picture only
makes sense if one can show that the presence of polymers is indeed suppressed,
which we will do by proving that they lead to a convergent cluster expansion
whenever Re h is large enough.
We will now show that there exists a function a(S ) 0 such that when
Re h is taken sufficiently large, (5.6)-(5.7) hold. Here, these two conditions
translate into
X
8S ;
jh (S )jea(S) < 1 ;
(5.21)
jh (S )jea(S) j (S; S )j a(S ) ;
(5.22)
S X
S def
where we remind the reader that (S; S ) = (S; S ) 1. Since there are
only a finite number of connected subsets S , (5.21) is always satisfied,
irrespective of the choice of a(). It is therefore the second condition, (5.22),
which will dictate our choice for a().
Observe that
(S; S ) 6= 0 if only if S \ [S ]1 6= ∅, where
[S ]1 def
= j 2 Zd : d(j; S ) 1 :
Therefore, the sum in (5.22) can be bounded by
X
S jh (S )jea(S) j (S; S )j j[S ]1 j j2max
[S ]
X
1 S 3j
jh (S )jea(S) ;
where now the sum over S 3 j is an infinite sum over all finite connected
subsets of Zd that contain the point j . A look at the previous inequality shows
that if one defines, for all S ,
a(S ) def
= j[S ]1 j ;
then, since both the weights and
a() are translation invariant,
X
X
max
j
h (S )jej[S]1 j = jh (S )jej[S]1 j ;
j 2[S ]1 S 3j
S 30
and if one can guarantee that
X
S 30
jh (S )jej[S]1 j 1 ;
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(5.23)
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5.7. APPLICATIONS
then (5.22) will be satisfied. Now, the weight h (S ) contains two terms: a
surface term e 2 j@e S j , and a volume term e 2hjS j . Observe that ej[S ]1 j is
also a volume term, since
jS j j[S ]1 j (2d + 1)jS j :
We therefore see that in order for the series in (5.23) to converge and be
smaller or equal to 1, the magnetic field will need to be taken sufficiently large
to compensate the term ej[S ]1 j . It will also be necessary to compensate for the
fast-growing number of sets S 3 0 as a function of their size. The surface term,
on the other hand, will be of no help 2 and we will simply bound it by 1. So,
grouping the sets S 3 0 by size,
X
S 30
jh (S )jej[S]1 j =
X
k1
X
k1
X
2 j@e S j ej[S ]1 j
e
2k Re h
e
(2 Re h 2d 1)k # fS
S 30
jS j=k
e
3 0 : jS j = k g :
Exercise 5.9. Using Lemma 3.51, show that there exists a constant
c(d) > 0 such that
# fS 3 0 : jS j = kg eck :
c=
(5.24)
Using (5.24),
X
S 30
jh (S )jej[S]1 j X
k1
e
(2 Re h 2d 1 c)k
(Re h; d) ;
and, if we define
x0 = x0 (d) def
= inf x > 0 : (x; d) 1
and let
Hx+0 def
= fz 2 C :
Re z
> x0 g ;
+
then the cluster expansion for log Z+
;;h converges for all h 2 Hx0 .
In the rest of this section, we always assume that h 2 Hx+0 . Using (5.4), we
can write the pressure in as an absolutely convergent series:
n
X
Y
1 XX
1
log Z+;;h = h +
'T (S1 ; : : : ; Sn ) h (Si ) :
jj
jj n1 S1 Sn i=1
We will now see how the terms of the cluster expansion can be rearranged to
extract the volume and surface contributions to the pressure in .
As explained in Section 5.5, the contributions to the pressure in a finite
volume come from the clusters, that is, the indecomposable n-tuples, X =
fS1 ; : : : ; Sn g. The support of the cluster X = fS1 ; : : : ; Sn g is defined to be
S1 [ [ Sn , and will also be denoted by [X ]; jX j will then denote the number
2 In Chapter 10, when studying the vicinity of
dominant ones.
h
= 0,
the surface terms will be the
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CHAPTER 5. THE CLUSTER EXPANSION
of points in the support of X . To ease notation, we will write X instead
of [X ] and X 3 i instead of [X ] 3 i. With these notations, we can write
X
1
log Z+;;h = h +
T (X ) ;
jj
X where the notation T (X ) was
P introduced in (5.15).
Since, when X , jX1 j i2 1fX 3ig = 1, we can write
X
X T (X ) =
1 T
(X )
j
X
i2 X j
X X
X 3i
=
Xn X
1
i2 X Zd
X 3i
jX j
T (X )
X
1
X 6 jX j
X 3i
o
T (X ) :
(5.25)
The difference between the two series is well defined, since both are absolutely
convergent. Namely, using Theorem 5.3 for the terms n 2,
n
X
Y
X XX
X
jT (X )j n
j'T (S1 ; : : : ; Sn )j jh (Sk )j (5.26)
Sn
k=1
n1 S1 3i S2
X Zd
X 3i
X
jh (S1 )jej[S1 ]1 j (Re h; d) 1 :
S1 3i
By translation invariance, the first sum over X in the right-hand side of (5.25)
doesn’t depend on i, and thus yields a constant-volume contribution to the
pressure. To see that the second sum is a boundary term, observe that if i 2 and X 3 i but X 6 , then there must exist at least one component Si 2 X
which intersects the boundary of . As a consequence, X \ @ ex 6= ∅, and we
can write, using (5.26) for the second inequality,
X X
1
X
T (X ) j@ ex j max
jT (X )j j@ ex j :
j 2@ ex j
X
j
X 3j
i2 X 6
X 3i
We thus obtain
X 1
1
O(j@ ex j)
log Z+;;h = h +
T (X ) +
jj
jj :
X 30 jX j
(5.27)
Now, taking the thermodynamic limit in (5.27) along a sequence of boxes
B(n), n ! 1, the boundary term vanishes and we get the following: for all
h 2 Hx+0 , the pressure exists and equals
(h) = h +
1 T
(X ) :
j
X
X 30 j
X
(5.28)
Remark 5.10. We can use (5.28) into (5.27), for a fixed region , to distinguish between the two main contributions to the partition function, those
coming from the bulk (volume), and those coming from the boundary:
Z+
;;h
=e
(h)jj+O(j@ ex j) :
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173
5.7. APPLICATIONS
This will be used often in Chapter 10.
The cluster expansion of the pressure, in (5.27), describes the contributions
to the pressure when the magnetic field is large. Namely, the ground state
!+ contributes an amount h to the pressure, while the contributions due the
excitations away from ! + are added successively by considering terms of the
series associated to clusters containing one polymer, two polymers, etc.:
(h) = h+
X
1
S1 30 jS1 j
h (S1 )
1 X
1
(S ) (S )+ : (5.30)
2! S1 ;S2 : jS1 [ S2 j h 1 h 2
d(S1 ;S2 )1
S1 [S2 30
It is immediate to check that the contribution to the latter series of a cluster X = fS1 ; : : : ; Sn g is of order e 2h(jS1 j++jSn j) . Thanks to the absolute
summability of the series (5.30), we can regroup all terms coming from clusters contributing to the same order e 2nh , n 1. In this way, we obtain an
absolutely convergent series for (h) h in the variable e 2h . It is then a
straightforward, albeit quickly very tedious, exercise to compute the first few
terms of the latter.
Lemma 5.11. When
isfies
(h) = h + e
h 2 Hx+0 ,
4d e 2h +
de
the pressure of the Ising model on Zd sat
( 21 + d)e 8d e 4h + O(e 6h ) : (5.31)
(8d 4)
Proof. A pointed out above, the contribution of a cluster X = fS1 ; : : : ; Sn g is
of order e 2h(jS1 j++jSn j) . Therefore, the only cluster contributing to the term
of order e 2h is the one composed of the single polymer f0g; this gives the
first coefficient since @e f0g = 2d. There are two types of clusters contributing
to the term of order e 4h : the clusters composed of a single polymer of size
2 containing 0, and the clusters made of two polymers of size 1, at least one
of which is f0g. Let us first consider the former: since there are exactly
2d polymers of size 2 containing the origin and, for each such polymer S ,
@e S = 2(2d 1), this yields a contribution
2d 1
e
2
4(2d 1)
= de
(8d 4) :
Let us now turn to the clusters made up of two polymers of size 1, at least one
of which is f0g. The first possibility is that both polymers are f0g. This yields
the contribution
1 e 24d = 1 e 8d :
2
2
The second possibility is that X = fS1 ; S2 g = ff0g; figg, with i 0. Since
there are two ways to choose S1 ; S2 for a given i, and 2d ways of choosing i,
we obtain a contribution of
1
2
2 2d 12 e
24d
= de 8d :
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CHAPTER 5. THE CLUSTER EXPANSION
def
Remark 5.12. By considering the variable z = e 2h , we have so far shown
that
2
3
(5.32)
(h) h = a1 z + a2 z + a3 z + : : :
2
x
0
converges in the disk jz j < e
. Lemma 5.11 gave the first two terms of
that expansion, a1 and a2 . It is of course possible to compute the coefficients
to arbitrary order in a straightforward manner, but the computations become
complicated once the order gets large.
However, we already know from the Lee-Yang theorem that the pressure
is analytic (as a function of zP
) in the whole unit disk U = fz 2 C : jz j < 1g,
and since it is represented by n1 an z n when jz j < e 2x0 , this series actually
converges on all U. It is thus interesting to note that the expansion (5.31)
converges not only for Re h > x0 , but for all Re h > 0.
Remark 5.13. Using the +-boundary condition was a simplification, since
in this case the weights of the sets S don’t depend on the boundary condition. Nevertheless, the same analysis could have been done with any other
boundary condition, with slight changes, and would have led to the same expansion (5.28), only the boundary term in (5.25) being affected by the choice of
boundary condition. An interesting consequence of the above analysis is therefore that, when jRe hj is large, the thermodynamic limit for the pressure exists
for arbitrary boundary conditions, even when the magnetic field is complex. Exercise 5.14. Prove that last statement. What changes must be made if
one uses non-constant boundary conditions?
To summarize, we have seen that considering the Ising model with Re h >
1 spin as perturbations from the
ground-state ! + . These perturbations are under control whenever the cluster
expansion converges. This led us to a series expansion for the pressure of
the model in the variable e 2h , converging when Re h is large enough. As
mentioned above, convergence for all Re h > 0 then follows from the Lee-Yang
Theorem. Nevertheless, the description adopted above has one major defect:
it does not allow to study the model near Re h = 0, where a first-order phase
transition occurs. In Chapter 10, we will see that the vicinity of Re h = 0 can
indeed be studied more thoroughly, but with different polymers and ideas from
the Pirogov-Sinai Theory.
0 large allows one to see the regions of
In the following exercise, we propose to analyze the Ising model in the same
regime as before, that is, with jRe hj large, but with a different choice for the
polymers.
Exercise 5.15. Consider the set defined in (5.19), and consider each of
its points as a distinct polymer: = f1 ; : : : ; j j g. Express the partition
function for the corresponding polymer model, and verify the condition for
the convergence of the cluster expansion. Is the condition on jRe hj better
or worse?
5.7.2
The virial expansion for the lattice gas
We have seen in Section 4.4 that the phase transition of the Ising model (in
d 2) can be reinterpreted in terms of the lattice gas with nearest-neighbor
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5.7. APPLICATIONS
interactions. Namely, the pressure of that lattice gas, p (), was shown to
be analytic everywhere except at , where it has a discontinuous derivative
if the temperature is sufficiently low. To express the pressure as a function
@p
of the particle density 2 [0; 1], we inverted the relation = @ , to obtain
= (), and defined pe () def
= p ( ()). That function was shown to be
analytic on the gas branch (0; l ), flat on the coexistence plateau [l ; g ], and
again analytic on the liquid branch (l ; 1) (see Figure 4.9 and Exercise 4.22).
In this section, we will consider the behavior of the model on the gas branch
for small values of the density, at which we expect the ideal gas law to be a
good approximation:
pe () =
+ O(2 )
( small) :
(Similar statements were shown to hold for the hard-core and van der Waals
gases.) Here, we will actually obtain a representation of pe as a convergent
series, called the virial expansion:
pe () = b1 + b2 2 + b3 3 + : : :
( small) :
Besides providing high-order corrections to the ideal-gas equation, this also
proves that the pressure behaves analytically in , at least for small densities.
We will also compute explicitly the first few coefficients of this expansion.
The canonical lattice gas at low density corresponds, in the grand canonical ensemble, to large negative values of the chemical potential (remember
Exercise 4.9). We have also seen in Section 4.4.1 that the grand canonical
def
pressure p () can be mapped, via i = 2ni 1, to the Ising model with an
inverse temperature 0 = 14 and magnetic field h0 = 2 (2d + ), and that their
pressures are related by
p () =
0 0 (h ) + 2 :
(5.33)
Remark 5.16. We must not forget that this relation had been established
using the full Hamiltonian of the Ising model (with !i !j , not !i !j 1). Therefore, we will include the term jEb j that has been ignored until now, whose
contribution to the pressure is d .
Since a large negative chemical potential corresponds to a large negative
magnetic field, we will compute the virial expansion using the analyticity results for the Ising model for large values of jRe hj obtained in the previous
section. Namely, using the symmetry ( h) = (h) and using the expan0
sion (5.32), in terms of the variable z 0 = e2h , with Re h0 < x0 :
0
0 (h ) =
This gives
h0 + a1 z 0 + a2 z 0 2 + a3 z 0 3 + : : :
p () =
X
n1
an z 0 n ;
(5.34)
(5.35)
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CHAPTER 5. THE CLUSTER EXPANSION
which is called the Mayer expansion. Since it is absolutely convergent,
we can differentiate the Mayer series term by term with respect to . Since
@z 0 k = kz 0 k , this yields
@
=
X
@p X
kak z 0 k a~k z 0 k (z 0 ) :
=
@ k1
k1
We will obtain the virial expansion by inverting this last expression, obtaining
z 0 = 1 (), and injecting the result into (5.35). Since 0 (0) = a1 = e 4d > 0,
the analytic Implicit Function Theorem (Theorem A.38) implies that can
indeed be inverted on small disk
PD C centered at the origin. We denote its
Taylor expansion by 1 () = k ck k . Assuming that the coefficients ck are
known (they will be computed below), we can write down the virial expansion.
Namely,
pe () =
=
=
X
n1
X
n1
X
n1
where
an f 1 ()gn
an
an
X
k1 1
1
X
m
X Y
kn 1 i=1
cki ki
X
m
Y
m=1 k1 ;:::;kn 1 i=1
k1 ++kn =m
~bm def
=
m
X
n=1
an
cki ki m
Y
X
k1 ;:::;kn 1 i=1
k1 ++kn =m
X
m1
~bm m ;
cki :
A similar computation can be used in the following exercise.
Exercise
P 5.17 (Computing the Taylor coefficients of an inverse function). Let
(z ) = k1 a~k z k be convergent and invertible inPa neighborhood of z = 0,
with a
~1 6= 0. Write its inverse 1 as 1 (z ) = k1 ck z k . Show that the
following relations hold:
m
X
n=1
a~n
X
m
Y
k1 ;:::;kn 1 i=1
k1 ++kn =m
(
cki =
1
0
1:
Then, compute the first coefficients of
c1 =
1
;
a~1
c2 =
a~2
a~2
; c3 = 2 52
3
a~1
a~1
if m = 1 ;
otherwise. :
a~3
;
a~41
etc.
As can be verified, using the coefficients ak computed in Lemma 5.11,
~b1 = 1 ; ~b2 = a22 = 12 + d de4 ; etc.
a1
We have thus shown
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177
5.7. APPLICATIONS
Theorem 5.18. At low densities, the pressure of the nearest-neighbor lattice gas satisfies
pe () = + 1 ( 12 + d de4 )2 + O(3 ) :
5.7.3
The Ising model at high temperature (h = 0)
In this section, we consider the Ising model in the absence of magnetic field,
at very high temperature, where thermal fluctuations are so strong that the
spins are nearly independent.
Again, since we are only interested in properties of the pressure, we choose
the most convenient boundary condition, which in this case turns out to be
the free boundary condition. Proceeding as in Section 3.7.3, we express the
partition function as follows (see Exercise 3.52):
= 2jj (cosh )jE j
Z∅
;;0
X
E 2Eeven
(tanh )jE j ;
(5.36)
where the sum is over all subsets of edges E E which satisfy the following
condition: the number of edges of E incident to each vertex i 2 is even.
Each set E 2 Eeven
can be identified with a graph (by simply considering it together with the endpoints of each of its edges), and this graph can
be decomposed into connected components. For simplicity, we will identify
each connected component with its set of edges. We therefore consider the
decomposition
E = E1 [ [ En :
We therefore obtain
Z∅
;;0
= 2jj cosh( )jE j
n
X
X
1 + n1!
n1 E E
1
where it is assumed that each
(Ei ; Ej ) def
=
(
1
0
n
X nY
(tanh )jEi j
En E i=1
o
Y
1i<j n
o
(Ei ; Ej ) ;
Ei satisfies the condition above, and
if Ei and Ej have no vertex in common ;
otherwise,
is again of hard-core type. The above representation of the partition function is
clearly well suited to the high temperature regime, since the weight (tanh )jEi j ,
associated to a polymer Ei , decays fast when is small.
We now leave it as an exercise, proceeding as in Subsection 5.7.1, to show
that the conditions for the convergence of the cluster expansion are satisfied
when is sufficiently small, thus proving that the pressure behaves analytically
at high temperature:
Theorem 5.19. There exists r0 > 0 such that the pressure,
analytic in the disk f 2 C : j j < r0 g.
7!
Exercise 5.20. Compute the first few terms of the expansion of
d log(cosh ) as a power series in the variable z = tanh .
(0), is
(0)
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CHAPTER 5. THE CLUSTER EXPANSION
Remark 5.21.
Even though we have only considered analyticity of the
pressure as a function of here, it is possible to extract a lot of additional
information on the model in this regime. In Section 6.7.4 we will use a variant of
the above approach to prove that there is a unique Gibbs state at all sufficiently
high temperatures, for a large class of models.
It follows from the results of Chapter 3 that the pressure is continuously
differentiable at h = 0 when < c (d). In the next exercise, the reader is
asked to show that it is in fact analytic, at least when is sufficiently small.
Exercise 5.22. Show that, in all dimensions, there exists 0 > 0 such that
for all 0 0 , the pressure h 7! (h) is analytic at h = 0.
5.7.4
The Ising model at low temperature (h = 0)
We now consider the Ising model at very low temperature in the absence of
magnetic field. Our goal is to establish, in this regime, analyticity of the pressure, 7! (0), and exponential decay of the truncated two-point correlation
function hi ; j i;0 as ki j k2 ! 1.
When h = 0 and is large, typical configurations are small perturbations
of the ground states ! + and ! . We have seen in Section 3.7.2 that, in d =
2, a convenient description of the configurations at very low temperatures is
provided by the Peierls contours, that is, the connected sets of edges of the dual
lattice of Z2 separating regions of +1 and 1 spins, obtained after applying
the deformation rule of Figure 3.10.
In this section, we introduce an analogous contour representation of the
Ising model in any dimension d 2, in a large box , with either + or
boundary condition. Since the deformation rule of Figure 3.10 was specific
to the two-dimensional case, we will not use it and will define contours in a
slightly different manner. This will allow us to express the partition function
as a polymer model, in which the polymers are the contours.
Remark 5.23. Observe that, in d = 2, analyticity of 7! (0) for large can be deduced directly from the analyticity at small values of (Theorem 5.19
above), using the duality transformation (3.71) described in Section 3.8.3.
However, there is no analogous transformation in d 3, and analyticity in
must be obtained by other means.
We again write the Hamiltonian in a way that emphasizes the role played
by pairs of spins with opposite signs:
H;;0 (! ) =
jEb j X
fi;j g2Eb
(!i !j
1) :
(5.37)
We consider the +-boundary condition in a region b Zd . Given ! 2 +
, we
use again (! ) to denote the set of vertices i at which !i = 1. Rather than
(! ) itself (which was relevant when considering a large magnetic field), we
will be interested only in its boundary.
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5.7. APPLICATIONS
Similarly to the two-dimensional case, we associate to each vertex
the closed unit cube centered at i:
def
Si = i + [ 21 ; 12 ]d ;
and let
M (! )
Seen as a subset of Rd ,
connected components:
@ M (!)
def
=
[
i2 (!)
Si :
i 2 Zd
(5.38)
can be decomposed into a set of maximally
@ M ( ! ) = 1 [ [ n :
def
Each element of the set 0 (! ) = f1 ; : : : ; n g is called a contour of ! . In
d = 2 (see Figure 3.9), contours can be identified with connected sets of dual
edges. In general dimensions, contours are connected sets of plaquettes, which
are the (d 1)-dimensional
faces of d-dimensional hypercubes of the form
x 2 Rd : kx ik1 21 with i 2 . The number of plaquettes contained
in a contour i will be denoted ji j. Observe that to each plaquette
of @ M (! )
Pn
corresponds a unique edge of @e (! ), and so j@e (! )j = i=1 ji j.
We first write
Z+
;;0
b X
= ejE j
where
Y
!2
+ 2 0 (!)
( ) def
=e
( ) ;
2 j j :
The final step is to transform the summation over ! into a summation over
sets of contours. Let denote the set of possible contours appearing in
configurations of +
.
Definition 5.24. A collection of contours 0 is admissible if there
0
0
exists a configuration ! 2 +
such that (! ) = .
S
Definition 5.25. A subset A Zd is simply connected if i2A Si is a
simply connected subset of Rd .
Exercise 5.26. Assuming that is simply connected, show that a collection 0 = f1 ; : : : ; n g is admissible if and only if its contours are
pairwise disjoint: i \ j = ∅ for all i 6= j .
Why is this not true anymore if we drop the assumption that is
simply connected?
Therefore, provided that be simply connected, we can rewrite Z+
;;0 as
the partition function of a polymer model:
b
e jE j Z+;;0 =
X
Y
0 2 0
( )
admiss.
=1+
n
o Y
X nY
1 X
(i )
(i ; j ) ;
n!
n1 1 2 n 2 i=1
1i<j n
X
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CHAPTER 5. THE CLUSTER EXPANSION
where
def
(
(i ; j ) =
if i \ j = ∅ ;
otherwise,
1
0
(5.39)
is once more of hard-core type. The above representation of the Ising model is
well suited at low temperature, since the weight (i ) = e 2 ji j , decays fast
when is large. This was the feature that made Peierls’ argument efficient.
For the rest of the section, we will actually consider complex temperatures:
2 C.
def
We then verify that (5.6) and (5.7) hold with a( ) = j j.
Exercise 5.27. Prove that there exists x0 = x0 (d) > 0 such that, for all such that Re > x0 ,
X
j ( )jejj < 1 ;
(5.40)
and, for each
2
,
X
j ( )j ejj j (; )j j j :
Hint: Use Lemma 3.51 to count the number of contours
contains a fixed point.
(5.41)
whose support
Pressure
We leave it as an exercise to provide the details for the proof of the following
result:
Theorem 5.28. (d 2). There exists x0 = x0 (d) > 0 such that the pressure, 7! (0), is analytic in the half-space f 2 C : Re > x0 g.
As in the previous applications, it is now straightforward to extract the successive contributions to the pressure from the cluster expansion.
Exercise 5.29. (d 2) Assume Re the expansion are
(0) = d + e
4d
> x0 (d).
+ de
Show that the first terms of
4(2d 1)
+ O(e 8d ) :
Remember that for d = 2, Onsager gave a closed expression for (0) (see the
the double integral at the beginning of Section ??), which can be easily shown
to be analytic in for small or large . The above result provides analyticity,
without an exact expression, but the coefficients of the expansion in terms of
z = e 2 can be computed, in principle, up to any order.
Magnetization and decay of the truncated two-point function
We now move on to the study of correlation functions at low temperature.
def
d
that =
Q Let b Z be simply connected, and A . Remembering
+
i2A i , we will express the correlation function hA i;;0 in a form suitable
for an analysis based on the cluster expansion. We already know that the
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5.7. APPLICATIONS
partition function Z+
;;0 can be expressed as a polymer partition function.
Our task is thus to do the same for the numerator
X
!2
+
!A e
H;;0 (! )
b X
= ejE j
!2
+
!A
Y
2 0 (!)
( ) :
0
Let ! 2 +
and let 2 (! ) be one its contours. Consider the configuration
+
! in which has as its unique contour: 0 (! ) = f g. The interior of (see Figure 5.2) is defined by
Int def
= i 2 : !i = 1
(! ) :
Figure 5.2: The interior of a (here two-dimensional) contour: the interior is the set of all black vertices.
The simple, but important observation is that, for any
0
!i = ( 1)#f 2 (!): i2Int g ;
! 2 + ,
that is, the sign of the spin at the vertex i is equal to +1 if there is an even number of contours surrounding i (in the sense that i belongs to their interior),
and 1 if this number is odd. It follows from this observation that
P
0
A (!) = ( 1) i2A #f 2 (!) : i2Int g :
In particular, if we define new weights A
for the contours, by
A ( ) def
= ( 1)#fi2A : i2Int g ( ) ;
we see that
since
!A
Y
2 0 (!)
( ) =
Y
2 0 (! )
A ( ) ;
X
X X
# f 2 0 (! ) : i 2 Int g =
1fi2Int g
i2A
i2A 2 0 (!)
X
=
# fi 2 A : i 2 Int g :
2 0 (! )
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CHAPTER 5. THE CLUSTER EXPANSION
We conclude that
hA i+;;0
P
=
Q
Q
0 2 0 A
( ) f; 0 g2
P
Q
Q
0 2 0 ( ) f; 0 g2
0 0 (; 0 )
0 0
A
Ξ
:
0
(; ) Ξ
(5.42)
Clearly, now, both the numerator and denominator in (5.42) have the form
of a polymer partition function (and differ only in the weight associated to
contours surrounding vertices in A). When Re > x0 (see Exercise 5.27), the
cluster expansion for log Ξ converges, and since jA
( )j = j ( )j for all ,
A
the same holds for log Ξ . We thus obtain
hA i+;;0 = exp
log ΞA
nX
= exp
X log Ξ
TA; (X )
X
X o
T (X ) ;
where the sums in the rightmost expression are over clusters of contours in
and T (X ) and TA; (X ) are defined as in (5.15) with weights given by
and A respectively. In particular, the contributions to both sums of all
clusters containing no contour surrounding a vertex of A cancel each other,
and we simply get
nX
hA i+;;0 = exp
o
X X A
(TA; (X ) T (X )) ;
where the notation X A means that X contains at least one contour such
that A \ Int 6= ∅. We leave it as an exercise to show that one can let " Zd
in the above expression:
Exercise 5.30. (d 2) Prove that
nX
hA i+;0 = exp
X A
TA; (X ) T (X )
o
;
(5.43)
provided that that Re is sufficiently large.
We now turn to two applications of this formula.
Magnetization at very low temperatures. We first use (5.43) to study
the magnetization at the origin, that is, we set A = f0g. In this case, we have
n X
h0 i+;0 = exp
X f0g
o
Tf0g; (X ) T (X )
;
(5.44)
where the condition X f0g now reduces to the requirement that at least one
of the contours in X surrounds 0. It is then a simple exercise, proceeding as
in the previous sections, to obtain a (very) low temperature expansion for the
magnetization.
Exercise 5.31. (d 2) Prove that, for all sufficiently large values of
h0 i+;0 = 1
2e 4d
4de
(8d 4)
+ O(e 8d ) :
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5.7. APPLICATIONS
Decay of the truncated 2-point function. As we saw in Exercises 3.52
and 3.53, the correlations of the Ising model decay exponentially fast at sufficiently high temperature (small )
cHT ( )kj ik1 ;
hi j i;0 e
In contrast, we know that at low temperature,
decay anymore since, by the GKS inequalities,
8i; j 2 Zd :
> c , the correlations don’t
hi j i+;0 hi i+;0 hj i+;0 = (h0 i+;0 )2 > 0 :
Here, we will study the truncated two-point function, which is the standard
physics terminology for the covariance between the random variables i and
j in the Gibbs state hi+;0 :
hi ; j i+;0 def
= hi j i+;0 hi i+;0 hj i+;0 :
Theorem 5.32. (d 2) There exist
for all 0 ,
jhi ; j i+;0 j e
Proof. Let us write, for
hand, by (5.44),
0 < 0 < 1 and cLT ( ) > 0 such that,
cLT ( )kj ik1 ;
8i; j 2 Zd :
e T (X ) def
A b Zd , = TA; (X ) T (X ).
A;
n X
hi i+;0 hj i+;0 = exp
eT
fig; (X ) +
X fig
X
X fj g
(5.45)
On the one
o
eT
fj g; (X ) :
On the other hand, by the general formula (5.43),
n X
hi j i+;0 = exp
Clusters
Ci
X fi;j g
o
eT
fi;j g; (X ) :
X fi; j g can be distributed into three classes:
def = X : X fig but X 6 fj g ;
Ci;j
Cj
def def n X
X fig
= X : X fig and X fj g :
Observe now that Tfi;j g; (X ) = Tfig; (X ) for all
Tfj g; (X ) for all X 2 Cj . This implies that
hi j i+;0 = exp
= X : X fj g but X 6 fig ;
eT
fig; (X ) +
+
= hi i+;0 hj i+;0 exp
X
X fj g
X
X 2Ci;j
n X
X 2Ci;j
X 2 Ci , and Tfi;j g; (X ) =
eT
fj g; (X )
o
eT
eT
eT
fi;j g; (X ) fig; (X ) fj g; (X )
eT
eT
eT
fi;j g; (X ) fig; (X ) fj g; (X )
o
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CHAPTER 5. THE CLUSTER EXPANSION
Now, for all
e T (X )j 2jT (X )j, and therefore
A, j
A;
hi j i+;0 hi i+;0 hj i+;0 exp
n
6
X
X 2Ci;j
o
jT (X )j :
The conclusion will thus follow once we prove that
X
X 2Ci;j
jT (X )j e
ckj ik1 ;
for some constant c = c( ) > 0. We now prove
P this claim. Assume
By (5.26), for any vertex v 2 Rd , since jX j 2X j j,
X
X 3v
jT (X )jejX j This implies that, for any
X
X 3v
jX jR
X
X 2Ci;j
jT (X )j for some
X 3v
jT=2 (X )j 1 :
R > 0,
jT (X )j e
Therefore, since each
X
2x 0 .
X 2 Ci;j
X
R2kj ik1
Rd
R
X
X 3v
satisfies
X
X 3v
jX j=R
jT (X )jejX j e
R :
jX j 2kj ik1 ,
jT (X )j X
R2kj ik1
Rd e
R
c = c(; d) > 0, for all large enough.
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e
ckj ik1 ;