Mathematical Physics, Analysis and Geometry - Volume 13

378 Pages·2010·8.22 MB·English

by  A. Boutet de Monvel, V. A. Marchenko, H. McKean (Editors)

Math Phys Anal Geom (2010) 13:1–18
DOI 10.1007/s11040-009-9063-1
Griffith–Kelly–Sherman Correlation Inequalities
for Generalized Potts Model
Nasir Ganikhodjaev · Fatimah Abdul Razak
Received: 22 April 2009 / Accepted: 8 July 2009 / Published online: 17 July 2009
© Springer Science + Business Media B.V. 2009
Abstract Inspired by the work of D.G.Kelly and S.Sherman on general
Griffiths inequalities on correlations in Ising ferromagnets, we formulate and
prove Griffith–Kelly–Sherman-type inequalities for the ferromagnetic Potts
model with a general number q of local states. We take as local state space for
c q−1
the q-state Potts model the set F ={−l, −l+1, · · · , l−1, l},where l = . The
2
c c c c
important properties of F for what follows are that |F | = q and F = −F .
Keywords Correlation inequalities · Potts model ·
Griffith–Kelly–Sherman inequalities · Gibbs measure
Mathematics Subject Classifications (2000) 82B20 · 82B26
1 Introduction
Statistical physics seeks to explain the macroscopic behavior of matter on the
basis of its microscopic structure. This includes the analysis of simplified math-
ematical models [3]. The Potts model [12] was introduced as a generalization of
the Ising model to more than two components (spins). Ising model considered
N. Ganikhodjaev (B) · F. A. Razak
Department of Computational and Theoretical Sciences,
Faculty of Science, IIUM, 25200 Kuantan, Malaysia
e-mail:
2 N. Ganikhodjaev, F.A. Razak
only up and down spins [8] whereas Potts model incorporates more possibilities
of spins and their interactions. The Potts model describes an easily defined
class of statistical mechanics models. At the same time, its rich structure is
surprisingly capable of illustrating almost every conceivable nuance of the
subject. The Potts model encompasses a number of problems in statistical
physics (see, e.g. [15]).
Correlation inequalities play an important role in many areas of statistical
mechanics. In addition to describing microscopic structure they also provide
information about macroscopic properties: for ferromagnetic spin systems
they give monotonicity of the critical temperature, inequalities for critical
exponents, etc [1]. In the recent work of N.Macris [9] it was shown that a
correlation inequality of statistical mechanics can be applied to linear low-
density parity-check codes. In this paper we prove Griffith–Kelly–Sherman
(GKS) inequalities [8] for the generalized (likely a generalization of the
classical Ising model in [8]) Potts model.
At present there are a lot papers and books where the authors proved
correlation inequalities for different models (see for example [1–11]) and it
is possible that our results follow from some of them . To the best knowledge
of the authors, formulated here Griffith–Kelly–Sherman inequalities for Potts
model are new and proof of these inequalities one can consider as new alter-
nate combinatoric proof. Finally note that after putting our manuscript to arxiv
[17], we received from Prof.G.R.Grimmett letter [7], where he derived our
inequalities using the FKG inequality for the random-cluster representation of
the Potts model.
2 Potts Model with Long-Range Binary Interaction
Let N denote the index set {1, 2, · · · , n}, consider the space of all spin config-
urations (σ1, σ2, .., σn) where each σi is allowed the values from 1 to q (q ⩾ 2).
A general configuration is denoted by γ and (σi)γ is the number of values
(1, · · · , q) which appears as the ith spin (component) in γ . Let be the set
of all possible configurations.
For each pair (i, j) of distinct indices in N the extended real number
Jij = J ji ⩾ 0 (1)
is given. The requirement J ji ⩾ 0 is that the system be ferromagnetic. The
Hamiltonian of the one-dimensional Potts model is the real valued function
on configurations, whose value at the configuration γ is
∑
Hγ = − Jijδ(σ
i)γ (σ j)γ (2)
1⩽i< j⩽n
where δ is the Kronecker’s delta defined as
{
1 if (σi)γ = (σ j)γ ,
δ(σ
i)γ (σ j)γ =
0 otherwise.
Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model 3
The Potts model with long-range interaction was considered by Fortuin and
Kasteleyn [2], where the model is defined on a lattice whose sites coincide the
node points of the network and whose edges coincide the resistors.
For q = 2 the model equivalent to the Ising model [5].
The Gibbs probability P on the space of configurations is defined by
−1
P(γ ) = Z exp (−β Hγ ), (3)
where
∑
Z = exp (−β Hγ ) (4)
γ
and
−1
β = (kT) > 0, (5)
where k is the Boltzman’s constant and T is the absolute temperature. For
brevity, β will be assumed to be 1 for the rest of the paper which gives the
probability as
−1
P(γ ) = Z exp (−Hγ ). (6)
The expected value of a random variable X on this probability space (, P)
is called its thermal average and is denoted by angular brackets:
∑
⟨X⟩ = E(X) = X(γ )P(γ ). (7)
γ
3 Centered Random Variables
Let σi denote the random variable whose value at γ is (σi)γ , that is, it’s range
is the following set F = {1, 2, · · · , q}, then
∑
⟨σi⟩ = (σi)γ P(γ ). (8)
γ
′
We introduce centered random variable σ
i whose values are derived from σi
such as
′
σ
i = σi − ⟨σi⟩. (9)
c
Proposition 1 For any given q and arbitrary i ∈ N, range F of the centered
′
random variables σ is the following set:
i
c
F = {−l, −(l − 1), · · · , l − 1, l};
q−1
where l = , that is, l is an integer or half-integer.
2
Proof Let
( j)
A
i = {γ ∈ : (σi)γ = j}, (10)
4 N. Ganikhodjaev, F.A. Razak
where i ∈ N and j ∈ F. So that, any ⟨σi⟩ can be written as
( ) ( ) ( )
(1) (2) (q)
⟨σi⟩ = 1 · P A
i + 2 · P Ai + · · · + q · P Ai .
Definition 1 For arbitrary permutation π ∈ Sq let us define transformation
Tπ : → by the following way: for any γ = (σ1, σ2, · · · , σn) assume
(Tπ)γ = {π(σ1), π(σ2), · · · , π(σn)}.
Remark 1 For any transformation Tπ defined above, P((Tπ)γ ) = P(γ ) for
( ) ( )
(π( j)) ( j)
arbitrary γ ∈ , that is P A
i = P Ai for any permutation π ∈ Sq and
c
any j ∈ F (also for j ∈ F ).
The remark above follows from the fact that Kronecker’s delta only takes
into account the similarity of spins. Since Tπ is a one-to-one transformation, it
is also measure preserving.
It follows that for any i ∈ N
( ) ( ) ( )
(1) (2) (q)
P A = P A = · · · = P A
i i i
and since P is a probabilistic measure, then for any i ∈ N and j ∈ F,
( )
( j)
P A = 1/q.
i
Therefore we have
⟨σi⟩ = (1 + 2 + · · · + q)/q = (q + 1)/2, (11)
′
consequently enabling us to find σ for any q values of spins by rewriting
γ
(9) as
′
σ
i = σi − (q + 1)/2, (12)
c
which implies that F = F − (q + 1)/2 hence the statements of Proposition 1
follows. ⊔⊓
c
Taking into account that changing the value of the spins from F → F does
c
not affect the Hamiltonian as well as the Gibbs probability and also that F =
c
−F , then for any i ∈ N,
∑
′
⟨σ ⟩ = 1/q j = 0. (13)
i
c
j∈F
2
′ ′ q −1
Thus for all k ⟨σ ⟩ = 0 and var(σ ) = .
k k 2
Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model 5
Griffiths [5] proved in Ising ferromagnets, i.e., q = 2, the following sets of
inequalities:
′ ′
⟨σ σ ⟩ ⩾ 0 for all k, m ∈ N. (14)
k m
′ ′ ′ ′ ′ ′ ′ ′
⟨σ σ σ σ ⟩ − ⟨σ σ ⟩⟨σ σ ⟩ ⩾ 0 for all k, m, p, r ∈ N. (15)
k m p r k m p r
(Note that k, m, p, r need not be distinct.)
The Griffiths inequalities hold for Potts model (2) with q > 2 also. They will
appear as a consequence of the extension below.
4 Generalized Potts Model
Let us consider the following generalization of Potts model. In addition to
the binary interactions, assume that there can also be multisite interactions as
well as external fields. Interest in systems with multisite interactions has been
stimulated by the seminal work of [14]. Such systems display varied critical
behaviour, and proliferate under the action of the real space renormalization
group [13].
Let N denote as above the index set {1, 2, · · · , n} and the spin variable σi at
c q−1
a lattice point i takes the values F = {−l, −l + 1, · · · , l − 1, l},where l = .
2
For each A = {i1, · · · , ik} ⊂ N where k ⩾ 2, let the real number JA ⩾ 0 be
given, and define the Hamiltonian by
∑ ∑
Hγ = − JAδ(σ A)
γ − Jiδ(σi)γ ,−l (16)
A⊂N i
where
∏
A ∅
σ = σi (σ ≡ 1)
i∈A
and the generalized Kronecker’s delta δ(σ A)
γ is
{
1 if (σi
1 )γ = · · · = (σik)γ
δ(σ A)
γ =
0 otherwise.
The Hamiltonian (16) generalize the following Hamiltonian [15]
∑ ∑ ∑
H = −L δσ
i,0 − K δσiσ j − K3 δσiσ jσk − · · · , (17)
i ⟨i, j⟩ ⟨i, j,k⟩
where σi = 0, 1, · · · , q − 1 specifies the spin states at the ith site and Kn, n ⩾ 3,
is the strength of the n-site interactions, and L is an external field applied to
the spin state 0.
Consideration of Potts models with multisite interactions has proved to be
fruitful in many fields of physics, ranging from the determination of phase
diagrams in metallic alloys and exhibition of new types of phase transition,
to site percolation [16].
Next model has the same local state space as above.
6 N. Ganikhodjaev, F.A. Razak
2
The ∇ − model Let L be a simple cubic lattice. The Hamiltonian of the
Blume-Capel model is
∑ ∑ ∑
2 2
H = 1/2 (sa − sb ) − g s
a − h sa (18)
⟨a,b⟩ a a
where the spin variable sa at a lattice point a takes the values 0, ±1, · · · , ±l, l is
an integer or half-integer, and the first sum is over pairs of nearest-neighbour
points of the lattice.
For l = 1/2 the model is equivalent to the Ising model. For l > 1/2 this
model exhibits much more interesting low-temperature behaviour than the
Ising model [13]. However it is easy to see that for l > 1/2 the Blume-Capel
model is not equivalent to a generalized Potts model.
Abelian ferromagnetic models Abelian model is defined as classical lattice
gas model where a set of particle types forms a finite abelian group [10] and
the finite volume Hamiltonian has following form:
∑
ˆ ˆ ˆ
H = − J(B)B; J(B) ⩾ 0, (19)
Bˆ ∈χˆ
where the summation is over all characters of the product group of particle
configurations χ in a finite volume . Defined above generalized ferromag-
netic Potts model provides an example of abelian model with cyclic group Zq,
where multisite interaction is given by
m−1 q−1
∏∑
1
δ(X(a1)X(a2) · · · X(am)) = exp{(2πik/q)[X(a j) − X(a j+1)]} ×
m−1
q
j=1 k=0
q−1
∑
× exp{(2πik/q)[X(am) − X(a1)]},
k=0
where m ⩾ 2. By discussing the arbitrary finite abelian group case one can
enlarge the family of multisite ferromagnetic models [10].
5 First Griffith–Kelly–Sherman Inequality
Below we consider following Hamiltonian
∑
Hγ = − JAδ(σ A)
γ (20)
A⊂N
where the sum is taken over all A with |A| ⩾ 2.
Let R be a list of indices from N,where R may contain repeated indices.
Then for any R define
∏
R ∅
σ = σi (σ ≡ 1).
i∈R
Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model 7
′ ′
Let R ⊂ N be the set of all elements in R. The difference between R and R
′
is that R may contain repeated indices while R may not since it is a set. When
′
there are no repeated indices in R we have R = R.
Remark 2 In the case of Ising model and also if l = 1 we can consider only sub-
set of N and it is not necessary to consider list of indices since corresponding
random variables take only values 0, ±1.
Let xA = exp(JA) ⩾ 1. Define
∏
Zγ = exp(−Hγ ) = xA (21)
A⊂N
which enables us to express the Gibbs probability on the space of configura-
∑
tions as P(γ ) = Zγ /Z , where Z =
γ Zγ . Thus the expected value of any
R
(σ )γ is given by
〈 〉 ∑ ∑( )
R R −1 R
σ = (σ )γ P(γ ) = Z σ Zγ . (22)
γ
γ γ
R
Theorem 1 In probability space (, P) defined by (20)–(22), we have ⟨σ ⟩ ⩾ 0
for any list R of indices from N.
Proof Notice that since we allow repeated indices in R, then R is com-
prised of odd and even groups of repeated indices. For example when R =
[1, 2, 3, 3, 4, 4, 4], it has three odd groups of indices which are [1], [2] and
[4,4,4]. It also has one even group of indices [3,3]. Define odd groups in R
as θi when i ∈ R is repeated an odd number of times. Similarly, define even
groups in R as ϵi when i ∈ R is repeated an even number of times. So now,
we can say that R = [1, 2, 3, 3, 4, 4, 4] is comprised of θ1, θ2, ϵ3 and θ4 thus
R = [θ1, θ2, ϵ3, θ4].
+
Define R ⊂ to be a set of configurations where multiplication of all spins
+ −
in R gives a positive value and similarly R ⊂ is a set of configurations
− 0
where multiplication of all spins in R gives a negative value. Also let R ⊂
be the set where the multiplication of spins are zero so that we have =
+ − 0 0
R ∪ R ∪ R . Note that we only have R when q = 2l + 1 is odd and it does
R R + −
not appear in ⟨σ ⟩ (since σ = 0), so we only need to consider R and R .
R
When R is only comprised of even groups of repeated indices then (σ )γ =
∏
ϵi − R
i∈R(σ )γ ⩾ 0 thus R = ∅ and ⟨σ ⟩ ⩾ 0 since P(γ ) ⩾ 0.
Otherwise, consider cases where at least one odd group of repeated indices,
θi, exist in R (which is true for all instances when |R| is odd). In these cases,
+ ′ −
for each γ ∈ R there exists a corresponding γ ∈ R due to symmetrical
c
properties of centered value variables in F . Any element of either subsets can
be transformed into a corresponding element of the other subset by choosing
8 N. Ganikhodjaev, F.A. Razak
any i ∈ R and multiplying σi with −1 or simply multiplying any one of the θi,
+ −
i ∈ R with −1. Consequently, for these cases, if q is even then |R | = |R | =
n + − 0
| |/2 = q /2 and if q is odd, then |R | = |R | = | − R |/2, so that
〈 〉 ∑ ( ) ∑ ( )
R R R ′
σ = σ P(γ ) + σ ′ P(γ ).
γ γ
+ ′ −
γ ∈R γ ∈R
( ) ( )
R R
Since σ ′ = − σ , we can write
γ γ
〈 〉 ∑ ( ) ∑ ( )
R R ′ −1 R
σ = σ [P(γ ) − P(γ )] = Z σ [Zγ − Zγ ′ ].
γ γ
+ +
γ ∈R γ∈R
+ −
When |R| is odd, the one-to-one correspondence between R and R
( )
R
can also be obtained simply by multiplying σ with −1. By this way the
γ
difference of spins are preserved hence the Gibbs measure is preserved,
+ −
consequently Zγ = Zγ ′ for any γ . In other words for |R| odd, Tπ : R → R ,
( ) ( )
R R
where Tπ is stated in Definition 1. Hence, since σ = − σ ′ and Zγ =
γ γ
〈 〉 ∑ ( )
R −1 R
Zγ ′ for all odd |R|, σ = Z
γ∈R+ σ γ [Zγ − Zγ ′ ] = 0 and Theorem 1
stands.
Let A ⊂ be a set of configurations, and then define
∑ ( )
R
ζ(R, A) = σ Zγ .
γ
γ∈A
For brevity if A = let ζ(R) = ζ(R,) . Thus we can write
∑( ) 〈 〉
R R
ζ(R) = σ Zγ = Z · σ ,
γ
γ∈
+ R − R
and when R = {γ ∈ : σ > 0} and R = {γ ∈ : σ < 0}, we have
∑ ∑
+ − R R
ζ(R) = ζ(R, R ) + ζ(R, R ) = (σ )γ Zγ + (σ )γ ′ Zγ ′
+ ′ −
γ ∈R γ ∈R
∑
R
= (σ )γ [Zγ − Zγ ′ ].
+
γ∈R
(1) (0)
Let B ⊂ N, where B ={γ ∈ : δ(σ B)
γ =1} and B ={γ ∈ : δ(σ B)γ = 0},
(1) (0)
then since = B ∪ B ,
∑ ( ) ∑ ( )
R R (1) (0)
ζ(R) = σ Zγ + σ Zγ = ζ(R, B ) + ζ(R, B ).
γ γ
(1) (0)
γ ∈B γ∈B
Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model 9
+ (1) R − (1)
Similarly, let R B = {γ ∈ : σ > 0 and δ(σ B)
γ = 1}, R B = {γ ∈ :
R + (0) R
σ < 0 and δ(σ B)
γ = 1}, R B = {γ ∈ : σ > 0 and δ(σ B)γ = 0} and
− (0) R
R B = {γ ∈ : σ < 0 and δ(σ B)
γ = 0} for any B ⊂ N, we can write
∑ ∑
R R
ζ(R,) = (σ )γ Zγ + (σ )γ ′ Zγ ′ +
+ (1) ′ − (1)
γ ∈R B γ ∈R B
∑ ∑
R R
+ (σ )γ Zγ + (σ )γ ′ Zγ ′
+ (0) ′ − (0)
γ ∈R B γ ∈R B
+ (1) − (1) + (0) − (0)
= ζ(R, R B ) + ζ(R, R B ) + ζ(R, R B ) + ζ(R, R B ).
∏
R ϵi
For cases where |R| is even and (σ )γ ̸=
i∈R(σ )γ (there exists θi, i ∈ R),
R
we seek to prove by induction on s, the number of JA > 0. To prove ⟨σ ⟩ ⩾ 0
R
we only need to prove that ζ(R) ⩾ 0 since ζ(R) = Z · ⟨σ ⟩ and Z > 0. For
s = 0 we have Zγ = Zγ ′ = 1, thus
∑ ( ) ∑ ( )
R R
ζ(R) = σ [Zγ − Zγ ′ ] = σ [1 − 1] = 0
γ γ
+ +
γ ∈R γ∈R
and Theorem 1 is satisfied. Note that P(γ ) = 1/Z for all γ hence we have uni-
∑
R R
form measure which renders ⟨σ ⟩ = 0 since
γ∈R(σ )γ = 0 due to centered
value properties.
Let ζs(R) be ζ(R) for any s number of nonzero existing interactions. Assume
ζs(R) ⩾ 0 for all s ⩽ k such that for any s we add JB
s > 0. Then for s = k + 1
let JB
k+1 > 0 be the additional interaction. Since we know that xBk+1 will only
(1) (1)
multiply all the terms in B
k+1 where Bk+1 = {γ ∈ : δ(σ Bk+1 )γ = 1}, the terms
(0) (0)
in B
k+1 (Bk+1 = {γ ∈ : δ(σ Bk+1 )γ = 0}) remains the same as it was in s = k.
Thus we have
( ) ( ) ( ) ( )
(1) (0) (1) (0)
ζk+1(R) = ζk+1 R, B
k+1 + ζk+1 R, Bk+1 = xBk+1 · ζk R, Bk+1 + ζk R, Bk+1 .
(1)
By induction hypothesis we have ζk(R) ⩾ 0, and if we have ζk(R, B
k+1) ⩾ 0
we shall be able to write
(1) (0)
ζk+1(R) = xB
k+1 · ζk(R, Bk+1) + ζk(R, Bk+1)
(1) (0)
> ζk(R, B
k+1) + ζk(R, Bk+1) = ζk(R) ⩾ 0.
since xB
k+1 > 1 and it is multiplied with a positive sum. Thus given
(1) R
ζk(R, B
k+1) ⩾ 0, we have ⟨σ ⟩ ⩾ 0 for any n number of vertices, q number
of spins and R in which any i ∈ R is also in N. ⊔⊓
N
Lemma 1 Let ζ
s (R) be ζ(R) where s is the number of nonzero JA, N =
{1, · · · , n} is the set of n vertices and R is a list where i ∈ R implies i ∈ N.
N
Given that ζ (R) ⩾ 0 for any n vertices and q number of spins then we have
s
N (1) (1)
ζ
s (R, B ) ⩾ 0 where B ⊂ N and B = {γ ∈ : δ(σ B)γ = 1}.
10 N. Ganikhodjaev, F.A. Razak
(1)
Proof Let B = {b1, b2 · · · bm} ⊂ N. For γ ∈ B , the spins are always similar
for all its vertices, such that σb
1 = σb2 = · · · = σbm, thus we seek to treat B as a
single vertex, say b 1. Firstly get B ∩ A for all existing JA’s, if B ∩ A = ∅, then
xA is left as it is, but if B ∩ A ̸= ∅ then we shall do some alterations.
For A’s where B ∩ A ̸= ∅ let CA = A − (B ∩ A). If there exist A’s with
similar CA, then we seek to group it together. Let C = CA ∪ b 1 where b 1 is
∏
∗
the first element of B then we set x
C = xA for all A’s with similar CA, to
represent them in a group as a single interaction. We can do this because they
(1)
will always appear together in ζs(R, B ). Thus if there exist similar CA’s for
different A’s, the number of existing interaction is reduced but the remaining
interaction has a larger size which does not matter since JA can even be ∞. If
∗
CA = ∅ then the x
b1 group if comprised of all of A ⊂ B. This group will appear
(1)
in every in term in ζ(R, B ) due to the fact that all spins in B are similar for
(1)
γ ∈ B .
∗
After replacing all the terms where B ∩ A ̸= ∅ with its corresponding x ,
C
then we will see that we have
∗
N (1) ∗ N ∗
ζ (R, B ) = x · ζ (R ) (23)
k b1 s
∗ ∗
where s ⩽ k, N = (N − B) ∪ b1 and R is obtain by simply replacing any
∗
i ∈ R which is also in B with b 1 (if none of i ∈ R is in B then R = R).
∗
R R
(σ )γ = (σ )γ since the spins in B are all similar, we are simply renaming the
vertices. A simple example is that initially we have B = {2, 3} thus b 1 = 2 and
(1)
R = [1, 2, 3, 4] in N = {1, 2, 3, 4} then we can see that the γ ∈ B is exactly
∗ ∗
similar to γ ∈ for cases where R = [1, 2, 2, 4] in N = {1, 2, 4} which can be
modified by renaming index 4 as index 3 and then it can be obtained just like
in the case where R = [1, 2, 2, 3] in N = {1, 2, 3}.
∗
N ∗ N
We can find ζ (R ) exactly the same way we obtain ζ (R) where N =
s s
∗ ∗ ∗
{1, · · · , n }, n = n − |B| + 1 and R is transformed accordingly to a new
∗
R where |R | = |R|, the difference is only that the vertices have different
position, but because any interactions are accounted for, this does not really
matter since the existence and size of interactions does not depends on the
∗
∗ N ∗
vertices being neighbours or not. Now that we know x ⩾ 1, ζ (R ) ⩾ 0
b1 s
N ∗
(since ζ (R) ⩾ 0 for any n including n and any R where i ∈ R implies i ∈ N),
s
N (1)
then we have ζ (R, B ) ⩾ 0. ⊔⊓
k
Example 1 In this example we seek to illustrate Lemma 1. Let N = {1, 2, 3},
q = 3, R = [1, 3] and B = {1, 2}. The only possible interactions are J12, J13,
J23, J123. Assume only x12 = 1 hence s = 3. We also have
(1) (1)
B = {(−1, −1, −1), (−1, −1, 1), (1, 1, −1), (1, 1, 1)} and ζ(R, B )
= 2(x13x23x123 − 1).
∗
Since C13 = {3}, C23 = {3} and C123 = {3}, assign x
13 = x13x23x123. Replace it in
(1)
ζ(R, B ), we have
(1) ∗
ζ(R, B ) = 2(x − 1).
13