On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree
In the paper, we study the q-state (where q = 3, 4, 5, ... ) Potts model with special external field on a Cayley tree of order k ≥ 2. For antiferromagnetic Potts model with such an external field on the Cayley tree of order k ≥ 6, the non-uniqueness of weakly periodic (non-periodic) Gibbs measures i...
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nasplib_isofts_kiev_ua-123456789-1405572025-02-09T15:31:27Z On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree Rahmatullaev, M.M. In the paper, we study the q-state (where q = 3, 4, 5, ... ) Potts model with special external field on a Cayley tree of order k ≥ 2. For antiferromagnetic Potts model with such an external field on the Cayley tree of order k ≥ 6, the non-uniqueness of weakly periodic (non-periodic) Gibbs measures is proved. The weakly periodic Gibbs measures for the Potts model with zero external field are also studied. It is proved that under some conditions imposed on the parameters of the model there can be not less than 2q - 2 such measures. 2016 Article On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree / M.M. Rahmatullaev // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 4. — С. 302-314. — Бібліогр.: 16 назв. — англ. 1812-9471 DOI : doi.org/10.15407/mag12.04.302 Mathematics Subject Classification 2000: 82B26 (primary); 60K35 (secondary) https://nasplib.isofts.kiev.ua/handle/123456789/140557 en Журнал математической физики, анализа, геометрии application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In the paper, we study the q-state (where q = 3, 4, 5, ... ) Potts model with special external field on a Cayley tree of order k ≥ 2. For antiferromagnetic Potts model with such an external field on the Cayley tree of order k ≥ 6, the non-uniqueness of weakly periodic (non-periodic) Gibbs measures is proved. The weakly periodic Gibbs measures for the Potts model with zero external field are also studied. It is proved that under some conditions imposed on the parameters of the model there can be not less than 2q - 2 such measures. |
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Rahmatullaev, M.M. On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree Журнал математической физики, анализа, геометрии |
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Rahmatullaev, M.M. |
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Rahmatullaev, M.M. |
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On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree |
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On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree |
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On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree |
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On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree |
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On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree |
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on weakly periodic gibbs measures of the potts model with a special external field on a cayley tree |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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On Weakly Periodic Gibbs Measures of the Potts Model with a Special External Field on a Cayley Tree / M.M. Rahmatullaev // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 4. — С. 302-314. — Бібліогр.: 16 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT rahmatullaevmm onweaklyperiodicgibbsmeasuresofthepottsmodelwithaspecialexternalfieldonacayleytree |
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2025-11-27T10:31:11Z |
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Journal of Mathematical Physics, Analysis, Geometry
2016, vol. 12, No. 4, pp. 302–314
On Weakly Periodic Gibbs Measures of the Potts
Model with a Special External Field on a Cayley Tree
M.M. Rahmatullaev
Institut of Mathematics
29 Durmon Yuli Str., 100125, Uzbekistan
E-mail: mrahmatullaev@rambler.ru
Received March 5, 2015, revised January 27, 2016
In the paper, we study the q-state (where q = 3, 4, 5, . . . ) Potts model
with special external field on a Cayley tree of order k ≥ 2. For antiferro-
magnetic Potts model with such an external field on the Cayley tree of order
k ≥ 6, the non-uniqueness of weakly periodic (non-periodic) Gibbs measures
is proved. The weakly periodic Gibbs measures for the Potts model with
zero external field are also studied. It is proved that under some conditions
imposed on the parameters of the model there can be not less than 2q − 2
such measures.
Key words: Cayley tree, Gibbs measure, Potts model, weakly periodic
measure.
Mathematics Subject Classification 2010: 82B26 (primary); 60K35 (sec-
ondary).
1. Introduction
The Potts model on the Cayley tree and its Gibbs measures were studied in [1]-
[4]. In [5], the ferromagnetic three-state Potts model on the Cayley tree of order
two was studied, and it was shown that there exists a critical temperature Tc > 0
such that for T < Tc there are at least three translation-invariant Gibbs measures
and an uncountable number of non-translation-invariant Gibbs measures. In [6],
the results of [5] were generalized to the Potts model with a finite number of
states on the Cayley tree of an arbitrary (finite) order.
In [7], for an antiferromagnetic Potts model on the Cayley tree it was shown
that the translation-invariant Gibbs measure is unique.
c© M.M. Rahmatullaev, 2016
On Weakly Periodic Gibbs Measures of the Potts Model
The Potts model with a countable number of states in a non vanishing external
field on the Cayley tree was considered in [8]. It was proved that this model has
a unique translation-invariant Gibbs measure.
Recently, in [9], there were described all translation-invariant Gibbs measures
for the Potts model with zero-external field on the Cayley tree of order k ≥ 2 .
It was shown that at sufficiently low temperatures their number is 2q − 1.
In [10, 11], a notion of weakly periodic Gibbs measure was introduced, and
some Gibbs measures were found for the Ising model.
Weakly periodic ground states and weakly periodic Gibbs measures (coin-
ciding with the translation-invariant ones) for the Potts model were studied in
[12].
In [13], the existence of weakly periodic (non- translation-invariant) Gibbs
measures of the Potts model was proved.
This paper is devoted to the weakly periodic (non-periodic) Gibbs measures
for the Potts model with special external field on the Cayley tree. The paper
has the following structure. In Sec. 2, we introduce the main definitions and the
background. In Sec. 3, we give the results obtained for weakly periodic Gibbs
measures. The proofs of all results are given in Sec. 4.
2. Definitions and the Background
Let τk = (V, L) be a uniform Cayley tree, where each vertex has k + 1 neigh-
bors with V being the set of vertices and L, the set of edges.
It is known that τk can be represented as Gk, which is the free product of
k + 1 cyclic groups of the second order (see [7, 14, 15]).
Fix an arbitrary element x0 ∈ V and correspond it to the unit element e
of the group Gk. Without loss of generality, we assume that the Cayley tree is
a planar graph. Using a1, . . . , ak+1, we numerate the nearest neighbors of the
element e, moving in the positive direction (see Fig.1). Now we numerate the
nearest neighbors of each ai, i = 1, . . . , k + 1 by aiaj , j = 1, . . . , k + 1. Since all
ai have the common neighbor e, we give aiai = a2
i = e to it. Other neighbors
are numerated starting from aiai in the positive direction. We numerate the set
of all nearest neighbors of each aiaj by aiajaq, q = 1, . . . , k + 1 starting from
aiajaj = ai in the positive direction. Iterating this argument, one gets a one-to-
one correspondence between the set of vertices V of the Cayley tree τk and the
group Gk.
The group representation given above is called the right representation, since
in this case if x and y are the nearest neighbors on the tree and g, h are the
corresponding elements of the group Gk, then g = hai or h = gaj for some i or j.
Similarly, one can define the left representation.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 303
M.M. Rahmatullaev
a a
3 1 a a
3 2
a
1 3
a
a
1 3
a a
2
a
1 1
a a
2
a
1 1
a a
3
a
1 2
a a
3
a
2 1
a
a a
2 3
a a a
2 3 1
a a
2 2
a
3
a a a
2 1 2
a a a
2 1 3
e
a
1
a
2
a
1
a
2
a
3
Fig. 1. Some elements of the group G2 on the Cayley tree of order two.
In the group Gk, let us consider the left (right) shift transformations defined
as follows. For g ∈ Gk, let us set
Tg(h) = gh, (Tg(h) = hg), ∀h ∈ Gk.
The set of all left (right) shifts in Gk is isomorphic to the group Gk. Any trans-
formation S of the group Gk induces the transformation Ŝ of the set of vertices
V of the Cayley tree.
Therefore, we identify V and Gk.
Note that the set {Tg : g ∈ Gk} is a group with respect to the operation
TgTh = Tgh, and it is called a group of transformations.
Theorem 1. The group of the left (right) shifts on the right (left) represen-
tations of the Cayley tree is the group of transformations of the Cayley tree (see
[7, 15]).
For an arbitrary point x0 ∈ V , we set Wn = {x ∈ V |d(x0, x) = n}, Vn =
n⋃
m=0
Wm, Ln = {< x, y >∈ L|x, y ∈ Vn}, where d(x, y) is the distance between
the vertices x and y in the Cayley tree, i.e., the number of edges in the shortest
path joining the vertices x and y.
We consider the model where the spin variables take values in the set Φ =
{1, 2, . . . , q}, q ≥ 3, and are placed at the vertices of the tree. Then a configuration
σ on V is defined as a function x ∈ V → σ(x) ∈ Φ. The set of all configurations
coincides with Ω = ΦV .
304 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
On Weakly Periodic Gibbs Measures of the Potts Model
The Potts model with external fields is defined by the Hamiltonian
H(σ) = −J
∑
〈x,y〉∈L
δσ(x)σ(y) −
q∑
i=1
∑
x∈V
α̃i,xδiσ(x), (1)
where J, α̃x = (α̃1,x, ..., α̃q,x) ∈ Rq, and δij is the Kronecker symbol.
If J > 0 (resp. J < 0), then the model is called a ferromagnetic (resp.
antiferromagnetic) Potts model.
We define a finite-dimensional distribution by a probability measure µ in the
volume Vn as
µn(σn) = Z−1
n exp
{
−βHn(σn) +
∑
x∈Wn
h̃σ(x),x
}
, (2)
where β = 1/T , T > 0 is a temperature, Z−1
n is a normalization factor, {h̃x =
(h̃1,x, . . . , h̃q,x) ∈ Rq, x ∈ V } is a collection of vectors, and
Hn(σn) = −J
∑
〈x,y〉∈Ln
δσ(x)σ(y) −
q∑
i=1
∑
x∈Vn
α̃i,xδiσ(x).
The distribution (2) is said to be compatible if the equality
∑
ωn∈ΦWn
µn(σn−1 ∨ ωn) = µn−1(σn−1) (3)
holds for all n ≥ 1 and σn−1 ∈ ΦVn−1 . Here, σn−1 ∨ ωn is the union of configu-
rations. In this case, there exists a unique measure µ on ΦV such that for all n
and σn ∈ ΦVn ,
µ({σ|Vn = σn}) = µn(σn).
The measure is called the split Gibbs measure corresponding to Hamiltonian
(1) and the vector-valued function h̃x, x ∈ V .
The following statement describes a condition for h̃x and α̃x ensuring the
consistency of µn(σn).
Theorem 2. Probability distributions µn(σn), n = 1, 2, . . ., in (2) are com-
patible iff for any x ∈ V the following equation holds:
hx = αx +
∑
y∈S(x)
F (hy, θ), (4)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 305
M.M. Rahmatullaev
where αx = (α1,x, α2,x, . . . , αq−1,x) ∈ Rq−1, αi,x = βα̃i,x − βα̃q,x; hx = (h1,x, . . . ,
hq−1,x), hi,x = h̃i,x− h̃q,x + (βα̃i,x−βα̃q,x), and F : h = (h1, . . . , hq−1) ∈ Rq−1 →
F (h, θ) = (F1, . . . , Fq−1) ∈ Rq−1 is defined as
Fi = ln
(
(θ − 1)ehi +
∑q−1
j=1 ehj + 1
θ +
∑q−1
j=1 ehj
)
,
and θ = exp(Jβ), S(x) is a set of direct successors of x.
P r o o f. The proof is similar to that of Theorem 5.1 in [4].
Let Gk/G∗
k = {H1, . . . , Hr} be the quotient group for a normal divisor G∗
k of
index r ≥ 1.
In this paper, we consider the case where αx does not depend on x, i.e.,
αx ≡ α = (α1, . . . , αq−1).
Definition 1. A collection of vectors h = {hx, x ∈ Gk} is said to be G∗
k
-periodic if hyx = hx for all x ∈ Gk, y ∈ G∗
k. A Gk− periodic collection is said to
be translation-invariant.
For x ∈ Gk, by x↓ we denote the unique point of the set {y ∈ Gk : 〈x, y〉} \
S(x).
Definition 2. A collection of vectors h = {hx, x ∈ Gk} is said to be G∗
k
-weakly periodic if hx = hij for x ∈ Hi, x↓ ∈ Hj , ∀x ∈ Gk.
Definition 3. A measure µ is said to be G∗
k -periodic or weakly periodic
if it corresponds to a G∗
k -periodic or a weakly periodic collection of vectors h,
respectively.
3. Weakly Periodic Measures
The level of difficulty in describing weakly periodic Gibbs measures depends
on the structure and index of the normal subgroup relative to which the period-
icity condition is imposed. From [4], we know that in the group Gk, there is no
normal subgroup of odd index different from one. Therefore, we consider normal
subgroups of even indices. Here, we restrict ourself to the case of indices two.
Let q ≥ 3 be arbitrary, i.e., σ : V → Φ = {1, 2, 3, . . . , q}. We describe the
G∗
k-weakly periodic Gibbs measures for a normal subgroup G∗
k of index two. We
note (see [4]) that any normal subgroup of index two of the group Gk has the
form
HA =
{
x ∈ Gk :
∑
i∈A
ωx(ai)−even
}
,
306 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
On Weakly Periodic Gibbs Measures of the Potts Model
where ∅ 6= A ⊆ Nk = {1, 2, . . . , k + 1}, and ωx(ai) is the number of letters ai in
the word x ∈ Gk. Let A ⊆ Nk and HA be the corresponding normal subgroup of
index two. We note that in the case |A| = k + 1, i.e., in the case A = Nk, weak
periodicity coincides with ordinary periodicity.
Consider Gk/HA = {HA, Gk \HA} the quotient group.
For simplicity, we set H0 = HA, H1 = Gk \HA. The HA are weakly periodic
collections of vectors h = {hx ∈ Rq−1 : x ∈ Gk} have the form
hx =
h1, x↓ ∈ H0, x ∈ H0
h2, x↓ ∈ H0, x ∈ H1
h3, x↓ ∈ H1, x ∈ H0
h4, x↓ ∈ H1, x ∈ H1.
Here hi = (hi1, hi2, . . . , hiq−1), i = 1, 2, 3, 4. By (4), we then have
h1 = (k − |A|)F (h1, θ) + |A|F (h2, θ)
h2 = (|A| − 1)F (h3, θ) + (k + 1− |A|)F (h4, θ)
h3 = (|A| − 1)F (h2, θ) + (k + 1− |A|)F (h1, θ)
h4 = (k − |A|)F (h4, θ) + |A|F (h3, θ).
(5)
We introduce the notation zij = exphij , λj = exp(αj), i = 1, 2, 3, 4, j =
1, 2, . . . , q − 1. The last system of equations can then be rewritten as
z1j = λj
(
(θ−1)z1j+
∑q−1
i=1 z1i+1∑q−1
i=1 z1i+θ
)k−|A|(
(θ−1)z2j+
∑q−1
i=1 z2i+1∑q−1
i=1 z2i+θ
)|A|
z2j = λj
(
(θ−1)z3j+
∑q−1
i=1 z3i+1∑q−1
i=1 z3i+θ
)|A|−1 (
(θ−1)z4j+
∑q−1
i=1 z4i+1∑q−1
i=1 z4i+θ
)k+1−|A|
z3j = λj
(
(θ−1)z2j+
∑q−1
i=1 z2i+1∑q−1
i=1 z2i+θ
)|A|−1 (
(θ−1)z1j+
∑q−1
i=1 z1i+1∑q−1
i=1 z1i+θ
)k+1−|A|
z4j = λj
(
(θ−1)z4j+
∑q−1
i=1 z4i+1∑q−1
i=1 z4i+θ
)k−|A|(
(θ−1)z3j+
∑q−1
i=1 z3i+1∑q−1
i=1 z3i+θ
)|A|
,
(6)
here j = 1, 2, 3, . . . , q − 1.
We consider the map K : R4(q−1) → R4(q−1) defined as
z′1j = λj
(
(θ−1)z1j+
∑q−1
i=1 z1i+1∑q−1
i=1 z1i+θ
)k−|A|(
(θ−1)z2j+
∑q−1
i=1 z2i+1∑q−1
i=1 z2i+θ
)|A|
z′2j = λj
(
(θ−1)z3j+
∑q−1
i=1 z3i+1∑q−1
i=1 z3i+θ
)|A|−1 (
(θ−1)z4j+
∑q−1
i=1 z4i+1∑q−1
i=1 z4i+θ
)k+1−|A|
z′3j = λj
(
(θ−1)z2j+
∑q−1
i=1 z2i+1∑q−1
i=1 z2i+θ
)|A|−1 (
(θ−1)z1j+
∑q−1
i=1 z1i+1∑q−1
i=1 z1i+θ
)k+1−|A|
z′4j = λj
(
(θ−1)z4j+
∑q−1
i=1 z4i+1∑q−1
i=1 z4i+θ
)k−|A|(
(θ−1)z3j+
∑q−1
i=1 z3i+1∑q−1
i=1 z3i+θ
)|A|
,
(7)
here j = 1, 2, 3, . . . , q − 1.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 307
M.M. Rahmatullaev
We introduce the notation
Im = {(z1, z2, . . . , zq−1) ∈ Rq−1 : z1 = z2 = ... = zm, zm+1 = ... = zq−1 = 1}, (8)
Mm = {(z(1), z(2), z(3), z(4)) ∈ R4(q−1) : z(i) ∈ Im, i = 1, 2, 3, 4}, (9)
here m = 1, 2, . . . , q − 1.
Lemma 1. 1) For any fixed m ≥ 1, λ > 0 and
λi =
{
λ if 1 ≤ i ≤ m
1 if m < i ≤ q − 1,
(10)
the set Mm is an invariant set with respect to the map K, i.e., K(Mm) ⊂ Mm.
2) For α = 0, the sets Mm are invariant with respect to the map K for all
m = 1, 2, . . . , q − 1.
Case α 6= 0. Let us consider the case α 6= 0 and λi given by (10). For
z ∈ Mm, we denote zi = zij , i = 1, 2, 3, 4; j = 1, 2, . . . ,m.. Then on the invariant
set Mm the system of equations (6) has the form
z1 = λ
(
(θ+m−1)z1+q−m
mz1+θ+q−m−1
)k−|A| (
(θ+m−1)z2+q−m
mz2+θ+q−m−1
)|A|
z2 = λ
(
(θ+m−1)z3+q−m
mz3+θ+q−m−1
)|A|−1 (
(θ+m−1)z4+q−m
mz4+θ+q−m−1
)k+1−|A|
z3 = λ
(
(θ+m−1)z2+q−m
mz2+θ+q−m−1
)|A|−1 (
(θ+m−1)z1+q−m
mz1+θ+q−m−1
)k+1−|A|
z4 = λ
(
(θ+m−1)z4+q−m
mz4+θ+q−m−1
)k−|A| (
(θ+m−1)z3+q−m
mz3+θ+q−m−1
)|A|
.
(11)
We introduce the notation
fm(z) =
(θ + m− 1)z + q −m
mz + θ + q −m− 1
.
It is easy to prove the following
Lemma 2. The function fm(z) is strictly decreasing for 0 < θ < 1, 1 ≤ m ≤
q − 1 and it is strictly increasing for 1 < θ.
Proposition 1. Let z = (z1, z2, z3, z4) be a solution of the system of equations
(11). If zi = zj for some i 6= j, then z1 = z2 = z3 = z4.
We consider an antiferromagnetic Potts model (i.e., 0 < θ < 1).
Let |A| = k. Then the system of equations (11) has the form
z1 = λ (fm(z2))
k
z2 = λ (fm(z3))
k−1 (fm(z4))
z3 = λ (fm(z2))
k−1 (fm(z1))
z4 = λ (fm(z3))
k .
(12)
308 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
On Weakly Periodic Gibbs Measures of the Potts Model
The solving of the system (12) can be reduced to the analyzing of the following
system of equations:
{
z2 = λ (fm(z3))
k−1 · fm(λ(fm(z3))k)
z3 = λ (fm(z2))
k−1 · fm(λ(fm(z2))k).
(13)
Introduce the notation
ψ(z) = λ (fm(z))k−1 · fm(λ(fm(z))k). (14)
Then we reduce the system of equations (13) to the form
{
z2 = ψ(z3)
z3 = ψ(z2).
(15)
The number of solutions of the system (15) coincides with the number of solutions
of the equation ψ(ψ(z)) = z.
Lemma 3. Let γ : [0, 1] → [0, 1] be a continuous function with a fixed point
ξ ∈ (0, 1). Assuming that the function γ is differentiable at ξ ∈ (0, 1) and that
γ′(ξ) < −1, we have the values x0, x1 such that the inequalities 0 ≤ x0 < ξ <
x1 ≤ 1 hold and γ(x0) = x1, γ(x1) = x0 (see [16], p. 70).
It is known (see [4], p. 109) that for antiferromagnetic case we have a unique
translation-invariant Gibbs measure corresponding to the unique solution of the
equation z = λfk
m(z). We let z∗ denote this solution.
Proposition 2. For k ≥ 6 and λ ∈ (λc1 , λc2), the system of equations (15)
has three solutions (z∗, z∗), (z∗2 , z
∗
3), (z
∗
3 , z
∗
2), where λci = bk
i , i = 1, 2, and
b1 = (k−1−√k2−6k+1)(1−θ)(θ+q−1)z
k−1
k∗
2(mz∗+θ+q−m−1)2
,
b2 = (k−1+
√
k2−6k+1)(1−θ)(θ+q−1)z
k−1
k∗
2(mz∗+θ+q−m−1)2
.
(16)
.
We have thus the following theorem.
Theorem 3. Let |A| = k, k ≥ 6, and λ ∈ (λc1 , λc2). Then for the antiferro-
magnetic Potts model with special external field (given by (10)) there are at least
two HA− weakly periodic (non-periodic) Gibbs measures, where λci = bk
i , i = 1, 2.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 309
M.M. Rahmatullaev
Case α = 0. In this case, the system of equations (6) on the invariant set
Mm,m = 1, 2, . . . , q − 1 can be reduced to the following system of equations:
z1 =
(
(θ+m−1)z1+q−m
mz1+θ+q−m−1
)k−|A| (
(θ+m−1)z2+q−m
mz2+θ+q−m−1
)|A|
z2 =
(
(θ+m−1)z3+q−m
mz3+θ+q−m−1
)|A|−1 (
(θ+m−1)z4+q−m
mz4+θ+q−m−1
)k+1−|A|
z3 =
(
(θ+m−1)z2+q−m
mz2+θ+q−m−1
)|A|−1 (
(θ+m−1)z1+q−m
mz1+θ+q−m−1
)k+1−|A|
z4 =
(
(θ+m−1)z4+q−m
mz4+θ+q−m−1
)k−|A| (
(θ+m−1)z3+q−m
mz3+θ+q−m−1
)|A|
.
(17)
The following proposition is similar to Proposition 1.
Proposition 3. Let m ∈ {1, 2, . . . , q − 1} be fixed and z = (z1, z2, z3, z4) be
a solution of the system of equations (17). If zi = zj for some i 6= j, then
z1 = z2 = z3 = z4.
Theorem 4. Let |A| = k and k ≥ 6. If one of the following conditions is
satisfied:
1) 4k
k+1+
√
k2−6k+1
≤ q < 4k
k+1−√k2−6k+1
0 < θ < θ2,
2) q ≤ 4k
k+1+
√
k2−6k+1
θ1 < θ < θ2,
then there are at least 2q − 2 weakly periodic (non-periodic) Gibbs measures,
where
θ1 =
4 k − kq − q − q
√
k2 − 6 k + 1
4k
, θ2 =
4 k − kq − q + q
√
k2 − 6 k + 1
4k
.
(18)
R e m a r k 1. The new Gibbs measures described in Theorem 3 and Theorem
4 allow us to describe a continuous set of non-periodic Gibbs measures different
from the previously known ones.
R e m a r k 2. If instead of (9) we consider Mq−1, then Theorem 4 coincides
with Theorem 3 from [13].
4. Proofs
P r o o f of Lemma 1. 1) For the fixed m, λi given by (10) and z =
(z(1), z(2), z(3), z(4)) ∈ Mm, it is clear that z(i) ∈ Im, i = 1, 2, 3, 4. From (8), we
have that z(i) = (zi, ..., zi, 1, ..., 1), where zi 6= 1, i = 1, 2, 3, 4. Using (7), we obtain
310 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
On Weakly Periodic Gibbs Measures of the Potts Model
z′1j =
(
θ+q−2+z1
θ+q−2+z1
)k−|A| (
θ+q−2+z2
θ+q−2+z2
)|A|
= 1, j = m,m + 1, ...q − 1
z′2j =
(
θ+q−2+z3
θ+q−2+z3
)|A|−1 (
θ+q−2+z4
θ+q−2+z4
)k+1−|A|
= 1, j = m,m + 1, ...q − 1
z′3j =
(
θ+q−2+z2
θ+q−2+z2
)|A|−1 (
θ+q−2+z4
θ+q−2+z4
)k+1−|A|
= 1, j = m,m + 1, ...q − 1
z′4j =
(
θ+q−2+z4
θ+q−2+z4
)k−|A| (
θ+q−2+z3
θ+q−2+z3
)|A|
= 1, j = m,m + 1, ...q − 1
.
Consequently, K(z) ∈ Mm.
The second part of the lemma is proved in a similar way.
P r o o f of Proposition 1
From the system of equations (11), we obtain
z1
z2
=
(
fm(z1)
fm(z4)
)k−|A|(fm(z2)
fm(z3)
)|A|−1 (
fm(z2)
fm(z4)
)
, (a1)
z1
z3
=
(
fm(z2)
fm(z1)
)
, (a2)
z1
z4
=
(
fm(z1)
fm(z4)
)k−|A|(fm(z2)
fm(z3)
)|A|
, (a3)
z2
z3
=
(
fm(z3)
fm(z2)
)|A|−1 (
fm(z4)
fm(z1)
)k−|A|+1
, (a4)
z2
z4
=
(
fm(z4)
fm(z3)
)
, (a5)
z3
z4
=
(
fm(z1)
fm(z4)
)k−|A|(fm(z2)
fm(z3)
)|A|−1 (
fm(z1)
fm(z3)
)
. (a6)
Let z = {z1, z2, z3, z4} be a solution of the system of equations (11) and
z1 = z2. Then from the strict monotonicity of fm(z) and from the equality
(a2), we obtain that z1 = z2 = z3. In this case, from (a4) we obtain z1 = z4,
consequently, z1 = z2 = z3 = z4.
Let z1 = z3. Then from the strict monotonicity of fm(z) and from the equality
(a2), we obtain that z1 = z2 = z3. In this case, from (a4) we obtain z1 = z4,
consequently, z1 = z2 = z3 = z4.
Let z1 = z4. Then from the strict monotonicity of fm(z) and from the equality
(a3), we obtain that z2 = z3. Then from (a5), we obtain the inequality
z2fm(z2) = z4fm(z4). (19)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 311
M.M. Rahmatullaev
We consider the function
φ(z) = zfm(z) = z
(θ + m− 1)z + q −m− 1
mz + θ + q −m− 1
and calculate its derivative
φ′(z) =
(θ − 1 + m)mz2 + 2(θ − 1 + m)(θ + q −m− 1)z + (q −m)(θ + q −m− 1)
(mz + θ + q −m− 1)2
.
From θ > 0, z > 0, 1 ≤ m ≤ q − 1 and q ≥ 2, we obtain that φ(z) is a strictly
increasing function. Consequently, (19) is true only when z2 = z4.
The other cases are proved in a similar way.
P r o o f of Proposition 2. For (14), it is easy to see that
1) ψ(z∗) = z∗,
2) the function ψ(z) is defined on R+,
3) ψ(z) is bounded and differentiable at z∗.
Then, by Lemma 1, for ψ′(z∗) < −1, equation (15) has three solutions of the
form (z∗, z∗), (z∗2 , z
∗
3), (z
∗
3 , z
∗
2). The inequality ψ′(z∗) < −1 is equivalent to the
inequality
b2 + (k − 1)
(θ − 1)(θ + q − 1)z
k−1
k∗
(mz∗ + θ + q −m− 1)2
b + k
(1− θ)2(θ + q − 1)2z
2 k−1
k∗
(mz∗ + θ + q −m− 1)4
< 0, (20)
where b = k
√
λ. Consequently,
(b− b1)(b− b2) < 0,
where b1, b2 are defined in (16). Proposition 2 is proved.
P r o o f of Theorem 4.
Consider the case 0 < θ < 1 and |A| = k. Then the system of equations (17)
has the form
z1 = (fm(z2))
k
z2 = (fm(z3))
k−1 (fm(z4))
z3 = (fm(z2))
k−1 (fm(z1))
z4 = (fm(z3))
k .
(21)
Reduce (21) to the system
{
z2 = (fm(z3))
k−1 · fm((fm(z3))k)
z3 = (fm(z2))
k−1 · fm((fm(z2))k).
(22)
After introducing the notation
ϕ(z) = (fm(z))k−1 · fm((fm(z))k), (23)
312 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
On Weakly Periodic Gibbs Measures of the Potts Model
the system of equations (22) can by written as follows:
{
z2 = ϕ(z3)
z3 = ϕ(z2).
(24)
Note that the following hold:
1) ϕ(1) = 1,
2) the function ϕ(z) is defined on R+,
3) ϕ(z) is bounded and differentiable at z = 1.
Therefore, by Lemma 3, for ϕ′(1) < −1, the system of equations (24) has three
solutions of the form (1, 1), (z∗2 , z
∗
3), (z
∗
3 , z
∗
2). The inequality ϕ′(1) < −1 is equiv-
alent to the inequality
k
(θ − 1)2
(θ + q − 1)2
+ (k − 1)
(θ − 1)
(θ + q − 1)
+ 1 < 0. (25)
Consequently ,
2k(θ − θ1)(θ − θ2) < 0,
where θ1, θ2 are defined in (12).
It is clear that if k < 5, then θ1, θ2 are complex numbers, if k = 5, then
θ1 = θ2 and inequality (25) has no solution.
We now assume that k ≥ 6. We suppose that θ1, θ2 are simultaneously nega-
tive. Then (25) has no solution. If θ1 ≤ 0, 0 < θ2 < 1, i.e.,
4k
k + 1 +
√
k2 − 6k + 1
≤ q <
4k
k + 1−√k2 − 6k + 1
,
then inequality (25) has the solution θ ∈ (0, θ2), which proves item 1 in Theorem
3. Let 0 ≤ θ1 < 1, 0 < θ2 < 1, then the inequality
q ≤ 4k
k + 1 +
√
k2 − 6k + 1
is satisfied.
Inequality (25) has the solution θ1 < θ < θ2. It is easy to see that θ1, θ2
are less or equal to 1. Thus, by Theorem 2, for every m and under conditions
of Theorem 4, we obtain two weakly periodic (non-periodic) Gibbs measures.
From (8), we note that m is the number of coordinates other than 1 of the
vectors from Rq−1. It is obvious that the number of these vectors is equal to∑q−1
m=1 Cm
q−1 = 2q−1 − 1. Consequently, when the conditions of Theorem 4 are
satisfied, we obtain the 2(2q−1−1) = 2q−2 weakly periodic (non-periodic) Gibbs
measures. Theorem 4 is proved.
Acknowledgment. The author thanks Professor U.A. Rozikov for useful
discussions.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 313
M.M. Rahmatullaev
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