Theoretical and Mathematical Physics, 149(1): 1312–1323 (2006)
GIBBS MEASURES FOR THE SOS MODEL WITH FOUR STATES ON A CAYLEY TREE U. A. Rozikov∗ and Sh. A. Shoyusupov∗
We analyze the SOS (solid-on-solid) model with spins 0, 1, 2, 3 on a Cayley tree of order k ≥ 1. We consider translation-invariant and periodic splitting Gibbs measures for this model. The majority of the constructed Gibbs measures are mirror symmetric.
Keywords: Cayley tree, configuration, Gibbs measure, periodic Gibbs measure
1. Introduction A Cayley tree Γk = (V, L) of order k ≥ 1 is an infinite tree (a graph without cycles) with exactly k+1 edges issuing from each vertex, V is the set of its vertices, and L is the set of edges. Let Φ := {0, 1, . . . , m}, m ≥ 1, and let the configuration σ ∈ ΦV , i.e., σ = {σ(x) ∈ Φ: x ∈ V }. We consider the Hamiltonian H(σ) = −J
|σ(x) − σ(y)|,
(1)
x,y∈L
where J ∈ R, and x, y means neighboring vertices. The model corresponding to (1) is called the SOS (solid-on-solid) model. The SOS model can be regarded as a generalization of the Ising model [1], which is obtained for m = 1. The SOS model is mirror symmetric, i.e., the value of the Hamiltonian does not change under the transformation σ(x) → m − σ(x). For the SOS model on Zd , it was proved in [2] that there is a T0 > 0 such that the structure of the thermodynamic phases is determined for T < T0 via the dominating ground states of the SOS model, namely, the Gibbs measure (GM) is unique for even values of m, and there are two periodic GMs for odd values of m. The SOS model on a Cayley tree was considered in [3], where a critical temperature TC was found for m = 2 such that for T < TC , there are three different translation-invariant splitting GMs (TISGMs) for this model. A complete description of periodic GMs was also given in [3]. We note that for m = 2, the SOS model coincides with the Blume–Capel model [4]. For the SOS model, the contour method on a Cayley tree was used in [5] to prove that there are m+1 different GMs at sufficiently low temperatures. Here, we study the SOS model for m = 3, which differs from the Blume–Capel model. Our results here supplement those in [5], namely, we find the exact value of the critical temperature for m = 3. Furthermore, we consider periodic and nonperiodic GMs. Because the number of translation-invariant measures (TIMs) in the case of Zd depends on the parity of m, the results in this paper make it possible to compare the numbers of TISGMs for m of different parity because the situation with m = 2 was considered in [3] and we consider m = 3 here. ∗
Institute of Mathematics, Uzbek Academy of Sciences, Tashkent, Uzbekistan, e-mail:
[email protected],
[email protected]. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 18–31, October, 2006. Original article submitted January 16, 2006; revised March 29, 2006. 1312
c 2006 Springer Science+Business Media, Inc. 0040-5779/06/1491-1312
2. Preliminaries 2.1. Cayley tree. As is known, the Cayley tree Γk can be represented as a free product of k+1 cyclic groups of the second order (see, e.g., [6]). We let Gk denote this product group. Two vertices x, y ∈ V are called neighbors if they are endpoints of an edge l ∈ L, and we write l = x, y in this case. The distance d(x, y), x, y ∈ V , on the Cayley tree is introduced as follows: d(x, y) = d if d is the minimum value for which there exist vertices x = x0 , x1 , . . . , xd−1 , xd = y such that x0 , x1 , . . . , xd−1 , xd are neighbors. The sequence π = {x = x0 , x1 , . . . , xd−1 , xd = y} ⊂ V realizing this minimum is called the n path from x to y. We set Wn = {x ∈ V : d(x, x0 ) = n} and Vn = m=0 Wm , where the vertex x0 ∈ V is fixed. 2.2. The configuration space and the model. Let the spin assume values in the set Φ := {0, 1, 2, 3}. A configuration σ on A ⊆ V is defined as a function x ∈ A → σA (x) ∈ Φ. The set of all configurations coincides with ΩA = ΦA . We set Ω = ΩV and σ = σV . A periodic configuration is defined as a configuration σ ∈ Ω that is invariant with respect to some subgroup G∗k ⊂ Gk . In other words, a configuration σ ∈ Ω is said to be periodic if σ(yx) = σ(x) for all x ∈ Gk and y ∈ G∗k . For a given periodic configuration, the index of the subgroup is called the period of the configuration. A configuration invariant with respect to all shifts on the tree is said to be translation invariant. The Hamiltonian H: Ω → R of the SOS model is given by formula (1). 2.3. Gibbs measure. We consider the standard σ-algebra B of subsets Ω generated by the cylindrical subsets and also the probability measures on (Ω, B). A probability measure µ is called a GM with the Hamiltonian H if the Dobrushin–Lanford–Ruelle condition holds, µ {σ ∈ Ω: σVn = σn } = n where νωVW
n+1
Ω
n µ(dω)νωVW
n+1
(σn ),
is the conditional probability,
n νωVW
n+1
(σn ) =
1 exp −βH(σn | ωWn+1 ) . Zn (ωWn+1 )
Here, H(σn | ωWn+1 ) = −J
|σn (x) − σn (y)| − J
x,y∈L x,y∈Vn
Zn (ωWn+1 ) =
|σn (x) − ω(y)|,
x,y∈L x∈Vn , y∈Wn+1
exp −βH(˜ σn | ωWn+1 ) .
σ ˜n ∈ΩVn
Because only neighbors interact, the GM has the Markov property, namely, if a configuration ωWn is given, then the configurations in Vn−1 , i.e., inside Wn , and in V \ Vn+1 , i.e., outside Wn , are (conditionally) independent. As is known, for an arbitrary β > 0, the set of all GMs forms a nonempty convex compact subset in the space of all probability measures (see, e.g., [1], [7].) We use the standard definition of a TIM (see, e.g., [7] and [8]), and we say that µ is a mirrorsymmetric measure if the values of the measure remain unchanged under a simultaneous change of variable σ(x) → m − σ(x) on all vertices x ∈ V . 1313
2.4. Compatibility conditions. We fix an x0 ∈ V . For x, y ∈ V , we write x < y if the path from x to y goes through x. A vertex y is called a direct descendant of x if y > x and if x and y are neighbors. Let S(x) denote the set of direct descendants of x. We note that every vertex x = x0 in Γk has k direct descendants and that x0 has k + 1 descendants. We considered a special class of GMs called Markov chains in [1] and splitting GMs (SGMs) in [3]. Let h: x → hx = (h0,x , h1,x , . . . , hm,x ) ∈ Rm+1 be a vector function of x ∈ V \ {x0 }. We consider a probability distribution µ(n) on ΩVn , hσ(x),x , (2) µ(n) (σn ) = Zn−1 exp −βH(σn ) + 0
x∈Wn
where σn ∈ ΩVn and Zn =
exp −βH(˜ σn ) + hσ˜ (x),x .
σ ˜ n ∈ΩVn
x∈Wn
A probability distribution µ(n) is said to be compatible if for arbitrary n ≥ 1 and σn−1 ∈ ΩVn−1 , we have
µ(n) (σn−1 ∪ ωn ) = µ(n−1) (σn−1 ),
(3)
ωn ∈ΩWn
where σn−1 ∪ ωn ∈ ΩVn . In this case, there is a unique measure µ on Ω such that µ {σVn = σn } = µ(n) (σn ) for all n and σn ∈ ΩVn . This measure is called the SGM corresponding to the Hamiltonian H and to the function x → hx , x = x0 . The following theorem gives a condition on hx guaranteeing that the distribution µ(n) (σn ) is compatible. Theorem 1 [3]. The probability distribution µ(n) (σn ), n = 1, 2, . . . , determined by formula (2) is compatible if and only if the equation h∗x = F (h∗y , m, θ), (4) y∈S(x)
where θ = eJβ and β = 1/T , holds for every x ∈ V \{x0 }. Here, h∗x = (h0,x −hm,x , h1,x −hm,x , . . . , hm−1,x − hm,x ), and F ( · , m, θ): Rm → Rm is a vector function, i.e., h = (h0 , h1 , . . . , hm−1 ), F (h, m, θ) = F0 (h, m, θ), . . . , Fm−1 (h, m, θ) , such that
m−1 Fi (h, m, θ) = log
|i−j| hj e + θm−i j=0 θ , m−1 m−j h e j +1 j=0 θ
i = 0, . . . , m − 1.
(5)
3. Translation-invariant measures It follows from Theorem 1 that for every h = {hx , x ∈ V } satisfying (4), there is a unique GM µ (with the restriction µ(n) given by formula (2)), and vice versa. But analyzing solutions of Eq. (4) with an arbitrary m is not easy. The case m = 2 was analyzed in [3]. Here, we assume that m = 3, i.e., Φ := {0, 1, 2, 3}. Remark 1. Because the number of TIMs for the SOS problem on Zd depends on the parity of m, a natural question arises about how the number of TIMs depends on the parity of m in the case of the SOS model on a Cayley tree. The situation with an even m, m = 2, was considered in [3], and we have m = 3 in our case. Therefore, the results in this paper allow comparing the number of TIMs on a Cayley tree for different values of m. 1314
Let h3,x ≡ 0 for all x ∈ V . For m = 3, it follows from (4) and (5) that h0,x =
log
eh0,y + θeh1,y + θ2 eh2,y + θ3 , θ3 eh0,y + θ2 eh1,y + θeh2,y + 1
log
θeh0,y + eh1,y + θeh2,y + θ2 , θ3 eh0,y + θ2 eh1,y + θeh2,y + 1
log
θ2 eh0,y + θeh1,y + eh2,y + θ . θ3 eh0,y + θ2 eh1,y + θeh2,y + 1
y∈S(x)
h1,x =
y∈S(x)
h2,x =
y∈S(x)
(6)
It is natural to begin with translation-invariant solution (6), i.e., to assume that hx = h ∈ R3 . We set zi = ehi,x = ehi , i = 0, 1, 2, and z3 = 1 for all x ∈ V . From (6), we obtain z0 = z1 = z2 =
z0 + θz1 + θ2 z2 + θ3 θ3 z0 + θ2 z1 + θz2 + 1 θz0 + z1 + θz2 + θ2 θ3 z0 + θ2 z1 + θz2 + 1 θ2 z0 + θz1 + z2 + θ θ3 z0 + θ2 z1 + θz2 + 1
k ,
(7a)
,
(7b)
.
(7c)
k k
We note that z0 = 1 and z1 = z2 = z satisfy Eq. (7a) irrespective of k and θ. Substituting z0 = 1 and z1 = z2 = z in (7b) and (7c) gives 2 k θ + (θ + 1)z + θ z= . (8) θ3 + (θ2 + θ)z + 1 We introduce the notation a = θk+1 ,
b=
1 − θ + θ2 , θ2
x=
z . θ
(9)
Lemma 1 (Cf. Lemma 10.7 in [1]). Equation (8) has a unique solution under the conditions x > 0, 2 2 k ≥ 1, and a, b > 0 if k = 1 or b ≤ (k + 1)/(k − 1) . If k > 1 and b > (k + 1)/(k − 1) , then there exist ν1 (b, k) and ν2 (b, k) such that the conditions 0 < ν1 (b, k) < ν2 (b, k) are satisfied and Eq. (8) has three solutions if ν1 (b, k) < a < ν2 (b, k) and has two solutions if a = ν1 (b, k) or a = ν2 (b, k). In this case, we have νi (b, k) =
1 xi
1 + xi b + xi
k ,
where x1 and x2 are the solutions of the equation x2 + [2 − (b − 1)(k − 1)]x + b = 0. Lemma 2. If J ≥ 0, then system (7) has a unique solution (1, z∗ , z∗ ). Proof. Let A = z0 + θz1 + θ2 z2 + θ3 , B = θ3 z0 + θ2 z1 + θz2 + 1, A1 = θz0 + z1 + θz2 + θ2 , and B1 = θ2 z0 + θz1 + z2 + θ. From (7), we obtain [(1 − θ3 )T − 1](z0 − 1) + (θ − θ2 )T (z1 − z2 ) = 0, (θ − θ2 )T1 (z0 − 1) + [(1 − θ)T1 − 1](z1 − z2 )] = 0,
(10)
1315
where T = (Ak−1 + · · · + B k−1 )/B k > 0 and T1 = (Ak−1 + · · · + B1k−1 )/B k > 0. 1 We note that, for θ ≥ 1 (J ≥ 0), the determinant of homogeneous system (10) is positive, ∆ = (θ − 1)[(θ2 − 1)T T1 + (1 + θ + θ2 )T + T1 ] + 1 > 0. Consequently, system (10) has a unique solution, z0 = 1, z1 = z2 . We have thus proved that system (7) has only solutions of the form (1, z, z) for J ≥ 0. Because 2 b = (1 − θ + θ2 )/θ2 ≤ 1 < (k + 1)/(k − 1) , Lemma 1 implies that system (7) has a unique solution of the form (1, z∗ , z∗ ). The lemma is proved. For θ = 1, we obtain the following corollary from (10). Corollary. We have z0 = 1 if and only if z1 = z2 . Let βC =
2(k − 1) 1 √ log . J k − 1 + k 2 + 14k + 1
Lemma 3. If J < 0, then system of equations (7) has a unique solution of the form z ∗ = (1, z∗ , z∗ ) ∗ ∗ ∗ ∗ ∗ (i.e., z0 = 1) for β ≤ βC , and it has exactly three solutions z− = (1, z1,− , z1,− ), z0∗ = (1, z1,0 , z1,0 ), and h∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ z+ = (1, z1,+ , z1,+ ) for β > βC , and in this case, we have 0 < z1,− < z1,0 < z1,+ and z1,i = e 1,i , i = −, 0, +. 2 Proof. The value of βC is a solution of the equation b = (k + 1)/(k − 1) . The other assertions in Lemma 3 follow from Lemma 1. ∗ ∗ , z0∗ , z+ is a mirror-symmetric solution of system (7). Consequently, we have the We note that z ∗ , z− following theorem.
Theorem 2. For the SOS model with J > 0 and m = 3, a mirror-symmetric translation-invariant splitting Gibbs measure (MSTISGM ) exists and is unique for every β > 0. For J < 0 and m = 3, if k ≥ 2 and 0 < β ≤ βC , then there is a unique MSTISGM µ∗ , and if k ≥ 2 and β > βC , then there are three MSTISGMs µ∗− , µ∗0 , and µ∗+ corresponding to h∗i = log zi∗ , i = −, 0, +. The next lemma gives a useful property of the solution hx = (h0,x , h1,x , h2,x , ) of system (6) for h0,x ≡ 0 and h1,x = h2,x . Lemma 4. For J < 0, k ≥ 2, and β > βC , if hx = (0, h1,x , h1,x ) is a solution of system (6), then for an arbitrary x ∈ V , we have ∗ ∗ z1,− ≤ z1,x ≤ z1,+ , (11) ∗ ∗ < z1,+ are defined in Lemma 3. where z1,x = eh1,x and z−
Proof. We set zx = z1,x . Relations (6) imply the formula zx =
k
θ + zxi , 1 − θ + θ2 + θzxi i=1
zxj > 0,
j = 1, k,
where xj , j = 1, k, are direct descendants of the point x ∈ V . Let ϕ(x, θ) = (θ + x)/(1 − θ + θ2 + θx), ψ(x, θ, k) = (ϕ(x, θ))k , and G(u1 , . . . , uk ) =
k i=1
1316
ϕ(ui , θ),
uj > 0,
j = 1, k.
We note that ψ(0, θ, k) ≤ G(u1 , . . . , uk ) ≤ ψ(∞, θ, k). Hence, for zx , we have ψ(0, θ, k) ≤ zx ≤ ψ(∞, θ, k). We now consider the function G(u1 , . . . , uk ) for ψ(0, θ, k) ≤ uj ≤ ψ(∞, θ, k). In this case, by analogy with the above, we obtain ψ(ψ(0, θ, k), θ, k) ≤ zx ≤ ψ(ψ(∞, θ, k), θ, k). Iterating this procedure, at the nth step, we obtain the inequality ψ (n) (0, θ, k) ≤ zx ≤ ψ (n) (∞, θ, k) ∗ is for an arbitrary n ≥ 1 and for x ∈ V \ {x0 }. The sequence ψ (n) (∞, θ, k), n = 1, 2, . . . , decreases, and z1,+ ∗ its lower bound. The limit of this sequence is a fixed point for ψ, and it consequently coincides with z1,+ . The lower estimate for z1,z in (11) is found similarly. The lemma is proved.
Lemma 5. For J < 0 and β ≤ βc , the measure µ∗ is a unique mirror-symmetric SGM. ∗ ∗ Proof. Under the conditions z1,− = z1,+ = z1∗ , we conclude from (11) that hx = (0, z1∗ , z1∗ ) is a unique solution. The lemma is proved.
4. Complete description of periodic SGMs Let Gk be a group representation for Γk (see Sec. 2.1).
k be a subgroup of the group Gk . A collection of vectors h = {hx : x ∈ Gk } is Definition 1. Let G
k .
k -periodic if hx = hyx for all x ∈ Gk and y ∈ G said to be G
k k -periodic collection of vectors is said to be G Definition 2. The measure corresponding to a G periodic. A Gk -periodic measure is said to be translation-invariant. Remark 2. Proceeding from the one-to-one correspondence between the set of vertices V of the Cayley tree and the elements of the group Gk , the measure µ, in terms of the Cayley tree by itself, can be said to
k -periodic if the value of the measure is preserved under the shift x → yx (on the Cayley tree) for all be G
k . We note that it appears natural to define a shift on a Cayley tree using the group x ∈ Gk and y ∈ G representation for the Cayley tree. In this section, we describe SGMs that are periodic relative to every normal divisor of a finite index. Let K be a normal divisor of index r for the group Gk . We also introduce the notation Gk /K = {K0 , K1 , . . . , Kr−1 }, K0 = K, for the quotient group. We set qi (x) = |S(x) ∩ Ki |,
i = 0, 1, . . . , r − 1,
N (x) = |{j: qj (x) = 0}|,
where | · | is the number of elements in a set. Let Q(x) = (q0 (x), q1 (x), . . . , qr−1 (x)). We note that for every x ∈ Gk , there is a permutation πx of the coordinates of the vector Q(x) such that πx Q(e) = Q(x),
(12) 1317
where e ∈ Gk is the unit element [10]. It follows from (12) that N (x) = N (e) for all x ∈ Gk . The K-periodic collection of vectors has the form {hx = hi ∈ R3 for x ∈ Ki , i = 0, 1, . . . , r − 1}. By (6) and in view of (12), the vectors hi , i = 0, 1, . . . , r − 1, satisfy the system of equations N (e)
hm =
qij (e)F (hπm (ij ) ; θ) − F (hπm (ij ) ; θ),
(13)
j=1
where m = 0, r − 1, j0 = 1, N (e), and N (e) = |{i1 , i2 , . . . , iN (e) }|. Let the functions F : R4 → R4 , F (h, θ) = (F0 (h, θ), F1 (h, θ), F2 (h, θ)) be given by the relations F0 (h, θ) = log
eh0 + θeh1 + θ2 eh2 + θ3 , θ3 eh0 + θ2 eh1 + θeh2 + 1
F1 (h, θ) = log
θeh0 + eh1 + θeh2 + θ2 , θ3 eh0 + θ2 eh1 + θeh2 + 1
F2 (h, θ) = log
θ2 eh0 + θeh1 + eh2 + θ . θ3 eh0 + θ2 eh1 + θeh2 + 1
(14)
Lemma 6. If θ = 1, then F (h, θ) = F (l, θ) if and only if h = l. Proof. Necessity. Let F (h, θ) = F (l, θ). Then [1 + θ2 + θ4 + θ2 t1 + θ(1 + θ2 )t2 ](z0 − t0 ) + + [θ2 (t2 − t0 ) + θ(1 + θ2 )](z1 − t1 ) + θ[θ − (1 + θ2 )t0 − θt1 ](z2 − t2 ) = 0, [θ2 t2 + θ(1 + θ2 )](z0 − t0 ) + [1 + θ2 + θt2 ](z1 − t1 ) + θ[1 − t1 − θt2 ](z2 − t2 ) = 0,
(15)
θ2 (z0 − t0 ) + θ(z1 − t1 ) + (z2 − t2 ) = 0, where zi = ehi and ti = eli , i = 0, 1, 2. We note that the determinant of homogeneous system (15) is positive for θ = 1, ∆ = (1 + θ3 t0 + θ2 t1 + θt2 )2 > 0. Consequently, Eq. (15) has a unique solution zi = ti , i = 0, 1, 2. The sufficiency is obvious. The lemma is proved. (2)
(2)
Let Gk be a subgroup of Gk consisting of words of even length. Clearly, Gk is a subgroup of index two. Theorem 3. For every normal divisor of a finite index K, each K-periodic GM coincides with either (2) a TIM or a Gk -periodic measure. Proof. Using (13), we obtain F (hπ(i1 ) , θ) = F (hπ(i2 ) , θ) = · · · = F (hπ(iN (e) ) , θ), whence by Lemma 6 hπm (i1 ) = hπm (i2 ) = · · · = hπm (iN (e) ) . (2)
(2)
Consequently, hx = hy = h1 if x, y ∈ S1 (z) and z ∈ Gk , and hx = hy = h2 if x, y ∈ S1 (z) and z ∈ Gk \Gk , where S1 (x) = {y ∈ V : x, y}. Hence, if h1 = h2 , then hx is a constant function, and if h1 = h2 , then hx (2) is a Gk -periodic function. The theorem is proved. 1318
Let K be a normal divisor of Gk of finite index. There arises a question about the condition on K for every K-periodic Gibbs measure to be a TIM. Let I(K) = K ∩ {a1 , . . . , ak+1 }. Theorem 4. If I(K) = ∅, then every K-periodic GM for the SOS model is a TIM. Proof. If x ∈ K, then xai ∈ K if and only if ai ∈ K. Because I(K) = ∅, there is an ai ∈ K. Consequently, K contains the subset Kai = {xai : x ∈ K}. By Theorem 3, hx = h1 and hxai = h2 . Because x and xai belong to the class K, we have hx = hxai = h1 = h2 . The theorem is proved. By Theorems 3 and 4, the problem of describing K-periodic GM for I(K) = ∅ reduces to describing fixed points of the map h = kF (h; θ), (16) and for I(K) = ∅ to describing the solutions of the system h = kF (l; θ), (17) l = kF (h; θ). We note that map (16) describes a TIM (see (7)). System (17) describes periodic measures of period two (2) or, more precisely, Gk -periodic measures. From (17), we obtain k t0 + θt1 + θ2 t2 + θ3 z0 = , θ3 t0 + θ2 t1 + θt2 + 1 z1 = z2 = t0 = t1 = t2 =
θt0 + t1 + θt2 + θ2 θ3 t0 + θ2 t1 + θt2 + 1 θ2 t0 + θt1 + t2 + θ θ3 t0 + θ2 t1 + θt2 + 1
k , k
z0 + θz1 + θ2 z2 + θ3 θ3 z0 + θ2 z1 + θz2 + 1 θz0 + z1 + θz2 + θ2 θ3 z0 + θ2 z1 + θz2 + 1 θ2 z0 + θz1 + z2 + θ θ3 z0 + θ2 z1 + θz2 + 1
, (18)
k , k , k .
Lemma 7. For J < 0 (θ < 1), system (18) is satisfied only for zi = ti , i = 0, 1, 2. 1/k
Proof. Let ui = zi
1/k
and vi = ti
, i = 0, 1, 2. Then (18) implies
[A + B0 (1 + θ2 + θ4 + θ2 uk1 + θ(1 + θ2 )uk2 )](u0 − v0 ) + + [B1 (θ2 (uk2 − uk0 ) + θ3 + θ)](u1 − v1 ) + + [B2 (θ2 − (θ3 + θ)uk0 − θ2 uk1 )](u2 − v2 ) = 0, [B0 (θ2 uk2 + θ3 + θ)](u0 − v0 ) + [A + B1 (θuk2 + θ2 + 1)](u1 − v1 ) +
(19)
+ [B2 (θ − θuk1 − θ2 uk0 )](u2 − v2 ) = 0, θ2 B0 (u0 − v0 ) + θB1 (u1 − v1 ) + (A + B2 )(u2 − v2 ) = 0, 1319
where A = (θ3 v0k + θ2 v1k + θv2k + 1)(θ3 uk0 + θ2 uk1 + θuk2 + 1)(1 − θ2 )−1 > 0, + · · · + vik−1 > 0, Bi = uk−1 i
i = 0, 1, 2.
A simple, but rather cumbersome, analysis shows that the determinant of system (19) is positive, and consequently zi = ti , i = 0, 1, 2. The lemma is proved. We now consider the antiferromagnetic case J > 0 (θ > 1). In this situation, by Lemma 2, system (18) has a unique solution (1, z1∗ , z1∗ , 1, t∗1 , t∗1 ). Let z0 = t0 = 1, z1 = z2 = z, and t1 = t2 = t. Then (18) implies z= t=
θ+t 1 − θ + θ2 + θt
θ+z 1 − θ + θ2 + θz
k , (20)
k .
Lemma 8. Let (z∗ , z∗ ) be a solution of system (20). If kz∗ (θ − 1) > 1, (θ + z∗ )(1 − θ + θ2 + θz∗ )
(21)
∗ ∗ ∗ ∗ ∗ ∗ then system (10) has three solutions (z− , z+ ), (z∗ , z∗ ), and (z+ , z− ), where z− = ψ(z+ , θ, k) and
ψ(x, θ, k) =
θ+x 1 − θ + θ2 + θx
k .
Proof. If inequality (21) is satisfied, then the solution z∗ is an unstable point for ψ. For every z ≥ 1, the ∗ iterations ψ (2n) (z, θ, k), ψ (2n) (z, θ, k) > z∗ , decrease monotonically. Hence, limn→∞ ψ (2n) (z, θ, k) = z+ ≥ z∗ exists and is a solution of the equation z = ψ(ψ(z, θ, k), θ, k).
(22)
∗ ∗ ∗ > z∗ holds. Similarly, we have z− = ψ(z+ , θ, k), Because z ∗ is an unstable point, the inequality z+ ∗ z− < z∗ , which is also a solution of Eq. (22). The lemma is proved.
We have thus proved the following theorem. Theorem 5. For the SOS model with m = 3, the following assertions are true for the Cayley tree relative to every normal divisor of finite index K: 1. In the ferromagnetic case (J < 0) and for J = 0, the K-periodic GM coincides with a TIGM. 2. In the antiferromagnetic case (J > 0), a. if I(K) = ∅, then the K-periodic GM coincides with a TIGM, and b. if inequality (21) holds and if I(K) = ∅, then there are three K-periodic GMs µ12 , µ∗ , and µ21 . In this situation, µ∗ is a TIM, and µ12 and µ21 are periodic measures with period two, i.e., they are measures.
(2) Gk -periodic
1320
5. Nonperiodic SGMs In this section, we consider the case J < 0, m = 3, β > βC . Using µ∗i , i = −, 0, +, we show that system (6) has uncountably many nonperiodic solutions. Let an infinite path π = {x0 = x0 < x1 < x2 < · · · } on the Cayley tree Γk be given. A t ∈ [0, 1] can be standardly associated with this path (see [11] and [12]). Let π1 = {x0 , x1 , . . . } and π2 = {y0 , y1 , . . . } be two infinite paths with x0 = y0 = x0 . We map the pair (π1 , π2 ) to the vector function hπ1 π2 : x ∈ V → hπx1 π2 satisfying (6). The paths π1 and π2 split Γk into three subgraphs Γk1 , Γk2 , and Γk3 if π1 = π2 and into two subgraphs Γk1 and Γk3 if π1 = π2 . The vector function hπ1 π2 is determined by
hπx1 π2
h∗ − = h∗0 h∗ +
for x ∈ Γk1 , for x ∈ Γk2 ,
(23)
for x ∈ Γk3 ,
∗ ∗ and the vectors h∗i = (0, log z1,i , log z1,i ), i = −, 0, +, are defined by the solutions of system (7). 3 Let h = (h0 , h1 , h2 ) ∈ R , let h = max{|h0 |, |h1 |, |h2 |},
and let the function h → F (h) be given by the two relations in (14). We introduce the notation f (x, y, z) ≡ f (x, y, z, a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 ) = log
a1 e x + a2 e y + a3 e z + a4 , b1 ex + b2 ey + b3 ez + b4
(24)
where ai ≥ 0,
bi ≥ 0,
i = 1, 2, 3, 4,
a1 + a2 + a3 + a4 > 0,
b1 + b2 + b3 + b4 > 0.
(25)
Lemma 9. The following estimates hold: a. If a1 a2 a3 a4 b1 b2 b3 b4 > 0, then ∂f b 1 a2 | |a1 b3 − b1 a3 | |a1 b4 − b1 a4 | ≤ max √|a1 b2 −√ √ √ , √ , √ , ∂x ( a1 b2 + b1 a2 )2 ( a1 b3 + b1 a3 )2 ( a1 b4 + b1 a4 )2 ∂f |a2 b3 − b2 a3 | |a2 b4 − b2 a4 | b 1 a2 | ≤ max √|a1 b2 −√ √ √ √ √ , , , ∂y ( a1 b2 + b1 a2 )2 ( a2 b3 + b2 a3 )2 ( a2 b4 + b2 a4 )2 ∂f b 1 a3 | |a2 b3 − b2 a3 | |a3 b4 − b3 a4 | ≤ max √|a1 b3 −√ √ √ , √ , √ . ∂z ( a1 b3 + b1 a3 )2 ( a2 b3 + b2 a3 )2 ( a3 b4 + b3 a4 )2 b. If a1 a2 a3 a4 b1 b2 b3 b4 = 0, then ∂f ≤ 1, ∂x
∂f ≤ 1, ∂y
∂f ≤ 1. ∂z
Proof. Let a, b, c, d ≥ 0, a + b > 0, and c + d > 0. We introduce the notation f1 (x) = log (aex + b)/(cex + d) . It can be easily verified that bc| √|ad −√ ( ad + bc)2 |f1 (x)| ≤ 1
for abcd = 0,
(26)
for abcd = 0. 1321
We consider f2 (x, y) = log (a1 ex + a2 ey + a3 )/(b1 ex + b2 ey + b3 ) . By inequality (26), we have y ∂f2 (x, y) b1 (a2 ey + a3 )| ≤ |a1 (b2 e + b3 ) − 2 . ∂x a1 (b2 ey + b3 ) + b1 (a2 ey + a3 )
(27)
2 Let t = ey and ϕ(t) = |a1 (b2 t + b3 ) − b1 (a2 t + a3 )|/ a1 (b2 t + b3 ) + b1 (a2 t + a3 ) . Then a3 b 2 − a2 b 3 a1 b 1 ϕ (t) = · 2 . (a2 t + a3 )(b2 t + b3 ) a1 (b2 t + b3 ) + b1 (a2 t + a3 )
Consequently, ϕ(t) increases monotonically if the inequality a3 b2 − a2 b3 ≥ 0 holds and decreases monotonically in the case a3 b2 − a2 b3 < 0. Because t > 0, inequality (27) implies that |a1 b3 − b1 a3 | ∂f2 (x, y) |a1 b2 − b1 a2 | ≤ max {|ϕ(0)|, |ϕ(∞)|} = max √ (28) 2 , √ 2 . √ √ ∂x a1 b 3 + b 1 a3 a1 b 2 + b 1 a2 Using (28), we similarly obtain the first inequality for ∂f (x, y, z)/∂x in assertion a in Lemma 9. The other two inequalities are proved similarly. The inequalities in assertion b are obtained from those in a if at least one of the parameters a1 , a2 , a3 , a4 , b1 , b2 , b3 , and b4 is zero; in this case, inequalities (25) hold. The lemma is proved. We note that every coordinate of the function F (see (14)) has form (24). Lemma 9 implies the next lemma. Lemma 10. The following estimates hold for every h = (h0 , h1 , h2 ) ∈ R3 : ∂F0 |1 − θ3 | ∂F1 |1 − θ2 | ∂F2 |1 − θ| ≤ ≤ , , ∂hj ∂hj ∂hj ≤ 1 + θ , 1 + θ3 1 + θ2 F (h, θ) − F (l, θ) ≤ 3
|1 − θ3 | h − l, 1 + θ3
h, l ∈ R3 .
(29a)
(29b)
Proof. Inequalities (29a) are consequences of those in Lemma 9. To prove (29b), we write F (h) − F (l) = max {|Fi (h) − Fi (l)|} ≤ i=0,1,2
∂Fi |h0 − l0 | + ∂Fi |h1 − l1 | + ∂Fi |h2 − l2 | ≤ ∂h1 ∂h2 i=0,1,2 ∂h0
≤ max ≤3
|1 − θ3 | h − l. 1 + θ3
Using Lemma 10 (see similar considerations in connection with Theorem 3 in [12]), we can prove the following theorem. Theorem 6. For arbitrary infinite paths π1 and π2 , there is a unique vector function hπ1 π2 satisfying relations (6) and (23). We map (π1 , π2 ) to the pair (t, s) ∈ [0, 1] × [0, 1]. It can be shown (see [11]–[13]) that hπ1 (t)π2 (s) are different for different (t, s) ∈ D = {(u, v) ∈ [0, 1]2 : u ≤ v}. According to Theorem 1, an SGM on Γk , denoted here by µ(t, s), can be associated with each collection hπ1 (t)π2 (s) . We have proved the next theorem. Theorem 7. For each (t, s) ∈ D, there is a unique SGM µ(t, s). Moreover, we have µ(0, 0) = µ∗+ , µ(0, 1) = µ∗0 , and µ(1, 1) = µ∗− . 1322
6. Discussion Describing the set of all Gibbs distributions corresponding to Hamiltonian (1) is one of the main problems, but it has not yet been completely solved, even for some rather simple Hamiltonians (see, e.g., [6], [10]– [14]). The simplicity of the Cayley tree allows describing a rather wide class of GMs for models without “good” symmetries (see [9] and [15]) and with competing interactions (see, e.g., [16]). Our results in this paper allow indicating the explicit values of β for which there are periodic GMs. On the other hand, in view of the results in [3] and in the present paper, it can be noted that the number of TISGMs is the same and is equal to three for m = 2 and m = 3. This demonstrates an essential distinction between the SOS models on Zd and on Γk because the number of TIMs in the case of Zd depends on the parity of m (see [2]), namely, a unique GM for even m and two periodic GMs for odd m. Acknowledgments. One of the authors (U. A. R.) is grateful to La Sapienza University (Rome, Italy) for the permanent support of visits in 2005 and also to Professors G. Gallavotti and M. Cassandro for the invitations to “La Sapienza.” The authors express their gratitude to the referee for the valuable remarks. This work was supported in part by NATO (Reintegration Grant No. Fel. RIG. 980771) and the Centre for Science and Technologies of Uzbekistan (Grant No. F.-1.1.2).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
C. J. Preston, Gibbs States on Countable Sets, Cambridge Univ. Press, Cambridge (1974). A. E. Mazel and Yu. M. Suhov, J. Statist. Phys., 64, 111–134 (1991). U. A. Rozikov and Yu. M. Suhov, “Gibbs measures for SOS models on a Cayley tree,” math.PR/0409047 (2004). R. Fernandez, “Contour ensembles and the description of Gibbsian probability distributions at low temperature,” http://www.univ-rouen.fr/LMRS/Persopage/Fernandez/resucont.html (1998). U. A. Rozikov, Lett. Math. Phys., 71, No. 2, 27–38 (2005). N. N. Ganikhodzhaev and U. A. Rozikov, Theor. Math. Phys., 111, 480–486 (1997). Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results [in Russian], Nauka, Moscow (1980); English transl., Pergamon, Oxford (1982). H. O. Georgii, Gibbs Measures and Phase Transitions (de Gruyter Stud. Math., Vol. 9), de Gruyter, Berlin (1988). Yu. M. Suhov and U. A. Rozikov, Queueing Syst., 46, 197–212 (2004). U. A. Rozikov, Theor. Math. Phys., 112, 929–934 (1997). P. M. Blekher and N. N. Ganikhodzhaev, Theor. Probab. Appl., 35, 216–227 (1990). U. A. Rozikov, Theor. Math. Phys., 118, 77–84 (1999). U. A. Rozikov, Siberian Math. J., 39, 427–435 (1998). N. N. Ganikhodzhaev, Theor. Math. Phys., 85, 1125–1134 (1990). J. Martin, U. A. Rozikov, and Yu. M. Suhov, J. Nonlinear Math. Phys., 12, 432–448 (2005). F. M. Mukhamedov and U. A. Rozikov, J. Stat. Phys., 114, 825–848 (2004); 119, 427–446 (2005).
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